Guest Essay by Kip Hansen — 3 January 2023
“People use terms such as “sure” to describe their uncertainty about an event … and terms such as “chance” to describe their uncertainty about the world.” — Mircea Zloteanu
In many fields of science today, the word “uncertainty” is bandied about without much thought, or at least expressed thought, about which meaning of “uncertainty” is intended. This simple fact is so well known that a group in the UK, “Sense about Science”, published a booklet titled “Making Sense of Uncertainty” (.pdf). The Sense about Science group promotes evidence-based science and science policy. The Making Sense of Uncertainty booklet, published in 2013, is unfortunately an only vaguely-disguised effort to combat climate skepticism based on the huge uncertainties in Climate Science.
Nonetheless, it includes some basic and necessary understandings about uncertainty:
Michael Hanlon: “When the uncertainty makes the range of possibilities very broad, we should avoid trying to come up with a single, precise number because it creates a false impression of certainty – spurious precision.”
A good and valid point. But the larger problem is “trying to come up with a single … number” whether ‘spuriously precise’ or not.
David Spiegelhalter: “In clinical medicine, doctors cannot predict exactly what will happen to anyone, and so may use a phrase such as ‘of 100 people like you, 96 will survive the operation’. Sometimes there is such limited evidence, say because a patient’s condition is completely novel, that no number can be attached with any confidence.”
Not only in clinical medicine, but widely across fields of research, we find papers being published that — despite vague, even contradictory, and limited evidence with admitted weaknesses in study design — state definitive numerical findings that are no better than wild guesses. [ See studies by Jenna Jambeck on oceanic plastics. ]
And, perhaps the major understatement, and the least true viewpoint, in the booklet:
“There is some confusion between scientific and everyday uses of the words ‘uncertainty’ and ‘risk’. [This first sentence is true. – kh] In everyday language, we might say that something that is uncertain is risky. But in scientific terms, risk broadly means uncertainty that can be quantified in relation to a particular hazard – and so for a given hazard, the risk is the chance of it happening.”
A Lot of Confusion
“The risk is the chance of it happening.” Is it really? William Briggs, in his book “Uncertainty: The Soul of Modeling, Probability & Statistics”, would be prone to point out that for there to be a “chance” (meaning “a probability”) we first need a proposition, such as “The hazard (death) will happen to this patient” and clearly stated premises, most of which are assumed and not stated, such as “The patient is being treated in a modern hospital, otherwise healthy, the doctor is fully qualified and broadly experienced in the procedure, the diagnosis is correct…”. Without full exposition of the premises, no statement of probability can be made.
I recently published here two essays touching on uncertainty:
Plus or Minus Isn’t a Question and Limitations of the Central Limit Theorem.
Each used almost childishly simple examples to make several very basic true points about the way uncertainty is used, misused and often misunderstood. I expected a reasonable amount of push-back against this blatant pragmatism in science, but the ferocity and persistence of the opposition surprised me. If you missed these, take a look at the essays and their comment streams. Not one of the detractors was able to supply a simple example with diagrams or illustrations to back their contrary (almost always “statistical”) interpretations and solutions.
So What is the Problem Here?
1. Definition In the World of Statistics, uncertainty is defined as probability. “Uncertainty is quantified by a probability distribution which depends upon our state of information about the likelihood of what the single, true value of the uncertain quantity is.” [ source ]
[In the linked paper, uncertainty is contrasted to: “Variability is quantified by a distribution of frequencies of multiple instances of the quantity, derived from observed data.”]
2. Misapplication The above definition becomes misapplied when we consider absolute measurement uncertainty.
Absolute error or absolute uncertainty is the uncertainty in a measurement, which which is expressed using the relevant units.
The absolute uncertainty in a quantity is the actual amount by which the quantity is uncertain, e.g. if Length = 6.0 ± 0.1 cm, the absolute uncertainty in Length is 0.1 cm. Note that the absolute uncertainty of a quantity has the same units as the quantity itself.
Note: The most correct label for this is absolute measurement uncertainty. It results from the measurement process or the measurement instrument itself. When a temperature is always (and only) reported in whole degrees (or when it has been rounded to whole degrees), it has an inescapable absolute measure uncertainty of ± 0.5°. So, the thermometer reading reported/recorded as 87° must carry its uncertainty and be shown as “87° ± 0.5°” — which is equivalent to “any value between 87.5 and 86.5” —there are an infinite number of possibilities in that range, all of which are equally possible. (The natural world does not limit temperatures to those exactly lining up with the little tick marks on thermometers.)
Dicing for Science
Let’s take a look at a simple example – throwing a single die and throwing a pair of dice.
A single die (a cube, usually with slightly rounded corners and edges) has six sides – each with a number of dots: 1, 2, 3, 4, 5 and 6. If properly manufactured, it has a perfectly even distribution of results when rolled many times. Each face of the die (number) will be found facing up as often as every other face (number).
~~~
This represents the distribution of results of 1,000 rolls of a single fair die. If we had rolled a million times or so, the distribution values of the numbers would be closer to 1-in-6 for each number.
What is the mean of the distribution? 3.5
What is the range of the result expected on a single roll? 3.5 +/- 2.5
Because each roll of a die is entirely random (and within its parameters, it can only roll whole values 1 through 6), for the every next roll we can predict the value of 3.5 ± 2.5 [whole numbers only]. This prediction would be 100% correct – in this sense, there is no doubt that the next roll will be in that range, as it cannot be otherwise.
Equally true, because the process can be considered entirely random process, every value represented by that range “3.5 ± 2.5” [whole numbers only] has an equal probability of coming up in each and every “next roll”.
What if we look at rolling a pair of dice?
A pair of dice, two of the die’s described above, rolled simultaneously, have a value distribution that looks like this:
When we roll two dice, we get what looks like an unskewed “normal distribution”. Again, if we had rolled the pair of dice a million times, the distribution would be closer to perfectly normal – very close to the same number for 3s and for 11s and the same numbers for 1s 2s as for the 12s.
What is the mean of the distribution? 7
What is the range of the result expected on a single roll? 7 ± 5
Because each roll of the dice is entirely random (within its parameters, it can only roll whole values 2 through 12), for the every next roll we can predict the value of “7 ± 5”.
But, with a pair of dice, the distribution is no longer even across the whole range. The value of the sums of the two dice range from 2 through 12 [whole numbers only]. 1 is not a possible value, nor is any number above 12. The probability of rolling a 7 is far larger than rolling a 1 or 3 or 11 or 12.
Any dice gambler can explain why this is: there are more combinations of the values of the individual die that add up to 7 than add up to 2 (there is only one combination for 2: two 1s and one combination for 12: two 6s).
Boxing the dice
To make the dicing example into true absolute measurement uncertainty, in which we give a stated value and its known uncertainty but do not (and cannot) know the actual (or true) value, we will place the dice inside a closed box with a lid. And then shake the box (roll the die). [Yes, Schrödinger’s cat and all that.] Putting the dice in a lidded box means that we can only give the value as a set of all the possible values, or, the mean ± the known uncertainties given above.
So, then we can look at our values for a pair of dice as the sum of the two ranges for a single die:
The arithmetic sum of 3.5 ± 2.5 plus 3.5 ± 2.5 is clearly 7 ± 5. (see Plus or Minus isn’t a Question).
The above is the correct handling of addition of Absolute Measurement Uncertainty.
It would be exactly the same if adding two Tide Gauge Measurements, which have an absolute measurement uncertainty of ± 2 cm, or adding two temperatures that have been rounded to a whole degree. One sums the value and sums the uncertainties. (Many references for this. Try here.)
Statisticians (as a group) insist that this is not correct – “Wrong” as one savvy commenter noted. Statisticians insist that the correct sum would be:
7 ± 3.5
One of the commenters on Plus or Minus gave this statistical view: “the uncertainties add IN QUADRATURE. For example, (25.30+/- 0.20) + (25.10 +/- 0.30) = 50.40 +/- SQRT(0.20^2 + 0.30^2) = 50.40 +/-0.36 … You would report the result as 50.40 +/- 0.36”
Stated in words: Sum the values with the uncertainty given as the “square root of the sum of the squares of the uncertainties”.
So, let’s try to apply this to our simple dicing problem using two dice:
(3.5 ± 2.5) + (3.5 ± 2.5) = 7 ± SQRT (2.5^2 + 2.5^2) = 7 ± SQRT(6.25 + 6.25) = 7 ± (SQRT 12.5) = 7 ± 3.5
[The more precise √12.5 is 3.535533905932738…]
Oh, my. That is quite different from the result of following the rules for adding absolute uncertainties.
Yet, we can see in the blue diagram box that the correct solution including the full range of the uncertainty is 7 ± 5.
So, where do the approaches diverge?
Incorrect assumptions: The statistical approach uses a definition that does not agree with the real physical world: “Uncertainty is quantified by a probability distribution”.
Here is how a statistician looks at the problem:
However, when dealing with absolute measurement uncertainty (or in the dicing example, absolute known uncertainty – the uncertainty is known because of the nature of the system), the application of the statistician’s “adding in quadrature” rule gives us a result not in agreement with reality:
One commenter to the essay Limitations of the Central Limit Theorem, justified this absurdity with this: “there is near zero probability that both measurements would deviate by the full uncertainty value in the same direction.”
In our dicing example, if we applied that viewpoint, the ones and sixes of our single dies in a pair would have a ‘near zero’ probability coming up together (in a roll of two dice) to produce sums of 2 and 12. 2 and 12 represent the mean ± the full uncertainty value of plus or minus 5.
Yet, our distribution diagram of dice rolls shows that, while less common, 2s and 12s are not even rare. And yet, using the ‘adding in quadrature’ rule for adding two values with absolute uncertainties, 2s and 12s can just be ignored. We can ignore the 3s and 11s too.
Any dicing gambler knows that this is just not true, the combined probability of rolling 2, or 3, or 11, or 12 is 18% – almost 1-in-5. Ignoring a chance of 1-in-5, for example “there is a 1-in-5 chance that the parachute will malfunction”, is foolish.
If we used the statisticians regularly recommended “1 Standard Deviation” (approximately 68% – divided equally to both sides of the mean) we would have to eliminate from our “uncertainty” all of the 2s, 3s, 11s, and 12s, and about ½ of the 3s and 4s. — which make up 34% (>1-in-3) of the actual expected rolls.
Remember – in this example, we have turned ordinary uncertainty about a random event (roll of the dice) into “absolute measurement uncertainty” by placing our dice in a box with a lid, preventing us from knowing the actual value of the dice roll but allowing us to know the full range of uncertainty involved in the “measurement” (roll of the dice). This is precisely what happens when a measurement is “rounded” — we lose information about the measured value and end up with a “value range”. Rounding to the “nearest dollar” leaves an uncertainty of ± $ .50 ; rounding to the nearest whole degree leaves an uncertainty of ± 0.5°; rounding to the nearest millennia leaves an uncertainty of ± 500 years. Measurements made with an imprecise tool or procedure produce equally durable values with a known uncertainty.
This kind of uncertainty cannot be eliminated through statistics.
Bottom Lines:
1. We always seem to demand a number from research — “just one number is best”. This is a lousy approach to almost every research question. The “single number fallacy” (recently, this very moment, coined by myself, I think. Correct me if I am wrong.) is “the belief that complex, complicated and even chaotic subjects and their data can be reduced to a significant and truthful single number.”
2. The insistence that all “uncertainty” is a measure of probability is a skewed view of reality. We can be uncertain for many reasons: “We just don’t know.” “We have limited data.” “We have contradictory data.” “We don’t agree about the data.” “The data itself is uncertain because it results from truly random events.” “Our measurement tools and procedures themselves are crude and uncertain.” “We don’t know enough.” – – – – This list could go on for pages. Almost none of those circumstances can be corrected by pretending the uncertainty can be represented as probabilities and reduced using statistical approaches.
3. Absolute Measurement Uncertainty is durable – it can be diluted only by better and/or more precise measurement.
4. Averages (finding means and medians) tend to disguise and obscure original measurement uncertainty. Averages are not themselves measurements, and do not properly represent reality. They are a valid view of some data — but often hide the fuller picture. (see The Laws of Averages)
5. Only very rarely do we see original measurement uncertainty properly considered in research findings – instead researchers have been taught to rely on the pretenses of statistical approaches to make their results look more precise, more statistically significant and thus “more true”.
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Author’s Comment:
Hey, I would love to be proved wrong on this point, really. But so far, not a single person has presented anything other than a “my statistics book says….”. Who am I to argue with their statistics books?
But I posit that their statistics books are not speaking about the same subject (and brook no other views). It takes quite a search to even find the correct method that should be used to add two values that have absolute measurement uncertainty stated (as in 10 cm ± 1 cm plus 20 cm ± 5 cm). There are just too many similar words and combinations of words that “seem the same” to internet search engines. The best I have found are physics YouTubes.
So, my challenge to challengers: Provide a childishly simple example, such as I have used, two measurements with absolute measurement uncertainties given added to one another. The arithmetic, a visual example of the addition with uncertainties (on a scale, a ruler, a thermometer, in counting bears, poker chips, whatever) and show them being added physically. If your illustration is valid and you can arrive at a different result than I do, then you win! Try it with the dice. Or a numerical example like the one used in Plus or Minus.
Thanks for reading.
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Statistical reasoning somehow never feels right, which is why people play lotteries.
Tom H ==> Ever hopeful — gambling has always been a weakness of mankind — at least as far back as history has been recorded. Even when the House (casino) makes it very plain that all the games and machines (and lotteries) are rigged to ensure that the House wins more than it loses, and that the gamblers lose more than they win, the gamblers still play.
“the gamblers still play.”
Hope springs eternal.
Life is a gamble, they say.
I thought it was “but a game”?
(Maybe a Vegas game?)
I was playing a nickel mechanical slot machine in the late 70s that was supposed to return 2 nickels for 2 cherries but in stead provided 3.
I played that slot for several hours a day, receiving free drinks while playing, with a 2 dollar buy in. When I finally lost the 2 dollars, having also always tipped the cocktail waitress for every drink, I would go home.
After a week or so they fixed the machine, probably when they collected the win and weighed the tubs and noticed the difference in “win”.
Just that difference made the machine break even of better for the player.
Another time I was playing Keno and miss marked my numbers, unintentionally, as 5 numbers, not the 6 I actually played. The Keno writer wrote up the ticked as 5 numbers, and I played and drank for quite a while until I hit 5 numbers and at that point the writer notices the mistake and did not pay the ticket and reissued it as a 5 number ticket. That is how tight the margins are. Vegas was not built on the customer winning.
Humans are not fundamentally rational creatures. They are only capable of rational behavior for short periods of time in order to achieve their irrational goals.
Some games allow one to lose money slowly
The gamblers must know they “paid for the casino”
They must enjoy gambling.
I find the interior of casinos in Las Vegas to be depressing.
However, I invested my liquid life savings ($129) for a 25% share of the Brooklyn Bridge, to diversify: I already owned a 1% share of Al Gore’s Manhattan Gondola line, to be launched when Manhattan streets are flooded from the rising seas of climate change — Wall Street executives will need some way to get to their offices!
… By the way, Al Gore told me 1,934 more one percent shares are available for $1.000 each
In Vegas, the casino owners like to tell the gamblers “This place wasn’t built on your winnings.”
Gambling is a tax paid by people who do not understand mathematics.
I have never met a gambler who admits to losing money but the house always wins.
My dad used to say, when warning about the folly of gambling, “You never see a bookmaker on a bike”
Chasmsteed ==> I once knew a man that lived high off of his gambling proceeds. He only played poker in casinos — the House takes a percentage cut of the pots but is not an active player. After a great deal of questioning (and gaining his trust) on the point: How do you win so much? — he finally admitted that he was a member of a “cheating team”. A group of of ten good poker players, regulars at all the local casinos, who would pick a high roller with lots of money to lose, and get random members of the Team to take a majority of the places at the table. With a system of informing each other of the values of their hands, they had a huge advantage. As a Team they would drive pots up and push the win to one of the members. They would all meet for breakfast at a distant coffee shop or diner (at least an hour away) and would divide the winnings.
With ten members and lots of casinos, they had never been identified as a group, thus escaped detection.
There are also casino shills who get paid to feed the pot—it is my understanding that if you take a seat at a poker table, you can ask who the shills are and they are supposed to raise their hands.
My son had a policy of quitting when he was ahead. In the big government casino in Montreal he quit one night when he was $900 up. In the lobby the security guards were absent, replaced by a bunch of guys in unmarked uniforms. The took his winnings. Needless to say, he got a good scare about the actual functioning of casinos and went back to honest labour.
Fran ==> Good story — one of my brothers was a “pool shark” playing table pool for money — the most usual tactic is to play only moderately well, lose a few games, then up the bet for the big score — clear the table in one go.
He quite when a couple of really bad guys didn’t like the game, and took him out back and promised to break both his thumbs if they ever saw him in a pool hall again. There was a little “near-breaking” of thumbs involved.
There is an interesting story about the discoverer of Earth’s radiation belts, Van Allen. The house was generous with regard to their loss. However, it was made clear to him that he was not welcome to come back. Perhaps they charged his winnings up to an expensive education. They fixed the balance in the roulette wheels.
Tom, don’t discount the entertainment value of gambling. A little fantasy pushes the boundaries of imagination. Sometimes a lottery ticket is a mini-vaction and as rejuvenating.
I’d amend that: Statistical reasoning somehow never feels right, which is why people play lotteries and the house always wins.
No, a pair of dice get a triangular distribution, not a bell-shaped curve….
DMac ==> Yes, more precisely, but I intentionally use the term “looks like” because it looks like an unskewed normal distribution, which I later show with the two overlaid.
Nobody knows how the climate system really works, nobody has a handle on those known unknowns, let alone those unknown unknowns.
There is very little in the way of certainty, that’s where blind faith and dogma come in.
Should be stated in all caps: AVERAGES ARE NOT THEMSELVES MEASUREMENTS! Especially if you take the yearly average of the monthly average of the daily average of Temperature, from thousands of places, over decades of time, then subtract one large number from another large number and claim that you have found a meaningful difference in temperature – accurate to one hundredth of a degree! Balderdash!
hiskorr ==> Averages of Averages of Averages. Almost never a good idea — almost always results in an incorrect view of data.
Yes, spot on Hiskorr.
If I can pose an analogy –
applying the authenticated mathematical and statistical disciplines & processes that Kip and others detail here to the schemozzle that is the field of “global temperature measurements” is like trying to work out what the ideal number of birds eye chilis is to put in a curry –
how much curry are you making?
what’s the size ranges of the selected chilis?
are they all the at same level of ripeness?
discard all seeds or use a few?
how hot is “hot”?
etc etc
You get the picture –
A normal distribution assumes scalar data and an infinite variation of throw values are possible,
Not so a dice, it is effectively an ordinal scale and only certain results are possible.
Imagine a questionnaire with a checkbox of 1 = male and 0 = female. If our average comes out at 0.6 we can conclude (and can only conclude) that 60% of the respondents were male.
We cannot conclude that 100% of the respondents were slightly female.
(Although marketing types do use such silly expressions.)
I suspect a protracted bunfight coming
chas ==> Ah, but dice are an example that readers can easily understand and suffice for the purposes of illustrating of the point.
When Einstein was discussing dice with Born, Schrodinger’s cat was already dead,
vuk => “♪♫ but the cat came back, the very next day….”
And
…as opposed to the abstract concept of male/ female gender?
Sigh…(insert smiling yellow circle)
With measurements only a limited number of uncertainty intervals are available as well. If some instruments are +/- 0.1C, some +/- 0.3C, and yet others +/- 0.5C then you don’t have a continuous spectrum of uncertainty intervals.
If you have 100 boards of various lengths all of which average 6′, 50 with a measurement uncertainty of +/- 0.08′ and 50 with a measurement uncertainty of +/- 0.04′, then what would be the measurement uncertainty of a board that is actually the average length of 6′?
A statistician or climate scientists would tell you it is the average measurement uncertainty. A carpenter would tell you it’s either +/- 0.08′ or +/- 0.04′.
The total uncertainty would be sqrt[ 50 (0.08)^2 + 50(0.04)^2] = 0.63
The average uncertainty would be 0.63/100 = .0063.
This is how statisticians and climate scientists get uncertainty values that are physically unrealizable.
If I, and most engineers I know, were to use these 100 boards to make a beam spanning a foundation (for instance) I would use the total uncertainty of +/- 0.63′ to make sure it would reach, I certainly wouldn’t use +/- 0.0063′.
Bottom line: average uncertainty is *NOT* uncertainty of the average. No matter how badly statisticians and climate scientists want it to be.
Actually, we would ONLY use the -0.08 as a guide, because that is the shortest possible length, and would determine the maximum span. Statistics have zero practical importance, only real-time in-situ measurements count.
Before you hit me with a complex bridge, note that the architecht and surveyor may use stats, but the actual engineer uses a tape measure.
Or else you end up with multi-billion embarassments like that US/ Canada bridge that is so far over cost, we forgot about the string of stupid design mistakes…
cilo ==> (I’m a bit late seeing this, I am reviewing the comment section as a postmortem.)
“…only real-time in-situ measurements count.” My son has been rebuilding a 150 year old home, and keeps trying to get away with measuring one side and extrapolating the other. When I am working with him, I repeatedly say “Just hold the board up there and see if it fits, is too short or too long”. I call it practical engineering.”
“Statisticians insist that the correct sum would be:7 ± 3.52“
Statisticians would insist you define what you mean by “±” in this context. This is usually meant to mean a confidence interval, which will have some percentage applied to it, say 95%.
In metrology, the preferred usage is to give a “standard uncertainty”, i,.e. the uncertainty expressed as a standard deviation, and use ± for expanded uncertainty.
By all means insist that ± is only ever used to represent 100% confidence, but that isn’t the definition used by thoise who dictate the expression of uncertainty, and I can;t see how it’s helpful in understanding dice rolls. How does it help someone to know that the result of rolling 100 dice could be anywhere between 100 and 600?
Because, knowing that, a person wouldn’t gamble with dice?
/snort/ — the expert is on his soapbox.
Do you have a point, or are you just trolling again? You do not need to be an expert to read what a document says.
“Stop whining”—CMoB.
I’ll take that as a no.
No 10 mK T uncertainties today?
Bellman ==> 1) See the defintion and references for Absolute Measurement Uncertainty.
2) I create an analogous situation with the dice — we do not know the actual true value, but we know the uncertainty range (think temperature rounded to whole degree)…our dice roll true value is unknown (because it is in a box) but we can express it as a value (the mean) plus or minus the uncertainty range.
With absolute measurement uncertainty, we are 100% confident in what the uncertainty is — that is definitional.
You keep showing that definition, and I don’t think it means what you seem to think it does.
is saying that it’s the uncertainty is expressed as an absolute value of the measurement, rather than as a fraction. Hence it’s expressed in the same units. In the example, the uncertainty is independent of the length. 6.0 ± 0.1 cm, 60.0 ± 0.1 cm and 0.6 ± 0.1 cm, all have the same absolute uncertainty, i.e. ± 0.1 cm. But they all have different relative uncertainties.1/60, 1/600, 1/6.
You seem to think that the word “absolute” is implying that the uncertainty must lie between the value ± 0.1cm, but it’s really representing a probable range, say 95%.
There is nothing in that definition that requires the uncertainty to be 100%.
Bellman ==> Absolute MEASUREMENT uncertainty deals with the known uncertainty in a measurement. It is often caused by the “least count” of the measuring device:
“Uncertainty appears because of the limits of the experimental apparatus. If your measuring device can measure up to 1 unit, then the least count of the measuring device is said to be 1 unit. You cannot get any more accurate than the least count.”
Thus, your result is a range “measured value units +/- 0.5 the least count unit.
There is no probability involved.
Think what happens when measurements are rounded to the nearest whole number (or any specified number of digits)
“Think what happens when measurements are rounded to the nearest whole number (or any specified number of digits)”
Then you have an uncertainty of at least ± 0.5. But this still has nothing to do with your misunderstanding of the term “absolute measurement uncertainty”.
“There is no probability involved.”
Of course there’s a probability involved. If the correct value lies between two values, and there’s no reason to suppose it lies in a special position, then it’s as equally likely to be anywhere within that interval. Hence you have a uniform distribution.
Nope. You are describing a uniform distribution. You simply do *NOT* know if there is a uniform distribution within the uncertainty interval.
Again, there is one, AND ONLY ONE, true value. There aren’t multiple true values. That one, AND ONLY ONE, true value is simply unknown but it has a probability of 1 of being the true value. Since a probability distribution has to integrate to one that means that all other values in the interval must have a probability of 0.
I know that many people consider uncertainty to have a probability distribution BUT that comes from only considering the variability of the stated values as defining uncertainty and not from actually understanding what measurement uncertainty is.
If you *KNOW* what the probability is for each value in an uncertainty interval then you really don’t have an uncertainty interval, you have a KNOWN not an UNKNOWN.
“You are describing a uniform distribution.”
Did you figure that out from me saying “Hence you have a uniform distribution.”?
“You simply do *NOT* know if there is a uniform distribution within the uncertainty interval.”
Yes I do. I’m making the assumption “and there’s no reason to suppose it lies in a special position” and the only uncertainty being described here comes from rounding.
“Again, there is one, AND ONLY ONE, true value.”
Hence why I didn’t say true values.
“That one, AND ONLY ONE, true value is simply unknown”
Hence why the rounded measurement has uncertainty.
“but it has a probability of 1 of being the true value.”
We are not talking about the probability of the ONE AND ONLY ONE true value. We are talking about the dispersion of the values that could reasonably be attributed to the ONE TRUE VALUE.
“I know that many people consider uncertainty to have a probability distribution…”
By many people, are you including every source you insisted I read on the subject.
“If you *KNOW* what the probability is for each value in an uncertainty interval then you really don’t have an uncertainty interval, you have a KNOWN not an UNKNOWN.”
Probability by definition is uncertain. If I know there’s a 10% chance the ONE TRUE VALUE might be a specific value, I do not know it is that value, I just know there’s a chance that it’s that value. Hence, I’m uncertain if it is that value.
“Yes I do. I’m making the assumption “and there’s no reason to suppose it lies in a special position” and the only uncertainty being described here comes from rounding.”
There is only ONE value in the interval that gives the true value. So how can there be multiple values that do the same?
Rounding is basically an exercise of maintaining resolution limits based on the uncertainty interval. If your uncertainty interval is wider than the resolution then what’s the purpose of having the stated value having more digits after the decimal point than the uncertainty?
You typically use higher resolution devices to make the uncertainty interval smaller – but that requires the instrument to actually have a smaller uncertainty interval than the resolution. It doesn’t do any good to use a frequency counter to measure a 1,000,000 hz signal out to 7 or 8 digits if the uncertainty interval for the counter is +-/ 100hz! The resolution doesn’t help much if it isn’t accurate!
“We are not talking about the probability of the ONE AND ONLY ONE true value. We are talking about the dispersion of the values that could reasonably be attributed to the ONE TRUE VALUE.”
And now we circle back to assuming all measurement uncertainty cancels and the dispersion of the stated values is the measurement uncertainty.
You keep denying you don’t do this but you do it EVERY SINGLE TIME!
“By many people, are you including every source you insisted I read on the subject.”
Nope. EVERY SINGLE TIME you see someone assigning a probability distribution to uncertainty it is because they assumed all measurement uncertainty cancels and the probability distribution of the stated values determines the uncertainty! EVERY SINGLE TIME.
You keep claiming you don’t ignore measurement uncertainty and then you do it EVERY SINGLE TIME!
“Probability by definition is uncertain. If I know there’s a 10% chance the ONE TRUE VALUE might be a specific value, I do not know it is that value, I just know there’s a chance that it’s that value. Hence, I’m uncertain if it is that value.”
Measurements are *ALWAYS* a best estimate. If you know a value in an uncertainty interval that has a higher probability of being the true value then *that* value should be used as the best estimate!
From the GUM, 2.2.3
“NOTE 3 It is understood that the result of the measurement is the best estimate of the value of the measurand, and that all components of uncertainty, including those arising from systematic effects, such as components associated with corrections and reference standards, contribute to the dispersion.” (bolding mine, tpg)
If you know the distribution of the uncertainty then that distribution should be used to provide the best estimate of the measurand.
What you simply, for some unknown reason, can’t accept is that you do *NOT* know the probability distribution of the uncertainty interval. It is UNKNOWN. It’s a CLOSED BOX.
It is not a Gaussian distribution. It is not a rectangular distribution. It is not a uniform distribution. It is not a Poisson distribution.
There is one, and only ONE, true value. It’s probability is 1. The probability of all other values in the uncertainty interval is 0. What is the name of that distribution?
And he still can’t grasp the concept of the true value!
Here’s a slightly clearer definition
https://www.bellevuecollege.edu/physics/resources/measure-sigfigsintro/f-uncert-percent/
Bellman ==> See above,
“the uncertainty expressed as a standard deviation”
This is only when you have a normal distribution and typically when the assumption can be made that all measurement uncertainty cancels. Then the standard deviation of the stated values is used to express the uncertainty.
I disagree with most that a measurement uncertainty interval implies some kind of probability distribution. Within that uncertainty interval one, and only one, value is the true value. All the rest are not. That means that one value has a probability of 1 of being the true value and all the rest have 0 probability of being the true value. The issue is that you don’t know which value has the probability of 1. That’s why it is called UNCERTAINTY! You don’t know and can never know which value is the true value.
“By all means insist that ± is only ever used to represent 100% confidence”
No one insists that measurement uncertainty intervals include *all* possible true values. The measurement uncertainty interval is used to convey how much confidence one has in the measurement. A small uncertainty interval implies you have used high precision, calibrated instruments in a controlled environment to make a measurement. A larger uncertainty interval implies just the opposite. But in neither case is it assumed that the interval is all-inclusive.
This also implies that the uncertainty interval may not include the true value. What kind of a probability distribution would you use to describe that? Certainly not a normal distribution or a uniform distribution, each of which implies you know the entire possible range of values and their frequency. What do you have if the frequency of all values is zero?
“In our dicing example, if we applied that viewpoint, the ones and sixes of our single dies in a pair would have a ‘near zero’ probability coming up together (in a roll of two dice) to produce sums of 2 and 12. 2 and 12 represent the mean ± the full uncertainty value of plus or minus 5. ”
Of course that’s not true if you are only have a sample of two. There’s about a 1 in 36 ~= 2.7% chance of rolling a 12. But now increase the sample size to 5. What’s the chance of rolling 30? It’s 1 in 6^5, about 0.013%. For 10 dice the probability of getting 60 is 0.0000017%.
You could roll ten dice every second for a year and there would still only be a 50/50 chance you would get your ten sixes.
You didn’t even bother reading Kip’s entire paper, did you? You just jumped to trying to prove him wrong.
Do a search on the word “million”.
Bellman ==> Read for understandidng ….not just looking for something to attack.
You are performing an entirely different experiment. Stick to two dice, as in the example, rolled multiple times. We are not summing subsequent rolls, only looking (actually, we “not looking” because we have them in a box with a lid) at each next roll.
Have you taken the challenge yet? I don’t think so —
If I’m misunderstanding your point, maybe you need to be clearer.
You said
I agreed with the statement that if you have a sample of two, it is not near zero that a 2 or 12 could be rolled. But pointed out that if you are were to take larger samples, it does become vanishingly small that all the dice would be the same value.
I am talking about a different experiment, one based on a larger samples size, because that is more relevant to my objection to your claims, that you should just add up all the uncertainties regardless of how many measurements you take.
“Have you taken the challenge yet? I don’t think so —”
As I said last time you asked, it feels like a rigged competition to me. Any example I give would be considered not childishly simple enough, or would be based on probability theory and statistics which you reject. .
“If we used the statisticians regularly recommended “1 Standard Deviation” (approximately 68% – divided equally to both sides of the mean) we would have to eliminate from our “uncertainty” all of the 2s, 3s, 11s, and 12s, and about ½ of the 3s and 4s. — which make up 34% (>1-in-3) of the actual expected rolls.”
Which is why you don;t use 1 standard deviation as your confidence interval.
Bell ==> I am not talking “confidence intervals”, am I?
Please, try actually reading the essay for understanding, until you are able to argue for my position, even though you don’t agree. Once you understand what I am saying, then you can ask questions or make objections.
[This intellectual ‘trick’ is taught in Debating classes all over the world. To be able win a debate, you must first really understand and be able to logically argue the opposing view. Then you will be able to counter it intelligently.]
So far, all your comments have been “I don’t understand” and the equivalent of the schoolyard insistence “Is not!”
bellman uses the same tactics as Nickpick Nick Stokes…
You talked about using 1 standard deviation, and said it was what was recommended by statisticians, but said this would eliminate some numbers from your uncertainty. If you are not using 1 SD as a confidence interval what are you using it for, and why do you think it would eliminate some numbers?
I’d like to argue for your position, but when you say that statisticians want to eliminate anything outside a 1 SD range, it’s difficult to have anything positive to say. You are just arguing against a strawman.
If you want people to use this debating technique, maybe you should start by trying to explain what statisticians mean by “regularly recommending 1 standard deviation.”, rather than just assuming they mean rejecting anything outside 1 standard deviation.
The GUM says:
2.3.1
standard uncertainty
uncertainty of the result of a measurement expressed as a standard deviation
2.3.5
expanded uncertainty
quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand
You are trying to confuse the issue. Standard uncertainty is usually understood to be one standard deviation which is what Kip says eliminates possible values from consideration. And that *IS* what statisticians typically mean when they say “uncertainty”.
I’m not confusing anything. Yopu do not use a standard uncertainty to eliminate anything outside it’s range. That would be crazy. By definition the standard uncertainty is 1 standard deviation of the measurement uncertainty. You expect values to lie outside it. If you can assume the distribution is normal you would expect 1/3 of measurements to lie outside the range. No one should say anything outside the range is impossible.
But that’s what Kip is suggesting in the part I quote:
Bellman ==> I guess you don’t read journals. Thousands of studies give results with a 1 SD as one of the many differing “uncertainties”, like this one.
Giving the results with 1 SD, is not the same thing as claiming you have eliminated all values outside that range.
Bellman ==> They literally ignore everything outside of the 1 SD. Look at the link….thousands of stuies do exactly the same thing.
“I’m not confusing anything. Yopu do not use a standard uncertainty to eliminate anything outside it’s range.”
Of course you do! Otherwise you wouldn’t use standard deviation, you would instead use the range of possible values as your uncertainty interval.
“By definition the standard uncertainty is 1 standard deviation of the measurement uncertainty.”
You’ve got it backwards, as usual. Measurement uncertainty is many times defined as 1 standard deviation. Not the other way around!
“If you can assume the distribution is normal you would expect 1/3 of measurements to lie outside the range. No one should say anything outside the range is impossible.”
Why do you *ALWAYS* assume everything is a normal distribution and work from there? The RANGE of a population is the minimum value and maximum value. Of course there is always a MINIMAL possibility that values outside the range but how does that jive with the statistical rule that the integral of the probability distribution should equal 1? Do all probability distributions extend to infinity?
“Otherwise you wouldn’t use standard deviation, you would instead use the range of possible values as your uncertainty interval.”
So are you saying the GUM is wrong to use standard uncertainty? If they think nothing can exist outside of the range of a standard uncertainty why talk about expanded uncertainty?
“So are you saying the GUM is wrong to use standard uncertainty? If they think nothing can exist outside of the range of a standard uncertainty why talk about expanded uncertainty?”
You just keep on cherry picking things you have absolutely no understanding of. Someday you *really* need to sit down and READ THE GUM for understanding. Read every single word and try to understand what it says!
“3.1.1 The objective of a measurement (B.2.5) is to determine the value (B.2.2) of the measurand”
“3.1.4 In many cases, the result of a measurement is determined on the basis of series of observations
obtained under repeatability conditions”
Standard uncertainty and expanded uncertainty are nothing more than indicators to others how well you have determined your measurements. If you use just standard uncertainty that carries with it certain expectation for what you will find if you repeat the measurement. Expanded uncertainty extends the interval in which a repeated result can be considered to be valid.
If I tell you my measurement is 70C +/- 0.5C using standard uncertainty then you have a certain expectation of where your measurement of the same thing would lie. What expectation would you have if I told you the expanded uncertainty was 70C +/- 1C?
Do you have even the slightest clue as to what the difference between standard and expanded uncertainty actually is and when the use of either is appropriate?
GUM:
2.3.1
standard uncertainty
uncertainty of the result of a measurement expressed as a standard deviation
2.3.5
expanded uncertainty
quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand
Again, you need to STOP cherry picking stuff you think you can use to prove someone wrong and actually STUDY the documents you are cherry picking from in order to understand what they are saying.
Now you are back to using the argumentative fallacy of Equivocation. Trying to change the subject to something else. You are so transparent!
“Remember – in this example, we have turned ordinary uncertainty about a random event (roll of the dice) into “absolute measurement uncertainty” by placing our dice in a box with a lid, preventing us from knowing the actual value of the dice roll but allowing us to know the full range of uncertainty involved in the “measurement” (roll of the dice).”
What does this mean? How does putting something into a box, turn “ordinary” uncertainty into “absolute measurement” uncertainty. Absolute measurement uncertainty just measn the uncertainty expressed as an absolute value, as opposed to the uncertainty expressed as a fraction of the measurement.
Bellman ==> Start reading for understanding. It means just what it says, and is a simulation of a measurement which has been rounded to the nearest whole number, in which case the actual value has been discarded (never to be recovered) and substituted with a central value and its uncertainty range.
“Absolute measurement uncertainty just means the uncertainty expressed as an absolute value” — the full definition includes that the uncertainty results from the measurement instrument itself or from the measurement process — and thus the amount of uncertainty is absolute — exactly this many millimeters or inches or degrees — every time.
Maybe your use of some other defintion is what leads you to object so much.
Bellman has no definitions other that what fits at the time so he can argue someone is wrong.
Kip,
“Start reading for understanding.”
How can you gain understanding by putting the dice in a box and not looking at the results of the throw? You have no quantitative information. You can only juggle Kippish rules.
Got any 20 mK T uncertainties to quote today?
Nick ==> Now I know that you are mainly trolling here today. Even you can understand that I have hidden the actual result of the dice roll to simulate what happens (as I explain) when a value is, for example, rounded — discarding the known actual measurement and substituting an central value with a known absolute uncertainty which results in a range within which can be found the unknown true value of the measurement.
I’m sure English is your native language.
Kip,
If you want to get any understanding from your example, you have to actually look at the dice. What else is there?
I did that. I looked at the sum of 10 dice throws. The range is 10 to 60; the mean should be 35. The sd of a single dice throw is sqrt(35/12)=1.708. The sd of the sum of 10 should be sqrt(350/12)=5.4.
So I simulated 100 sums of 10 random throws. I got the following totals:
25 1
26 2
28 3
29 5
30 4
31 4
32 4
33 11
34 5
35 8
36 11
37 8
38 8
39 2
40 7
41 4
42 4
43 2
44 1
46 3
48 2
49 1
No totals at all in the range 10-24 or 50-60.
Now indeed 35 is a good measure of the mean, and most of the results lie within one predicted sd, 30 to 40. All but 3 lie within the 2 sigma range, 24 to 46. This is what stats would say is the 95% confidence range. These are useful descriptors of what happens when you sum 10 dice throws.
All you are telling us is that the range is between 10 and 60. This is far less informative.
Nick ==> Use the example as given….we are not taking sums of rolls….You are rolling two dice? What? THEN summing values of 10 rolls? Can’t quite figure you out.
You’ve shifted from a simple absolute measurement uncertainty example to something about the probabilities of extreme values?
You want to take sums, and find probabilities, that’s your business … but take the challenge!
“we are not taking sums of rolls”
That is exactly what you are doing in most of your example. Taking two rolls and adding them. 2+5 etc. Of course it doesn’t matter whether you roll two dice together or at separate times (they will never be exactly together anyway).
“No totals at all in the range 10-24 or 50-60.”
That just means that you didn’t make enough rolls. You stated yourself that “The range is 10 to 60”.
“All you are telling us is that the range is between 10 and 60. This is far less informative. “
Is not variance based on the range? How is variance not informative? It’s just the square of the standard deviation.
All you’ve really shown here is that you need a significant portion of population in order to determine the physical range, the variance, and the standard deviation. That’s part of the problem with the global average temperature. It’s based on a poor sample of the population.
Why is the standard deviation and range ever given for the average global temperature?
Nick ==> Quite sure I already explained this to you.
“Any dice gambler can explain why this is: there are more combinations of the values of the individual die that add up to 7 than add up to 2 (there is only one combination for 2: two 1s and one combination for 12: two 6s). ”
That’s essentially one of the many explanations for how entropy works at the molecular level. When the number of dice and their possible movements is 10 raised to the power of a godsquillion then the arrow only ever points in one direction.
Misunderstandings arise when somebody thinks throwing a die many millions of times will produce an average of precisely 7.000000000000000000000000 etc etc.
It won’t (very probably). It will just be one of the vastly huge number of possibilities that are extremely close.
Nearly irrelevantly, I remember the quote Liet Kynes made to Duke Leto in Frank Herbert’s “Dune”:
“You never talk of likelihoods on Arrakis. You speak only of possibilities.”
Michael Hart ==> Great — plus 100 for the Dune reference.
“Hey, I would love to be proved wrong on this point, really. But so far, not a single person has presented anything other than a “my statistics book says….”. Who am I to argue with their statistics books? ”
Have you tried books on metrology? Here’s the GUM, which is supposed to be the standard document on the subject.
https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf/cb0ef43f-baa5-11cf-3f85-4dcd86f77bd6
Equation (10) is the general equation for combining independent uncertainties.
Those boring old “Statistics Books” cover every example in the post. This is starting to channel Rand Paul:
“But just because a majority of the Supreme Court declares something to be “constitutional” does not make it so.”
bob, are you suggesting that every ruling handed down by a SCOTUS majority is ‘constitutional’?
Yes, by definition. You might not like it. I might not like it. And unlike the rules found and firmed up decades/centuries ago, in those boring old “Statistics Books”, they can be changed by subsequent SCOTi. But when they are handed down, they are indeed “constitutional.
Not at all!
Except perhaps when the matter involves a controversy between two branches of government, the explicitly stated policy of the court is to avoid addressing Constitutionality if it is at all possible to decide the case on some other aspect, which it almost always is. The court assumes, for instance, that any action of the legislature is valid, even when “unconstitutional on its face” unless the suit is expressed in such a way that the court has no path around that conclusion. Usually, however, the court will not choose to put such as case on it docket.
There is the possible exception that occasionally the court itself, regardless of what the parties to the suit may ask, wants to apply the Constitution.
Give me a Statisics Book” used in Statistics classes that actually covers measurement uncertainty. I’ve got five different ones here and there isn’t one example in any of them where “stated value +/- uncertainty” is used for analysis. All the examples use “stated value” only and then calculated a standard deviation for those stated values and call it “uncertainty”.
Tim G ==> Maybe that’s why they use alternate definitions that are not what I am talking abut…..
I’m positive that is the case. As someone else on the thread pointed out, you don’t learn about measurement uncertainty in math classes unless you are in a physical science or engineering curriculum. And even then you only learn it in the labs if they bother to teach it there! I actually learned more about uncertainty as a working carpenter apprentice and mechanic/machinist than I did in the engineering curriculum.
Bellman ==> Waiting for you to take The Challenge.
“my challenge to challengers: Provide a childishly simple example, such as I have used, two measurements with absolute measurement uncertainties given added to one another. The arithmetic, a visual example of the addition with uncertainties (on a scale, a ruler, a thermometer, in counting bears, poker chips, whatever) and show them being added physically. If your illustration is valid and you can arrive at a different result than I do, then you win! Try it with the dice. Or a numerical example like the one used in Plus or Minus.”
Surely, a person as certain as you are of the correctness of your position on this issue can knock together a simple ‘obvious to a sixth-grader’ example that could be laid out on the table or shetched on the classroom white board that would convince all the readers here that you are right.
I can create examples endlessly….you ought to be able to do just one?
The trouble with “childishly simple examples” is that they become too childish and simple. Your example is just of tossing two dice, but you want to extrapolate that to all averages of any size sample.
And the disagreement here, is not about what’s correct or not, it’s about what’s useful. If you want a range that will enclose all possible values, no matter how improbable, then what you are doing is correct. But a smaller range that encloses 95% of all values might be more useful.
I’m really not sure what simple explanation of the probability distribution, CLT or whatever you would convince you. I’ve already given the example of throwing 10 or 100 dice and seeing how likely it is you would get a value close to your full range value. But you just reject that as a statistical argument.
Bellman ==> I am not averaging in the dicing examples — I am only doing arithmetic.
Take the Challenge! On any of your points, for heavens sake.
“If you can’t explain it to a six year old, you don’t understand it yourself.”
(attributed, but maybe not actually said by, Albert Einstein)
It doesn’t matter if you average or sum the dice. You explained that in your post on the CLT. Got rather annoyed that a video had taken the time to explain it if I remember correctly.
I’ve tried to explain to you as simply as I can why you are are wrong. You just won’t accept any example that disagrees with your misunderstandings.
I pointed out what happened when you average or sum 10 dice. You just complained I was doing a different experiment and you were only interested the roll of two dice.
Bellman ==> Still no example….
OK. As an example you put 100 dice in a box. Shake it around. According to you the expected value is 350, with an absolute uncertainty of ± 250, so the sum of the 100 dice could be anywhere between 100 and 600. So I give you 40 to 1 odds on the sum being greater than 500. All you know is that’s 1/5 of all the possible values, so do you think it’s a good bet or not. How would you use your ability to add uncertainties to tell you how likely it is that the sum is greater than 500?
By contrast, I say the standard uncertainty of each die is 1.7, and using the CLT I conclude that the sum of the 100 dice is going to be close to normal, with a mean of 350 and a standard deviation of sqrt(100) * 1.7 = 17. The 95% interval on the sum is ±33.3, so I would expect only about 2.5% of all sums to be greater than 350 + 33 = 385. Not even close to the 500 target. So this does not look like a good bet to me. I’ve only got a 1 in 40 chance of getting higher than 385.
In fact 500 is 350 + 150, around 8.8 standard deviations from the mean. The probability that you will get 500 or higher is 7 * 10^(-19). That’s a very small probability. Even if I offered you 1000000 to 1 odds, it is still a terrible bet.
To test this I used R to generate 10000 rolls of 100 dice. The mean of the sums was 350.02. The standard deviation was 17.06. The range of all values was 285 – 410.
Here’s the frequency of the rolls in blue, alongside Kip’s uncertainty range marked by the red lines.
This is the longest row of insults ever, s’long’s I bin here…
Bell, my Man, repeat your experiment with actual dice. I care not what language you use, you still saying “PRN”.
Pseudo-random number, man, PSEUDO… the best any computer can do.
Or has things changed while I wasn’t looking? It certainly is the only
P.S. Good luck finding a pair of honest dice. Yours never show a 1, frexample?
Pseudo random numbers are perfectly useful for Bellman’s example. Unless you’re writing CIA code, it’s a bogus criticism.
Malarky! I can tell you from experience in role playing games using dice that you will *ALWAYS* see the entire range of possible values over time. If the range is 100 to 600 then sooner or later you WILL see a 100 and a 600.
The fact that the psuedo-random generator never spit out either a 100 or 600 is experimental proof that something is wrong with the experiment!
If you’ve ever seen someone rolling 600 with 100 dice thet were cheating.
Was it you or your brother who was insisting that if you tossed a coin a million times you were almost certain tosee a run of 100 heads?
“Malarky! I can tell you from experience in role playing games using dice that you will *ALWAYS* see the entire range of possible values over time. If the range is 100 to 600 then sooner or later you WILL see a 100 and a 600.”
You seem overamped, which his why you are responding to a point I never made. My only claim was that PRN’s were as useful as TRN’s for this application.
Yes, I’m using pseudo random numbers, as does Kip.
No, I doubt that has much if an effect on my test, given the results look like you’d expect.
No, I have intention of making 1000000 rolls, to confirm this. Apart from anything there would be far more human errors.
Not sure what you mean about never showing a 1. I did the test quickly last night and it’s possible I made a mistake in the code, but if so it’s remarkable that the sum was nearly exactly expected value.
“Not sure what you mean about never showing a 1.”
Sooner or later ALL possible values in the range should appear at least once. The fact that your range is limited is proof that something is wrong in your experiment.
I’ll tell you what. Grab 100 6 sided dice and keep throwing them until you get all 1s. Please don’t post until you’ve done it.
bellman doesn’t believe in systematic uncertainty. All dice are perfect.
Stop lying about me.
The assumption was fair dice. If all your dice are loaded so they always come up as 6, then obviously you will always roll 600 on a hundred rolls. Complaining that my experiment wasn’t assuming loaded dice is missing the point.
Note, that even if the dice are rigged so that 6s come up 5/6 of the time, you could still be rolling your hundred dice every second for of every day for a year before you have a reasonable chance of getting 600.
No one was complaining that your experiment didn’t use loaded dice.
You didn’t allow for loaded dice in your experiment. That’s a totally different thing!
The issue is that since you have a CLOSED BOX you can’t tell if you have a loaded dice or not!
Do you understand what the term “CLOSED BOX” actually means?
*I* was the one that pointed out to you that you didn’t do enough rolls to properly develop the probability distribution. And now you are trying to lecture me on how many rolls are needed?
Bellman ==> Well, you demonstrate something …. at least. That a sum of lots of things has a different probability than the sum of a few things.
Not the point, and does not qualify.
If you have found some difference in rolling two dice a million times …….
How does that show us anything about rounding measurements and substituting the central value of the range?
“Does not qualify”
I’m shocked to discover this contest is as rigged as I said it was. The only childish examples permitted are those that agree with Kip’s argument.
Why should my example be limited to just two dice? The whole point is that uncertainty of the average decreases with sample size. A sample of 2 is a very small sample which won’t much difference between plain adding of uncertainties and adding using quadrature.
Kip wants to use the example of 2 dice to make a spurious claim that you must never add in quadrature, and then apply this logic to the average of a large sample, but won’t allow counterexamples if they use larger samples.
No one is saying it should be limited to just two dice.
It *should* be the same experiment, however. DON’T OPEN THE BOX!
Two dice allows for determining the actual range with a limited number of throws. Using more dice means a *big* increase in the number of rolls needed to see all values. As a quick guess for 100 six-sided dies It would be something like 6^100/100. A million rolls won’t even come close. 6^100 is something like 6×10^77 so you would need something like 6×1075 rolls to get all values.
TG: “No one is saying it should be limited to just two dice.”
KH: “If you have found some difference in rolling two dice a million times …….”
“DON’T OPEN THE BOX!”
Not much of an experiment if you can’t look at the result. Really, what new idiocy is this? “I’ve got an experiment that will prove my point, but it won;t work if you open the box, so you will just have to take my word that it works.”
“Using more dice means a *big* increase in the number of rolls needed to see all values.”
Gosh. Almost as if that’s my point. And also why Kip insists on limiting his childish example to 2 dice.
“As a quick guess for 100 six-sided dies It would be something like 6^100/100.”
Drop the divide by 100 and you would be correct. The expected number of rolls to get any result with probability p is equal to 1/p.
“A million rolls won’t even come close. 6^100 is something like 6×10^77 so you would need something like 6×10[^]75 rolls to get all values.”
Again, I’m not sure why you are dividing by 100, but it’s irrelevant. You just don’t get how big 10^77 is.
Get everyone on the planet to throw these dice every millisecond of every second of every minute of every hour of every day for a billion years, and you are still not remotely close to getting that number. And by not even close, I mean you would have to repeat the exercise something like 10^50 times to get the required number.
If you ever see it happen, assume you are living in a simulator, or someone was cheating.
“Not much of an experiment if you can’t look at the result.”
THAT’S THE WHOLE POINT OF THE CLOSED BOX!
The uncertainty interval provides no result allowing the development of a probability distribution!
IT’S A CLOSED BOX!
“You just don’t get how big 10^77 is.”
ROFL!! Why do you think I pointed it out to you? Rolling a million times is not nearly enough!
You are the one that tried to use your limited number of rolls to describe the range and variance of the population, not me!
You seem to be under the impression that if you can’t see something it doesn’t exist. You have dice in a box. You can’t see the dice, but you still know there is a probability distribution, you still know that some results are more probable than others.
This is really the crux of the problem. Kip says
And I say he’s wrong. We can know more than just the set of all possible values, we can know how much more likely some values are than others. This doesn’t matter too much with just two dice, but it matters a lot more with a bigger sample of dice, where the full range is covering values that are virtually impossible.
It’s just nonsense to suggest that the “only” thing we can say about the sum of 100 dice in a closed box is that they could be anything between 100 and 600.
You ignored the question. How would your understanding of uncertainty allow you to say if the bet was good or not?
How do you settle the bet when the box is CLOSED?
Uncertainty means YOU DON’T KNOW! You keep wanting to come back to you knowing what the true value is out of an uncertainty interval!
“Uncertainty means YOU DON’T KNOW! “
I ordered parts for my bike last week. When I aksed when they would be delivered and assembled, the bike store guy said I Don’t Know”. I was about to walk out the door, when I decided to swab him down a little.
“Will they be ready and assembled by tomorrow?”
Hell no. one of the parts is still on a boat”
“Will they be ready and assembled in a month?”
I can’t remember when a similar order took that long.
“Will they be ready and assembled in a week?”
If I had to bet even odds, I’d bet on it.
Now, questions for you:
A confession. It never happened. This story has been told to industry schools on statistics I’ve attended, twice. They are a response to ridiculous assertions on uncertainty. like yours.
What point do you think you are making?
The point of your example is that you DO NOT KNOW! The whole point of uncertainty is that you don’t know. It is a CLOSED BOX!
blob is completely off in the weeds this week.
“You seem to be under the impression that if you can’t see something it doesn’t exist.”
And now we circle back to your reading comprehension problems.
The issue is that you do not know! The issue is not whether something exists or not.
If there is systematic bias on any of the dice you don’t know what it is and yet it will definitely impact any distribution of values you might get.
Uncertainty is a CLOSED BOX. There might be something in there but you have no idea what.
Look at it this way. You have ten dice. 9 of them are 1″ in diameter and one (the true value) is 1/4″ in diameter. . There is one, and ONLY ONE, dice that will ever fall out of the box if you drill a 1/4″ hole in the bottom of the box- the true value. The only problem is that you DO NOT KNOW WHICH DICE IT IS.
So how do you drill a 1/4″ hole in that uncertainty box if everything is unknowable?
His experiment is fatally flawed. He opened the box to see what value was shown.
And since his experiment has a range far smaller than the possible range, the experiment is fatally flawed.
Could you explain how to perform the experiment without looking in the box. I can only see to approaches to the experiment- do the maths or look at the result. You reject both.
Kip *did* the math. You just don’t like the results.
Kip developed an uncertainty interval within which you can’t develop a probability distribution because you can’t see inside the box.
YOU are stuck in your box that an uncertainty interval has to have a probability distribution stating what the probability is for each and every value. In other words you think you can identify where in the uncertainty interval the true value lays.
BUT THE WHOLE POINT OF AN UNCERTAINTY INTERVAL IS THAT YOU DO *NOT* KNOW!
He did the maths as did I. The only reason we get different results, is because I’m interested in finding a range that covers the majority of results, whereas he wants something that covers every possible result.
I’m doing exactly what all your authorities do. You used to do it. You were the one who said the uncertainty of the sum of 100 thermometers with uncertainty ±0.5°C would be ±5.0°C. Now you seem to want to throw out every metrology textbook and insists that the only uncertainty range allowed is one covering all bases.
Well, the author once again throws two different things into the same pot in order to construct a conflict that doesn’t exist among the experts. 1. Kip understands absolute uncertainty as nothing other than the range of values. In other words, the set of all fundamentally possible outcomes, regardless of the probability of their occurrence in practice.
2. The statistical uncertainty, on the other hand, takes into account the probability of the possible outcomes occurring in practice. There the uncertainty or, complementarily, the accuracy range for a given probability is given: E.g. with probability 95% the result is between the values say Cl and Cu. The higher the probability is set close to 100%, the further apart Cl and Cu are, and the interval (Cl, Cu) is including each interval with lower probability. The interval with the probability 100% (“absolutely sure”) is the highest and identical to the interval of the absolute inaccuracy of 1.
But which of the two uncertainty metrices 1. or 2. is more relevant in practice? Kip thinks 1., the statisticians and measuring scientists usually take 2.. Let’s take Kip’s example with the two dice in the box, where you take the sum of the pips as the result regardless of the combination of the individual pips. Kip has already argued with one and two dice and said that with two dice the absolute uncertainty 7 ± 5 is the more relevant specification and not the interval (3.5 ± 2.5) + (3.5 ± 2.5) = 7 ± SQRT (2.5² + 2.5² ) = 7 ± SQRT(6.25 + 6.25) = 7 ± (SQRT 12.5) = 7 ± 3.5, which only defines a probability range, but does not include the rarer but nevertheless possible results.
OK, so what? As said, both intervals are valid; they simply give different definitions of uncertainty. Therefore there is no dispute. But why do statisticians usually prefer definition 2. as an indication? Is it really the case that – as Kip suspects – one wants to “disguise” the entire (absolute) range of uncertainty and therefore chooses 2.? Or do you take 2. because 1. is already known, but is usually uninteresting?
For our judgment, let’s increase the number of dice in the box from 2 to 200. The smallest possible sum of pips is 200, namely if all 200 dice have the pip 1, the highest possible is 1200 if all have 6. The probability of the occurrence of the result 200 or 1200, i.e. the limit points of absolute uncertainty, is (1/6)^200 = 2.3*10^(-156) (this is less likely than finding again a single specific atom after mixing it in the entire rest of the universe). The most probable result is 3.5 × 200 = 700. Thus, Kip’s absolute uncertainty is 700 ± 500, whereby everyone is already wondering what the practical relevance it is to keep the hopelessly improbable cases included by the uncertainty limits of ± 500 …
Altogether we have 1001 possible results for the sum of the pips of all dice: from the one comb
ination 1+1+1+…+1 = 200 for the sum of 200, through the 200 combinations for the sum 201, which is given by the series 2+1+1+.. +1 = 1+2+1+…+1 =… = 1+1+1+…+2 = 201 until again exactly one for 6+6+6+…+6 = 1200.
In order to motivate an individual judgment as to which interval specification (1. or 2.) has more practical relevance, let the sum result of a “box test” – but now with the 2000 dice in it – be linked to a wager! With 2000 dice, there are 10001 possible sums of pips (2000 to 12000). Let each player pay 100 Dollars into the pot and by that allow him to give a guess about the resulting sum by writing down a list of 300 (about 3%) of the 10,001 possible numerical results. The pot is won by the player (several winners share it) who has noted the result of the “box test” under the 300 numbers written down.
With Kip’s absolute uncertainty range from 2000 to 12000, would you dare to play the wager? Which 3% of possible sum results would you choose and why?
“The range of all values was 285 – 410″
First, that alone should tell you that something is off in your example.
Second, how do you know what the values are when you have a closed box? Uncertainty means you don’t KNOW the values in the box! The box remains closed.
You continue to go down the primrose path of assuming that you know the probability distribution for all the values in the uncertainty interval. YOU DON’T! If you did there wouldn’t be any uncertainty!
“ so do you think it’s a good bet or not.”
You continue to misunderstand. Is that deliberate?
You don’t open the box so you will never know what the sum is! Your bet can never be completed!
“I conclude that the sum of the 100 dice is going to be close to normal,”
Meaning you assume, as always, that there is no systematic bias at play and that all distributions are normal.
Keep trying. You haven’t met the challenge yet!
Kip is correct. You can’t even admit that the average uncertainty is not the uncertainty of the average.
Could somebody give Tim a shove, his needles stuck.
I don’t care how many time you are going to repeat this nonsense. I’m just going to remind everybody it is completely untrue.
You keep claiming you don’t believe all distributions are normal and that average uncertainty is not uncertainty of the average and that the standard deviation of the sample means is not the uncertainty of the mean BUT *every* *single* *time* you post something you circle right back to those. All distributions are Gaussian (i.e. all measurement error cancels), average uncertainty is the uncertainty of the average (i.e. you can ignore measurement uncertainty and just use the stated values), and that the standard deviation of the sample means is the uncertainty of the mean.
You can whine otherwise but your own words belie your claims.
“…BUT *every* *single* *time* you post something you circle right back to those.”
No I don’t. You just see what you want to see.
“All distributions are Gaussian (i.e. all measurement error cancels),”
How many more times? All measurements do not cancel, and to get some cancellation you do not need a Gaussian distribution.
“average uncertainty is the uncertainty of the average (i.e. you can ignore measurement uncertainty and just use the stated values)”
Complete non sequitur. Average uncertainty is not the uncertainty of the average. If you add all uncertainties to get the uncertainty of a sum (as Kip proposes), then you will find the uncertainty of the average is equal to the average uncertainty. But if you add the uncertainties in quadrature as I suggest (for independent uncertainties) then the uncertainty of the average will be less than the average uncertainty.
And this does not mean you can necessarily ignore measurement uncertainty when taking an average, I just think it becomes less relevant unless you there is a major systematic error in your measurements.
“and that the standard deviation of the sample means is the uncertainty of the mean.”
Again, only if you assume there are no other sources of uncertainty than that from random sampling.
“No I don’t. You just see what you want to see.”
Of course you do. It’s why you think you can develop a probability distribution for all the values in an uncertainty interval. Then you can say it all cancels and you can use the stated values!
“How many more times? All measurements do not cancel, and to get some cancellation you do not need a Gaussian distribution.”
Then why do you use average and standard deviation? If the distributions are not normal then those statistical descriptors tell you nothing about the distribution.
“Complete non sequitur. Average uncertainty is not the uncertainty of the average.”
Then why do you say it is? Why do you say a board of average length measured with a defined uncertainty can have an uncertainty that is the average uncertainty?
You say you don’t do this but you come right back to it EVERY SINGLE TIME!
“And this does not mean you can necessarily ignore measurement uncertainty when taking an average, I just think it becomes less relevant unless you there is a major systematic error in your measurements.”
And here we are again, you assuming that all uncertainty cancels! That uncertainty always has a Gaussian distribution.
You keep denying you don’t do this but you come right back to it EVERY SINGLE TIME!
“Again, only if you assume there are no other sources of uncertainty than that from random sampling.”
Then why do you never propagate measurement uncertainty onto the sample means and then from the sample means to the estimated population mean?
You keep denying you don’t do this but you come right back to it EVERY SINGLE TIME!
Its the hamster wheel, around and around it spins…
He can’t help it! It’s the only way he can justify the stated value as being the true value. If the uncertainty is a Gaussian distribution then the mean, the stated value, is the true value!
Then he can use the variation of the stated values to determine his standard deviation – known to him as his “uncertainty”.
He just plain can’t help himself no matter how much he denies he doesn’t.
Looks to me like he doesn’t even realize it, must be psychological.
Pointless to argue with you when you just ignore everything I say, and say I claimed the opposite. I really begin to worry about your cognitive faculties sometimes.
E.g.
Bellman: “Average uncertainty is not the uncertainty of the average.”
Gorman: “Then why do you say it is?”
It doesn’t matter how many times I say it isn’t and point out why it isn’t. Gorman just comes back with “why do you say it is”?
“Why do you say a board of average length measured with a defined uncertainty can have an uncertainty that is the average uncertainty?”
And here you see why he doesn’t understand this. He can’t understand there’s a difference between the uncertainty of a measurement of an average board, and the uncertainty of the average length of the board.
Bellman: ” I just think it becomes less relevant unless you there is a major systematic error in your measurements.”
Gorman: “And here we are again, you assuming that all uncertainty cancels!”
Here, less relevant means assume all uncertainties cancel.
Followed by the mindless claim “That uncertainty always has a Gaussian distribution.”. I keep trying to explain that you don’t need a Gaussian distribution for uncertainties to cancel, it just passes from one ear to another with nothing interfering with it’s passage.
“Then why do you never propagate measurement uncertainty onto the sample means and then from the sample means to the estimated population mean?”
I keep propagating measurement uncertainties, it’s just they are usually much smaller than the uncertainty caused by sampling, and so tend to become irrelevant.
“ I keep trying to explain that you don’t need a Gaussian distribution for uncertainties to cancel”
Then why do you keep saying that the uncertainty has a Gaussian distribution?
“I keep propagating measurement uncertainties, it’s just they are usually much smaller than the uncertainty caused by sampling, and so tend to become irrelevant.”
Uncertainties ADD for independent, multiple measurands. They *can’t* be less than your sampling uncertainty unless you are doing something very, very wrong!
“Then why do you keep saying that the uncertainty has a Gaussian distribution?”
I don;t keep saying it. It’s just the voices in your head.
Sometimes the distribution is normal, sometimes it isn’t. How much clearer can I be than that.
“Uncertainties ADD for independent, multiple measurands.”
Define “ADD” and define “for”. When and how are you adding them?
My two main objections here are to Kip arguing that when you add values you can only do plain adding, and not adding in quadrature. And your old assertion that once you’ve obtained the uncertainty of the sum you do not divided that by sample size when taking the mean.
It’s really tiring trying to keep track of who believes what, as nobody will answer a direct question and only speak in vague terms.
Do you now agree with Kip that you cannot add in quadrature?
Do you still disagree with Kip that you should divide the uncertainty of the sum by N when looking at the uncertainty of the mean?
“My two main objections here are to Kip arguing that when you add values you can only do plain adding, and not adding in quadrature.”
Taylor covers this in depth. If you don’t know whether there is cancellation then direct addition of uncertainties is appropriate. In any case it *always* sets an upper bound on the uncertainty.
If it is possible that there will be cancellation then adding in quadrature is appropriate. It can be considered a lower bound on the uncertainty.
READ THAT CAREFULLY! Adding in quadrature is used when there is *not* complete cancellation!
How do you get cancellation of uncertainty from dice rolls?
“It’s really tiring trying to keep track of who believes what, as nobody will answer a direct question and only speak in vague terms.”
You’ve been given answers, DIRECT ANSWERS, to your questions over and over and over and over again, ad infinitum! The fact that you simply won’t read them or even try to understand them is *YOUR* problem, not anyone elses. *YOU* make yourself the victim, not us.
“Do you now agree with Kip that you cannot add in quadrature?”
You don’t even understand, even after being told that you need to work out EVERY single problem in his book (answers in the back), when you add in quadrature and when you add directly.
“Do you still disagree with Kip that you should divide the uncertainty of the sum by N when looking at the uncertainty of the mean?”
I can’t find where Kip said the average uncertainty is the uncertainty of the mean. *YOU* are the only one that keeps claiming that even though you say you don’t!
“Taylor covers this in depth. If you don’t know whether there is cancellation then direct addition of uncertainties is appropriate. In any case it *always* sets an upper bound on the uncertainty. ”
Finally a straightish answer. Exactly if there is no cancellation, i.e. the errors are not independent direct addition is correct. It there is cancellation, e.g all errors are independent then adding in quadrature. Between those two it’s more complicated, but the uncertainty is always somewhere in between, i.e. direct addition is an extreme upper bound.
As I say this is not what Kip is saying. He insists that anything other than direct addition is wrong even when the uncertainties are independent, e.g when throwing dice.
“Adding in quadrature is used when there is *not* complete cancellation!”
I don’t think you meant to have that *not* there. But I’m still not sure what you mean by “complete cancellation”. Cancellation is never complete in the sense that you can assume that every error cancel out. If it did there would be no need to add in quadrature. The idea of that is that some errors cancel when you add, the total uncertainty still grows when adding, it just grows at a slower rate compared to the sum.
“How do you get cancellation of uncertainty from dice rolls?”
If you want to think of dice rolls as being like errors, you need to think of them as having negative and positive values, so tike each roll and subtract 3.5 from it. Then you get a representation of error with the values running from -2.5 to +2.5. If I roll, say a 2 and a 5 on the dice, that becomes -1.5 +1.5 = 0. Complete cancellation. If I roll 2 and 6, then we have -1.5 + 2.5 = +1.0. partial cancellation. If I roll 4 and 6, we have +0.5 + 2.5 = 3.0. No cancellation as such, but still less than the maximum we would have if we just rolled a 6.
Looking at all possible rolls of two dice, and just subtracting 7. The most likely value is 0, and the next most likely values are -1 and + 1. The least likely values are -5 and +5, each with just a 1/36 chance.
So yes, the errors tend to cancel.
“You’ve been given answers, DIRECT ANSWERS, to your questions over and over and over and over again, ad infinitum! The fact that you simply won’t read them or even try to understand them is *YOUR* problem, not anyone elses. *YOU* make yourself the victim, not us.”
Lets see how direct you are with the answers:
Q: “Do you now agree with Kip that you cannot add in quadrature?”
A: “You don’t even understand, even after being told that you need to work out EVERY single problem in his book (answers in the back), when you add in quadrature and when you add directly. ”
I’ll take that as a no. You do accept it’s appropriate to sometimes add in quadrature. You could have said that without all the insults, but I’m glad someone has finally answered the question.
Q: “Do you still disagree with Kip that you should divide the uncertainty of the sum by N when looking at the uncertainty of the mean?”
A: “I can’t find where Kip said the average uncertainty is the uncertainty of the mean. *YOU* are the only one that keeps claiming that even though you say you don’t!”
I didn’t ask if he said the average uncertainty is the uncertainty of the mean. Just if you agreed that you can divide the uncertainty of the sum by sample size to get the uncertainty of the average. Agreed, if you insist on direct addition they become the same. But Kip is the one insisting on direct addition.
The point about dividing the uncertainty by N has been made several times. Most recently in this comment
https://wattsupwiththat.com/2023/01/03/unknown-uncertain-or-both/#comment-3661678
Note that, the “average of many rounded values will carry the same original uncertainty value”. Is saying the uncertainty of the average will be the average uncertainty.”.
But it’s the division by n I’m still asking about. You were very adamant for a long time that that is something you should never do. I’m just wondering if you disagree with Kip here.
“ Exactly if there is no cancellation, i.e. the errors are not independent direct addition is correct”
It’s *NOT* an issue of independence. Single measurements of different things are independent by definition. The issue is whether the uncertainties represent a Gaussian distribution and therefore cancel. 1. Systematic bias ruins the Gaussian distribution assumption. 2. All field measurements have systematic bias. 3. If all measurements have systematic bias then assuming all the uncertainties cancel and the stated values can be used to determine uncertainty is wrong.
You simply do not live in the real world!
“Between those two it’s more complicated, but the uncertainty is always somewhere in between, i.e. direct addition is an extreme upper bound.”
I’ve asked you before and not received an answer. If we each measure the temperatures at our locations at 0000 UTC using our local measuring instrument, how Gaussian our our uncertainty intervals and how much will the uncertainties cancel?
Be brave. Give an answer. Will RSS be the appropriate statistical tool to use to determine the uncertainty of the sum of the two temperatures?
“It’s *NOT* an issue of independence.”
Possibly I’m conflating systematic errors with lack of independence. It might depend on exactly how you treat them.
“Single measurements of different things are independent by definition.”
That isn’t necessarily true. For example with temperatures it’s [possible that different weather conditions could cause a systematic shift in the measurement error. I think that’s what Pat Frank claims in his uncertainty analysis.
“The issue is whether the uncertainties represent a Gaussian distribution and therefore cancel.”
How many more times, it doesn’t matter what the distribution is.
“Systematic bias ruins the Gaussian distribution assumption.”
It doesn’t. A systematic error would preserve the shape of the distribution, but changes the mean.
“I’ve asked you before and not received an answer. If we each measure the temperatures at our locations at 0000 UTC using our local measuring instrument, how Gaussian our our uncertainty intervals and how much will the uncertainties cancel?”
I’ve no idea how Gaussian the uncertainty intervals are. You would have to test the equipment of rely on the manufacturers specifications.
If the uncertainties are random and independent the standard uncertainties will cancel in the same way as they always do regardless of the distribution, i.e. the single uncertainty divided by root 2.
If there’s the same systematic bias in both stations that won’t cancel, by definition of systematic.
“Be brave. Give an answer.”
I keep giving you answers but you just don;t like them.
“Will RSS be the appropriate statistical tool to use to determine the uncertainty of the sum of the two temperatures?”
Depends on the nature of the uncertainty and how detailed an analysis you are doing. What is the purpose of finding the sums of two temperatures, bearing in mind temperature is intensive and so the sum has no meaning? What do you want the individual temperature to represent? E.g. are you only interested in the temperature at the location of the station, or do you think it represents a broader area?
Really worrying about the measurement uncertainty seems pointless if all you are going to do is add two stations at different locations. Uncertainty becomes important when the measurements have some purpose, and with means that’s usually because you are testing for significant differences.
“ Cancellation is never complete in the sense that you can assume that every error cancel out. If it did there would be no need to add in quadrature. The idea of that is that some errors cancel when you add, the total uncertainty still grows when adding, it just grows at a slower rate compared to the sum.”
If one is skewed left and one skewed right because of systematic bias then how does quadrature work?
Using quadrature assumes a Gaussian distribution for each. Taylor specifically mentions this in his tome. I’ve given you the exact quote.
“Using quadrature assumes a Gaussian distribution for each.”
It does not. Here’s a page one of you lot insisted I read.
https://apcentral.collegeboard.org/courses/ap-statistics/classroom-resources/why-variances-add-and-why-it-matters
See the proof of why variances add. Nothing in it requires knowing the distribution, only the variance and mean. The only requirement are that these are finite and that the variables are independent.
You are kidding right? This page addresses the variance of random variables!
This was only meant to give you a feel for the fact that uncertainties add just like variances do.
If a random variable can be described by a mean and standard deviation then the implicit assumption is that it is Gaussian, or at least symmetric around a mean. If it isn’t then the mean and standard deviation is basically useless and the use of the quadratic formula is inappropriate, as Taylor specfically states.
You can continue to try and justify the global average temperature as being statistically valid but you are going to lose every time. The global temperature record is not Gaussian or symmetric around a mean, it is riddled with systematic bias (both from device calibration and microclimate impacts as well has human tampering), and variances are all over the map because of seasonal differences. ANOMALIES DO NOT HELP ELIMINATE ANY OF THIS.
“If one is skewed left and one skewed right because of systematic bias then how does quadrature work? ”
Magic.
But let’s see if the magic works. I generate two sets of figures. Sequence x is the exponential distribution with rate = 1/2. Sequence y is the negative of an exponential distribution with rate 1/3. Both were shifted to make their mean 0.
SD of x is 2, SD of y is 3. Using quadrature you expect the SD of x + y to be sqrt(2^2 + 3^2) ~= 3.6.
I generate 1000000 pairs and look at the standard deviation of the sum, and I get 3.6, to 2 significant figures.
To 4 sf, the expected SD 3.606, and the experimental result was 3.600.
Every thing you post trying to rationalize how the global average temp means something is magical thinking.
“Both were shifted to make their mean 0.”
And now we circle back around. Got to make everything symmetric (usually Gaussian) so you can assume everything cancels.
Uncertainty intervals with systematic bias do *NOT* have their mean at zero!
“And now we circle back around. Got to make everything symmetric (usually Gaussian) so you can assume everything cancels. ”
This constant moving of goal posts is exhausting. You asked”
“If one is skewed left and one skewed right because of systematic bias then how does quadrature work?”
Adding your own answer
“Using quadrature assumes a Gaussian distribution for each. Taylor specifically mentions this in his tome. I’ve given you the exact quote.”
I demonstrated that adding in quadrature works with two distributions one skewed to the left one to the right. So now you change the rules and demand that they don’t have a mean of zero. As well as making some inane suggestion that the two distributions were symmetrical. They were not symmetrical distributions, one was skewed to the left one to the right, and they weren’t even mirror opposits of each other.
If I had given them different means, the standard deviations would have still followed the rule of adding in quadrature, the only difference would be the sum would have a different mean.
Yes, this is what happens with systematic error, which is why adding with quadrature is used for independent random uncertainties.
Richard Feynman rather than Einstein.
“The trouble with “childishly simple examples” is that they become too childish and simple. Your example is just of tossing two dice, but you want to extrapolate that to all averages of any size sample.”
A correct theory will work for 2 dice or for 200 dice. It doesn’t matter. The amount of elements just creates more drudge work in doing the sums.
” But a smaller range that encloses 95% of all values might be more useful.”
So what? How do you *know* that? The word “might” is the operative word here. There *is* a reason why the statistical description of a skewed distribution is better served by – minimum, first quartile, median, third quartile, and maximum. Please notice that minimum and maximum *is* the range. The range is a direct part of the variance and the variance is an indirect description of the next expected value.
“I’m really not sure what simple explanation of the probability distribution, CLT or whatever you would convince you.”
Why do you cling so tight to the CLT? The CLT tells you NOTHING about the population probability distribution. It only tells you how close you are to the population mean and in a skewed distribution the population mean is not very descriptive. This stems from you viewing *all* probability distributions as Gaussian. You just can’t seem to punch your way out of that box!
“A correct theory will work for 2 dice or for 200 dice.”
Which is my point. Kip’s theory, that you can ignore all probability and just use the full range as a meaningful measure of uncertainty, clearly doesn’t work for 200 dice.
“The CLT tells you NOTHING about the population probability distribution.”
Which is why you don’t use it to tell you about the population distribution. It’s not what it’s purpose is.
(Actually, I think you could use the CLT to learn a lot about the population distribution, but that would require more work than I care to go into at the moment.)
“It only tells you how close you are to the population mean and in a skewed distribution the population mean is not very descriptive.”
Only because you never understand the reasons for wanting to know the mean or any other statistic.
“This stems from you viewing *all* probability distributions as Gaussian.”
Keep on lying.
I mean, I’ve literally just given you an example involving 6 sided dice, that do not have a Gaussian distribution.
“Which is my point. Kip’s theory, that you can ignore all probability and just use the full range as a meaningful measure of uncertainty, clearly doesn’t work for 200 dice.”
That is EXACTLY what the uncertainty interval is! You don’t ignore probability – there just isn’t ONE!
You do this so you can ignore the measurement uncertainty by assuming it cancels – even though you claim you don’t!
The uncertainty interval for 200 dice *IS* larger than for 2 dice and that *is* a meaningful number for uncertainty!
“Which is why you don’t use it to tell you about the population distribution. It’s not what it’s purpose is.”
Then why do you use it to determine the measurement uncertainty of the average when it is the population distribution and the associated measurement uncertainty that determines the measurement uncertainty of the average?
You say you don’t do this but you wind up doing it EVERY SINGLE TIME!
“(Actually, I think you could use the CLT to learn a lot about the population distribution, but that would require more work than I care to go into at the moment.)”
The CLT can only tell you how close you are to the population mean. There isn’t anything else you can use it for. The CLT and the standard deviation of the sample means won’t tell you if the population mean is from a multi-nodal distribution, from a skewed distribution, a distribution with long tails (kurtosis), etc.
“Only because you never understand the reasons for wanting to know the mean or any other statistic.”
There is a REASON why every single textbook in the world will tell you that mean and standard deviation are not applicable statistical descriptors for a skewed distribution.
From “The Active Practice of Statistics” by David Moore:
“The five-number summary is usually better than the mean and standard deviation for describing a skewed distribution or a distribution with strong outliers.” Use y-bar and s only for reasonably symmetric distributions that are free of outliers.”
If the distribution of temperatures along a longitude line from the south pole to the north pole is not Gaussian then why is the mean used as a good statistical descriptor? The distribution will be skewed because the south pole is much colder than the north pole. Do you have even the faintest of inklings as to why that is?
“The uncertainty interval for 200 dice *IS* larger than for 2 dice and that *is* a meaningful number for uncertainty!”
Then demonstrate how you would use it. I have a closed box with 200 dice, and I’ll never know the score. But I “measure” the sum by calculating the expected value is 200 * 3.5 = 700. And I calculate the the uncertainty as being ± 200 * 2.5 = ± 500. So I have a measurement of 700 ± 500. What does that practically tell me? Does it tell me the true value is as likely to be 200 as 700?
“Then why do you use it to determine the measurement uncertainty of the average when it is the population distribution and the associated measurement uncertainty that determines the measurement uncertainty of the average?”
Because I don’t think the uncertainty of the average is the distribution of the population. There are scenarios where that might be more important information, and as discussed many times prior there can be times where it is misleading to simply quote a mean with a ± without making it clear if that ± refers to the population or the mean. But if I want to know, say the global temperature mean, I’m only interested in the uncertainty of that mean, not of the population. That’s because the main purpose of the mean in most statistical tests, is to determine if it differs from different population or is changing over time. It’s the uncertainty of the mean that matters in that case, not the range of values used to make up the mean.
“The CLT can only tell you how close you are to the population mean.”
But there can more than one mean.
Say you are trying to determine if a die is fair or not. One approach would be to roll it a number of times and look at the mean of all your throws. If this was significantly different than the expected 3.5 you could conclude the die was not fair.
But the converse isn’t necessarily true. The die could have an average score of 3.5 but still be biased in other ways, such the one Kip used in the last essay. One way of testing for that would be to throw the die a large number of times and look at the average number of 6s. If that average was significantly different from the expected 1/6, you know the die is not fair, even though the average is 3.5.
In both cases you use the CLT to determine the significance of the result.
“There is a REASON why every single textbook in the world will tell you that mean and standard deviation are not applicable statistical descriptors for a skewed distribution. ”
Most books I’ve seen explain how to determine the standard deviation for different distributions, including skewed ones. Take a Poisson distribution, standard deviation equals sqrt(mean). Why do you think people want to know that if it’s not a useful result.
mean and standard deviation aren’t so useful if all you want to do is describe the distribution, but that’s not what standard deviation is being used for here.
The same old lies pop out again from da bellcurveman.
From the GUM:
2.3.1
standard uncertainty
uncertainty of the result of a measurement expressed as a standard deviation
2.3.5
expanded uncertainty
quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand
As usual, you are cherry picking with no understanding of what you are talking about.
Kip, I think that people’s views about uncertainty, probability, and statistics depends to some degree on educational/professional background. I would love to learn the path you took through life that brought you to your current views. For my part I began my intellectual life (post B.Sc.) as a physicist, but I had learned almost no probability at this point, very little statistics, and had extremely rudimentary views of uncertainty.
Where I finally obtained mature views about this topic is through engineering, especially metrology and manufacturing; and I am still learning about this topic which is why I plan to fetch Matt Briggs book on uncertainty to see what I might learn there. Let me examine just a couple of points because I don’t want to get into the postion of being squashed over making a comment look like an essay.
It would be a lousy approach if this is what we do, but in metrology we would reword this as “we can provide a best value for some parameter, but it has associated uncertainty”. In most cases the single number is useless without the uncertainty.
Now, this uncertainty value we supply doesn’t cover all possible cases (the fundamental issue of your dice example). There is further uncertainty which we attempt to handle more fully with a coverage factor.
I won’t do more at this point than point to the Guide to Uncertainty in Measurements (GUM). However, an additional problem in your examples of absolute uncertainty also involve bias, especially unrecognized bias, in measurements. This comes up in the efforts to improve absolute values for universal constants. In this regard Henron and Fischoff (Am J Phys, 54, 1989) found that the physicist/metrologists were very poor at imagining potential bias in their schemes. This led to various stated best values of universal constants where the quoted uncertainty made for a not credible best value when compared to the efforts of others.
Kevin ==> Thank you for your interesting comment. You are right, one’s view on and understanding/misunderstanding of “uncertainty” is anchored in their educational and professional experience. A fully indoctrinated statistician finds it nearly impossible to understand absolute measurement uncertainty — because it isn’t a probability — it is absolute.
Yes, do get Briggs’ book — pricey but worth it. He is a pragmatist’s statistician.
If you haven’t read it, my series on The Laws of Averages covers most of my view on this issue and uses some climate examples.
Did you catch my other essays on uncertainty?
stated value +/- uncertainty.
I learned about this in my first electrical engineering lab. We only got to use 10% components and no one could get the same answers.
Yes, interestingly, even my graduate physics courses never touched uncertainty. Even the statistics classes usually dealt with exact numbers, often integers, with little regard for measurement error. I was first introduced to uncertainty, significant figures, and rounding off in undergraduate inorganic chemistry, and then later in a land surveying class that B.S. Geology majors were required to take. While the undergraduate calculus series usually devoted a chapter in integral calculus to error, after that, it was never mentioned again. The unstated assumption was that all numbers were exact.
That was back in the day when most calculations were done with a slide rule, and one was doing good to get three reliable significant figures. Often, the measurement device provided more significant figures than the slide rule could handle, so the uncertainty was lost in the calculations. Unfortunately, that blind side has survived to today.
Indeed, chemists do a much better job at teaching uncertainty than the physicists do, or did. The physics curriculum has not changed much since I took it 50+ years ago. Analytical chemistry would be just about pointless without an estimate of precision or uncertainty.
BS geologists don’t get that much statistical training. Or the math required for even Eng. Stat. 101. Even at Mines or Polytech schools. OTOH, all engineering majors get exposed in about year 2. Petroleum engineers usually take a second course, since the oil and gas biz is chancy in every respect. Petroleum engineers also actually use what they’ve learned.
Kip:
This example from the NIST fundamental physical constants database gives the value of the electron mass:
https://physics.nist.gov/cgi-bin/cuu/Value?me|search_for=abbr_in! ,
along with the uncertainty of the value***. Notice there is no mention of any probability distribution associated with the uncertainty.
To assume there is some kind of distribution attached is simply wrong (to quote the mosh), all the interval tells you is that the true value of m_e is expected to lie somewhere within:
(9.1093837015 ± 0.0000000028) x 10-31 kg
(Note that m_e is a measured constant, which differs from other fundamental constants that have exact values with zero uncertainty. The electron charge e is an example of one of these.)
***NIST uses the term “standard uncertainty”, which is a bit off from the JCGM GUM terminology. Because there is no mention of a coverage factor, I would assume these are not expanded uncertainties.
Yes, the dice example is apparently using a coverage factor of one (1.0) which is incapable apparently of reaching all the important parts of the distribution. An expanded coverage is warranted. This is what I was heading toward in my comment. Thank you for this comment.
Thanks, Kevin.
Kevin ==> There is no need or application of coverage factor in this simple physical example of dice. The distribution is a range of All Possible values. It certainly meets all the “important parts” of the real world distribution.
I know that there are a lot of rather strange creatures living in the jungles of statistical approaches to uncertainty — but none of them are needed in my examples.
But by all means use them if it helps you to understand these excruciatingly simple facts about absolute measurement uncertainty.
Even an “expanded” coverage won’t reach *all* values.
True, but one sufficiently large will reach enough for any particular purpose — especially to demonstrate that a specific estimate may not be fit for purpose.
I should also note that the tiny uncertainty of m_e is not the result of NIST averaging 100 billion electron mass measurements; rather it reflects that NIST is very, very good at what they do, using the best laboratory equipment possible.
“Notice there is no mention of any probability distribution associated with the uncertainty.”
It states that it’s the standard uncertainty. they define standard uncertainty as
Then they define uncertainty
https://www.physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html
This is why you are called “bellcurveman”.
The fact remains that you cannot assume that you know ANY probability distribution associated with a given uncertainty interval (unless you are using standard climastrology pseudoscience, of course).
You are the only person who has ever called me “bellcurveman”. You seem to think it is some sort of an insult.
It doesn’t matter if there is a normal distribution or some other distribution. If you saying it is a standard uncertainty you are saying there has to be some sort of distribution, and that it’s possible that the error might be greater than the quoted uncertainty. It is not as you claim that the true value is expected to be within the range of the standard uncertainty.
No I think it is highly amusing given that you pound everything into a Gaussian curve.
Where have I done that? Nature tends to produce Gaussian distributions, courtesy of the CLT, but that doesn’t mean I assume all distributions are Gaussian.
Nature doesn’t tend to produce Gaussian distributions courtesy of the CLT. Statistics using the CLT tends to produce Gaussian distributions of sample means around the population mean. That tells you nothing about the distribution of the population. Even skewed populations can produce a Gaussian distribution of sample means around the population mean. That doesn’t imply at all that the distribution itself is Gaussian or that the mean is even useful in describing the population!
My point was, that often in nature random things are often roughly normal in distribution.Things are more likely to be close to the average height, weight etc, and fewer are are at the extremes. That was where the idea for the normal distribution came from in the first place.
I would guess that the reason so many populations tend towards the normal is because of the CLT. There are thousands of possible causes that will effect somethings value, but they tend to cancel out leading to values that are closer to the mean.
Of course there are many other natural distributions in nature, hence why I said “tends”.
“I would guess that the reason so many populations tend towards the normal is because of the CLT”
You are STILL confusing the distribution of sample means with the probability distribution of the population.
THEY ARE DIFFERENT THINGS. The CLT does *not* guarantee anything about the population distribution.
“There are thousands of possible causes that will effect somethings value, but they tend to cancel out leading to values that are closer to the mean.”
Malarky! Are temperatures from the equator to the north pole a Gaussian distribution? Do the temps at the equator cancel out temps at the north pole?
Are temps from the south pole to the north pole a Gaussian distribution? Do temps at the south pole cancel out temps at the north pole?
Here is what the UAH has for this question:
This just isn’t true. Read this site.
https://aichapters.com/types-of-statistical-distribution/#
The CLT does not cause populations to tend towards normal. That is a totally bizarre interpretation of the CLT.
The CLT under the right assumptions will have sample means from any distribution to converge to a normal distribution. With sufficiently large sample size and a sufficient number of samples, the mean of the sample means will be an estimate of the population mean. The standard deviation of the sample means multiplied by the sqrt of the sample size will provide an estimate of the population Standard Deviation.
The more the standard deviations of the individual samples vary, the less accurate the estimates for the population become.
Here is the problem. Knowing the population mean and standard deviation will not show the actual shape of the population distribution. In essence the CLT will not let you estimate (infer) the kurtosis and skewness.
“This just isn’t true. Read this site”
Why do you keep insisting I read random sites of the internet, which describe basic statistical facts in basic detail, without explaining what you want point you think it’s making?
How does any of that site justify your claim that what I said “just isn’t true”? What part are you saying isn’t true?
And this is all becoming a massive distraction for something that was just a casual aside.
Why do you need to read these sites? Because they contradict your assertions!
bellman: “I would guess that the reason so many populations tend towards the normal is because of the CLT”
The CLT has nothing to do with population value distributions, only with the tendency for sample means to cluster around the population mean.
If you can’t get the simple things right then you can’t get the more complex ones right either.
“Why do you need to read these sites? Because they contradict your assertions!”
But they almost never do. It’s just you haven’t understood them. Where in
https://aichapters.com/types-of-statistical-distribution/#
does it refute my assertion?
“The CLT has nothing to do with population value distributions, only with the tendency for sample means to cluster around the population mean.”
(Memo stop trying to add interesting asides in my comments, it only allows people use them for distraction.)
It isn’t hill I’m prepared to die on, but I do think it’s at least possible that the reason populations often tend to be relatively normal is connected to the CLT. Individuals in populations can be made up of multiple variables and the sum of multiple variables will tend towards a normal distribution. I could be completely wrong, it’s not an important point, and I’ll leave it there.
I use references so you know that what I say is not just opinion. If you can’t read a reference and learn from it, then that explains a lot.
The CLT is not used in nature. It is used by people to develop a normal distribution by using statistical calculations.
My point is they are not telling me anything I didn’t know already. If you want to use a reference to reinforce your point, please quote the relevant part rather than expecting me to guess which part you think supports your case.
This assumption is incorrect. Nature produces lots of log-normal distributions.
https://academic.oup.com/bioscience/article/51/5/341/243981
If you take two sets of 100 random numbers and add the together, you get a normal distribution. If you multiply them you get a log-normal distribution, all lumped up on the left with a long tail to the right.
Assuming a normal distribution, then taking the standard deviation, ignores the tail.
“Nature produces lots of log-normal distributions.”
At least in the world of subsurface rheology and geology they do. But we just evaluate the log values (or the even more complicated relations such as that between porosity and permeability) as normal, and then transform them when done. I’m guessing that other “natural” disciplines do so as well.
It’s the only way he can justify ignoring measurement uncertainty so that he can use the standard deviation of the stated values as the uncertainty.
Will you ever stop lying about me.
No one is lying about you. If you don’t like your quotes being thrown back at you then stop making assertions that are indefensible.
Exactly right.
You’re right it should be bellcurvedballman
Just accept you are never going to do better than Bellend. I chose my pseudonym expecting someone would use that.
“If you saying it is a standard uncertainty you are saying there has to be some sort of distribution”
You are *NOT* saying that at all!
What do you think your quote above is actually saying:
“This implies that it is believed with an approximate level of confidence of 68 % that Y is greater than or equal to y – u(y), and is less than or equal to y + u(y), which is commonly written as Y= y ± u(y).”
It does *NOT* say anything about where in the interval the probability of Y is the greatest! If you can’t say that then how can you have a probability distribution?
“It does *NOT* say anything about where in the interval the probability of Y is the greatest!”
Firstly, as it is talking about a normal distribution in that example, it is saying you believe Y is more likely to be closer to the measured value, than towards the edges of the standard uncertainty range.
Secondly, It doesn’t matter what the distribution is, I am simply pointing out that there has to be one, whether you know what it is or not, based on the fact that it is claimed there is a standard uncertainty and that implies you know what the standard deviation is, and you can’t have a standard deviation without a distribution.
See above.
And what is the probability distribution of an error band spec for a digital voltmeter?
See page 26 of this industry treatment of just this, to see his referenced probability distribution.
https://download.flukecal.com/pub/literature/webinar-uncertainty-presentation-Dec%202011.pdf
Look at page 25 of this document.
u_c = sqrt[ u1^2 + u2^2 + … + un^2]
No averaging.
Page 30. multiple measurements of the same UUT
Page 35. create a distribution from the scatter of the stated values.
Page 49. We see once again u_c = sqrt[ u1^2 + u2^2 + … + un^2]
Again – NO AVERAGING
Page 58. Increasing the number of measurements has a diminishing effect on Um, the expanded uncertainty.
Meaning that increasing the number of observations doesn’t decrease the uncertainty through the division by the number of observations.
——————————
Did you *really* think you were going to fool anyone into believing the claim by the statisticians and climate scientists on here that you can decrease uncertainty through averaging more observation? That the standard deviation of the sample means is the true uncertainty of the mean?
Again, you and Tim Gorman seem to be competing to see who can best respond to what I didn’t say. My response was to your specious:
“And what is the probability distribution of an error band spec for a digital voltmeter?”
I showed you an industry example of exactly that. Time to deflect once more?
Maybe you and Mr. Gorman can rise out of your respective lairs and get a little fresh air and sunshine during the day.
From the GUM, blob:
Standard uncertainty does NOT tell you a probability distribution!
Apparently Fluke thought that they had this “extensive knowledge of the probability distribution characterized by the measurement result y and its combined standard uncertainty uc(y)”. After all, they’re in the business, as a highly reputable corporation.
And even if they don’t have the “exactly known levels of confidence” currently only available from The Imaginary Guy In The Sky, the distribution they arrived at – you know, the one that you inferred didn’t exist – would be quite usable commercially.
What happens 3, 6, 12 + months after calibration, blob?
A recalibration, based on use and/or Fluke experience? I don’t really know. I don’t know your point either. Please expand.
RUOK? I was serous about sunshine and exercise. If they won’t let you do it, contact your guardian.
You’re just another clown show, blob.
Without even reading that Fluke link I can tell exactly what you didn’t understand—Fluke was no doubt giving the behaviour of the A-D conversion, which can be studied and documented in gory statistical detail.
But this is not the only element of voltmeter uncertainty, there are others! Including:
Temperature
Voltmeter range
Input voltage
And another big one—calibration drift. This is why I asked about months, the DVM uncertainty grows the farther you get from the last calibration. But since you have zero real experience with digital lab equipment, the clue went zooming right over your head.
You cannot average your way around these.
Lesson ends.
You spaced on the section of Type B uncertainties. All of the sources that you claim, and others that you missed, were included. Yes, they were bundled, and engineering judgment was used.. But Fluke, unlike you, is a commercial enterprise, and engineers it’s uncertainty calculations.
Bigger pic, just more deflection from your faux inference that digital voltmeter manufacturers do not derive error distributions for their products. I showed you a picture of one.
Deny reality, good job, blob.
Please continue in your delusions.
“engineering judgment was used”
In other words UNCERTAINTY intrudes!
There was no implying that digital voltmeter manufacturers don’t deliver error distributions for their products. The implication is that the error distribution will change over time. The manufacturer has no control over the environment the unit is used in or in how it is treated. Thus the calibration and the error distribution can change over time.
Why do you fight so hard to deny that simple fact?
He did a web search and found a Fluke spec PDF and proceeded to skim it for loopholes through the lens of his a priori assumption that a standard uncertainty gives/implies a probability distribution, without understanding what he was reading.
I confronted him with the facts that end-use conditions greatly affects the uncertainty of DMM measurements, and he went into an incoherent rant about how Fluke engineers know more than I in the post you replied to.
Jim even went through the document and showed what the guy missed while skimming. He had no coherent answer here as well.
This is religion, not science & engineering.
The number of so-called statisticians on here trying to justify the “global average temperature” that can’t even recognize a standard bi-modal distribution is just amazing. And sad.
The point? The point is that measurements in the field are not made in a manufacturers lab. What the manufacturer puts down for the instrument calibration and uncertainty NEVER survives the field.
If it *did*, then why would you ever need a recalibration?
I agree with everything you said. Please show me where I ever said otherwise.
Your whole argument against the assertions was based on the Fluke engineers and their measurement lab.
And now you are trying to save face rather than just admit that your rebuttal was useless.
Yep, a lame backpedal attempt.
As both Willis and bdgwx say, AGAIN, please don’t respond to what I didn’t say – or didn’t infer.
What was my assertion? Answer: that digital voltmeters have distributed error. They do, as the Fluke engineers showed us. That’s all.
BTW, 3rd attempt to get you off the dime on Pat Frank’s admission that you don’t know how to find the uncertainty of averages. Yes, he’s pretty well been laughed out of superterranea w.r.t. uncertainty determination, but he’s about your last hope.
Another noisy whiner, ask me if I care what you think.
And you deftly danced around and avoided the actual question I asked, blob, which is to tell your vast listening audience what the probability distribution is for any given DVM uncertainty interval.
You then did a mindless web search and ended up at a Fluke spec sheet, which you didn’t even bother read beyond a brief skimming, and cherry-picked something you thought was an answer.
Fail.
Try again, blob!
You can do it!
He won’t. He’s back-pedaling so fast he’s going to wind up on his butt. Actually he already has.
Pat Frank has answered every single criticism and never been rebutted.
Uncertainty in initial conditions compounds throughout iterative processes. It might be an inconvenient truth for you and many to accept but it is the truth nonetheless. Anyone that has ever been on a motorcycle that goes into tank-slapper mode can tell you all about that!
“Pat Frank has answered every single criticism and never been rebutted.”
Well, no. Pretty much 180 out from reality.
But the point is that you’re still shuckin’ and jivin’ on the fact that even Dr. Frank, as isolated from the world as he is, has no truck with your out and out faux assertions on how average and trend uncertainty can not be reduced by more data. You are smart enough to realize this, which is why you deflect from addressing it.
Where is YOUR air temperature uncertainty analysis, blob. I don’t see it.
You don’t even understand what you are talking about.
Calculating the population mean from sample means gets you closer and closer to the population mean as you increase the size of the samples.
THAT ISN’T THE UNCERTAINTY OF THE MEAN. The average value of a set of measurements with uncertainty can be significantly inaccurate, even if you have the entire population!
You wouldn’t last a day in any machine shop that I’ve worked in. The answer to the boss “I took a lot of measurements and averaged them to find out how close to accurate the product is” would get you shown the door.
TG: “Pat Frank has answered every single criticism and never been rebutted.”
He refuses to tell me where he got the formula u(Y) = sqrt[N*u(x)/(N-1)] when Y = Σ[x_i, 1, N] / N and u(x) = u(x_i) for all x_i. Instead he responds with arrogant hubris and ad-hominems.
Tell me again how the average uncertainty is the uncertainty of the average and that is why Frank is wrong!
The issue is between you and Dr. Frank. He is the one who admitted that you and others were willfully FOS. But to bone throw, Dr. Frank is also radio silent after being called out by Bellman and admitting that Bellman was correct (a first I believe). The worminess of both of you channels Profiles In Courage Kevin McCarthy calling out T**** one day and sneaking off to goober smooch him soon after.
You’re hope free. You won’t even admit the algebra errors that are assiduously demonstrated to you, step by step. Thank The Imaginary Guy In The Sky that you have extremely limited cred here, and none elsewhere.
blob felt the need to put DJT into his latest insane word salad—again.
TDS is never a pretty picture.
And FTR, it took Pat only about 3 posts to see right through bellcurvewhinerman’s act and tossed him into the looney bin.
You followed the exchange until it met your prejudgments. Not to the point where Dr. Frank admitted his mistake. Here are his words, “My mistake”. I’m looking forward to your whining around that.
And the channel I provided was factual. Which is why you jst whine “TDS” without addressing it.
Like the good and proper sophist that you are, you threw the context into the rubbish (assuming you even understood the context to begin with).
YOU are the clown who inserts President Trump into each an every word salad rant, not I, clown.
Are the Fluke engineers psychic and know the use conditions ahead of time of all the instruments they manufacture? Must be…
That multiple measurements of the same thing with the same device is needed!
“Firstly, as it is talking about a normal distribution in that example, it is saying you believe Y is more likely to be closer to the measured value, than towards the edges of the standard uncertainty range.”
That is *NOT* what it says.
It plainly says:
Y is greater than or equal to y – u(y)
Y is less than or equal to y + u(y)
No where in that is it implied that the uncertainty is a normal distribution!
“I am simply pointing out that there has to be one”
There does *NOT* have to be one. It doesn’t have to be Gaussian. It doesn’t have to be uniform. It doesn’t have to be anything other than the true value has a probability of 1 of being the true value (by definition!) and all the rest have a probability of 0 of being the true value. What kind of a distribution is that?
“t is claimed there is a standard uncertainty and that implies you know what the standard deviation is”
No! You’ve been given quote after quote and example after example that shows that standard uncertainty is *NOT* a probability distribution for measurements of different things.
You keep mixing up the standard deviation of the stated values with the uncertainty of those stated values. The standard deviation of the stated values is the standard uncertainty ONLY when 1. you are using multiple measurements of the same thing, 2. you can determine that no systematic bias exists, and 3. the measurement uncertainty cancels (i.e. it forms a Gaussian distribution). You can’t even just *assume* that 3. applies when you have multiple measurements of the same thing. You also need to show that each measurement was taken under the same environmental conditions. You have to *prove* that 3. applies.
Your assumption that everything is a Gaussian distribution is just not supportable. Yet it is what you fall back on every single time.
“That is *NOT* what it says.
It plainly says:
Y is greater than or equal to y – u(y)
Y is less than or equal to y + u(y)
No where in that is it implied that the uncertainty is a normal distribution!”
What bit of, “if the probability distribution … is approximately normal” don’t you understand?
Here’s the full quote again with the key words emphasized.
If the distribution normal, then all the talk about the 68% confidence level is irrelevant.
“There does *NOT* have to be one.”
Please learn some statistics. If there is a standard deviation there has to be a distribution. If there isn’t a distribution (and I’m not sure what that would even mean) how can there be a standard deviation. Again, you might not know what that distribution actually is, but there has to be one.
“It doesn’t have to be anything other than the true value has a probability of 1 of being the true value (by definition!) and all the rest have a probability of 0 of being the true value.”
That’s still a probability distribution.
But it isn’t what we are talking about here. It isn’t talking about the actual value of Y, it’s talking about the “distribution of values that could reasonably be attributed to the value of the quantity Y”.
“No! You’ve been given quote after quote and example after example that shows that standard uncertainty is *NOT* a probability distribution for measurements of different things. ”
No I haven’t. You just keep asserting it then claiming you’ve given me quote after quote.
IF is the operative word here. If means you need to prove the probability distribution is Gaussian. It also means you need multiple measurements of the same thing in order to derive the probability distribution.
None of this is applicable because you don’t have repeated measurements of the same thing.
Yes, that entire section is saying if the distribution is normal you can assume there is a 68% chance that Y lies within 1 SD of the measurement value.
It does not mean you have to prove it’s Gaussian, it can just be an assumption, nor does it mean that if the distribution is not Gaussian you do not have a distribution.
In a skewed distribution are 68% of the values within 1 standard deviation of the mean?
“if the distribution is normal”
“It does not mean you have to prove it’s Gaussian”
cognitive dissonance at its finest. First it has to be normal then it doesn’t matter.
Probably not. Hence the use of the phrase “if it is normal”.
“Probably not. Hence the use of the phrase “if it is normal”.”
ROFL!!! So once again we circle back to your assumption that all measurement uncertainty is Gaussian and cancels.
How would you KNOW if the distribution is skewed or not? How do you KNOW the probability value for each value in the interval?
*YOU* just assume that the stated value is the mean of the Gaussian distribution of uncertainty and therefore the stated value is always the true value!
You keep claiming you don’t do this but it shows up in what you post EVERY SINGLE TIME!
This is just getting sad.
TG: “In a skewed distribution are 68% of the values within 1 standard deviation of the mean?”
BM: “Probably not. Hence the use of the phrase “if it is normal”.”
TG: “So once again we circle back to your assumption that all measurement uncertainty is Gaussian and cancels.”
You really cannot see the words I write without passing them through your “you think everything is Gausian” filter.
“You really cannot see the words I write without passing them through your “you think everything is Gausian” filter.”
When you say that values in the uncertainty interval that are further away from the stated value have a smaller probability of being the true value YOU ARE STATING THAT THE PROBABILITY DISTRIBUTION OF THE UNCERTAINTY INTERVAL IS GAUSSIAN.
You can run but you can’t hide. You *always* circle back to believing that uncertainty is Gaussian and cancels, EVERY SINGLE TIME.
You simply can’t break out of that meme.
“When you say that values in the uncertainty interval that are further away from the stated value have a smaller probability of being the true value YOU ARE STATING THAT THE PROBABILITY DISTRIBUTION OF THE UNCERTAINTY INTERVAL IS GAUSSIAN.“
You keep conflating numerous things I may or may not have said, without providing any context.
“You simply can’t break out of that meme.”
You keep repeating the same phrases regardless of what I say, such as
yet think I’m the one locked in a meme.
“You keep conflating numerous things I may or may not have said, without providing any context.”
I didn’t figure you would actually address the issue.
So now we are going to deflect eh? A uniform distribution still doesn’t recognize the existence of systematic bias. It is a symmetric distribution around a mean where complete cancellation can be assumed.
Again, a triangular distribution is a symmetric distribution around a mean so you can assume complete cancellation.
You *ASSume* this so you can play like all measurement uncertainty cancels and you can use the stated values as the true value of the measurement. You can then use statistical analysis on the stated values to get a mean and standard deviation.
Once again, the CLT only gives you a normal distribution for the sample means. It says nothing about the uncertainty of the population mean.
Once again, Taylor is talking about the stated values, not the measurement uncertainty. Why can’t you get this straight?
Unfrackingbelivable.
For anyone still reading – Tim, has spent hundreds of comments in this thread alone, saying that only Gaussian distributions cancel, only Gaussian distributions can be added in quadrature, Taylor insists that all his uncertainties are Gaussian, and attacking me for “ASSuming” all distributions are Gaussian. And now, no, it turns out he didn’t mean Gaussian at all, he just meant symmetrical, and I’m the one deflecting because I didn’t guess what he really meant.
e.g.
“Again, a triangular distribution is a symmetric distribution around a mean so you can assume complete cancellation.”
Pivot from “you can only assume uncertainties cancel when you have a Gaussian distribution” to, “you can assume complete cancellation if you have a triangular distribution”.
But he’s still wrong. Distributions do not have to be symmetrical to cancel or be added in quadrature. What he really, really means is the distribution has to have a mean of zero.
I know I’ll regret this, but you tripped over this in your GUM cherry picking to find the answer you desire (which is pseudoscience, BTW):
You CANNOT derive/assume/imply any probability distribution from a GUM standard uncertainty!
As I said elsewhere, I am not saying you can derive a probability distribution from a standard deviation. My objection was to people saying there was no probability distribution. My point is there has to be some distribution even if you don’t know what it i, and that that somehow meant you couldn’t apply the usual rules for propagating independent uncertainties.
Equation 10 does not require you know the probability distribution of any of the uncertainties, just the standard uncertainty.
The point about different distributions, isn’t about how you propagate the uncertainty, it’s about what you can say regarding confidence intervals.
Of course, if you are adding or averaging a large number of values the CLT implies the combined uncertainty will be close to a normal distribution.
I knew I would regret it…back to the real issue:
—TG
I’m still asking about sums not averages. Just keep deflecting.
Sorry, hit my limit for wading in the mudflats of trendology…
Yes, that’s the reason you won’t answer.
Seriously, if you don’t want people to see how much of an effort you are making to avoid answering the question it would much better for you just to keep quite.
Don’t care what you think or see, trendologist.
And “keep quite” yerself hypocrite.
“As I said elsewhere, I am not saying you can derive a probability distribution from a standard deviation. My objection was to people saying there was no probability distribution. My point is there has to be some distribution even if you don’t know what it i, and that that somehow meant you couldn’t apply the usual rules for propagating independent uncertainties.”
Are you omnipotent? How do you KNOW there has to be some distribution?
I’ll ask again. What is the probability distribution where one value has a probability of 1 and all the rest have a probability of 0?
You keep making claims about not ignoring measurement uncertainty and then you circle right back around to saying that there has to be a probability distribution that allows you to ignore it – EVERY SINGLE TIME.
“How do you KNOW there has to be some distribution?”
Because there’s a standard deviation.
“I’ll ask again. What is the probability distribution where one value has a probability of 1 and all the rest have a probability of 0?”
It’s the probability distribution you’ve just described. 1 at one specific value 0 elsewhere.
“You keep making claims about not ignoring measurement uncertainty and then you circle right back around to saying that there has to be a probability distribution that allows you to ignore it”
What makes you think saying there is a probability distribution means I’m ignoring uncertainty? It’s the uncertainty that means there is a probability distribution, or possibly the other way round. If the distribution was the 1 and 0 you describe above there would be no uncertainty.
There will be a probability distribution, but we don’t know what it is.
In a lot of cases, we can’t know what it is, or was.
In other cases, we can know, but it doesn’t matter enough to find out, hence the use of tolerances.
“Equation 10 does not require you know the probability distribution of any of the uncertainties, just the standard uncertainty.”
Equation 10 ASSUMES a Gaussian distribution. If you don’t have a Gaussian distribution then RSS simply doesn’t work! It might give you a LOWER BOUND, but the actual uncertainty will undoubtedly be higher than that!
TG said: “Equation 10 ASSUMES a Gaussian distribution.”
Patently False. There is nothing in equation 10 or the more general law of propagation of uncertainty E.3 that mandates normality. In fact, the GUM even states there is no assumption of normality implied by E.3.
TG said: “ If you don’t have a Gaussian distribution then RSS simply doesn’t work!”
Due to the above this obviously isn’t true either.
It should be noted that the NIST uncertainty machine happily accepts any distribution even to the point of allowing users to upload a custom distribution.
“Due to the above this obviously isn’t true either.”
According to Taylor and Bevington it *is* true.
Are you saying they are liars or just ignorant?
You can keep asserting this all you want. I’ve already shown you why this isn’t correct. And you have not produced any evidence that GUM assumes anything of the sort.
“but the actual uncertainty will undoubtedly be higher than that!”
Then you should be able to produce a proof or a demonstration.
Taylor: “Chapter 5 discusses the normal, or Gauss, distribution, which describes measurements subject to random uncertainties.”
“If the measurements of x and y are independent and subject only to random measurement uncertainties, then the uncertainty ẟq in the calculated value of q = x + y is the sum in quadrature or quadratic sum of the uncertainties ẟx and ẟy.”
Bevington: “There are several derivations of the Gaussian distribution from first principles, none of them as convincing as the fact that the distribution is reasonable, that it has a fairly simple analytic form, and that it is accepted by convention and experimentation to be the most likely distribution for most experiments. In addition, it has the satisfying characteristic that the most probable estimate of the mean u from a random sample of observations x is the average of those observations x_bar”.
“In Chapter 2 we defined the mean u of the parent distribution and noted that the most probable estimate of the mean u of a random set of observations is the average x_bar of the observations. The justification for that statement is based on teh assumption that the measurements are distributed according to the Gaussian distribution.”
The GUM is *no* different. In order to do statistical analysis no systematic bias must exist, otherwise you won’t get a Gaussian distribution.
“B.2.15
repeatability (of results of measurements)
closeness of the agreement between the results of successive measurements of the same measurand carried out under the same conditions of measurement” (bolding mine, tpg)
Multiple measurements of the same thing! Those measurements are assumed to form a Gaussian distribution that can be analyzed to form a standard deviation and used as a measure of uncertainty. The measurement uncertainties of each observation is assumed to be cancelled leaving only the stated values.
Exactly how many examples in the GUM do you find where a single measurement is given as “stated value +/- uncertainty”?
The use of repeated measurements to obtain observations suitable for statistical analysis is ubiquitous in the GUM.
G.1.2 In most practical measurement situations, the calculation of intervals having specified levels ofconfidence — indeed, the estimation of most individual uncertainty components in such situations — is at best only approximate. Even the experimental standard deviation of the mean of as many as 30 repeated
observations of a quantity described by a normal distribution has itself an uncertainty of about 13 percent” (bolding mine, tpg)
You can whine about it all you want, it won’t change the truth. When Bevington says that measurements with systematic bias are not amenable to statistical analysis he is *not* lying. Your continued attempt to prove he is lying is just laughable.
Do you ever actually read any of these quotes for meaning rather than just seeing some random words you think prove your point?
None of these claim that you need a distribution to be Gaussian to add in quadrature.
Lets go through them:
No mention of Gaussian.
Says that a Gaussian distribution is convincing and is the most likely distribution. It does not say a distribution needs to be Gaussian to add in quadrature.
Is talking about maximum likelihood, not propagation of errors. That’s explained in Chapter 3, leading to equation 3.14, the standard formula for propagating random independent errors. Nothing in the derivation of it uses the assumption of normality, just as in Taylor’s derivation, where he specifically says the formula does not depend on the distribution.
“The GUM is *no* different. In order to do statistical analysis no systematic bias must exist, otherwise you won’t get a Gaussian distribution.”
You are really getting confused here. You can have a systematic bias in a Gaussian uncertainty distribution, it’s just going to change the mean. Saying no systematic bias must exist is not saying the distribution must be Gaussian.
Says nothing about distributions, or propagating of uncertainty. It’s just defining what repeatability means.
“Multiple measurements of the same thing! Those measurements are assumed to form a Gaussian distribution that can be analyzed to form a standard deviation and used as a measure of uncertainty”
That’s your assumption. As I said before and was shot down, it may be reasonable to assume measurements are likely to form a Gaussian distribution, but that isn’t a requirement.
“You can whine about it all you want, it won’t change the truth.”
I know, because the truth is determined by the maths and confirmed by experiment. Even if you could find some odd quote that seemed to deny the maths, it wouldn’t be proof anything. If you seriously belief that adding in quadrature doesn’t work unless the uncertainty distributions are all Gaussian, then ever find some proof of that, or demonstrate it.
Others have thought it, thinking you were attempting to give some credence to your claims, such as with “bigoilbob.”
The evidence that not once do statisticians or climate scientists (including bellman) ever use a 5-number statistical description of temperature data let alone kurtosis or skewness numbers.
It’s always Gaussian and mean. Not even standard deviation.
What is the shape of the temperature distribution along a longitude line from the south pole to the north pole at any instantaneous time?
My guess is that none of the CAGW adherents here have the slightest idea.
Nor even the number of points in the average!
“Degrees of freedom? BAH! Throw that stuff away.”
Refers to me, but won’t respond to my corrections of faux statements. Apparently, I’m getting promoted into Nick Stokes left handed compliment territory. He is referenced in as many posts in which he does not contribute as in those where he does.
BTW, do you still think that wind power is a quadratic function?
Poor blob, doesn’t get the respect he demands that others give him.
“Meaning of uncertainty
If the probability distribution characterized by the measurement result y and its standard uncertainty u(y) is approximately normal (Gaussian), and u(y) is a reliable estimate of the standard deviation of y,
”
u(y) will *NOT* have a normal distribution unless it is a standard deviation derived from stated values only.
“This implies that it is believed with an approximate level of confidence of 68 % that Y is greater than or equal to y – u(y), and is less than or equal to y + u(y), which is commonly written as Y= y ± u(y).”
Read this CAREFULLY, for real meaning! It does *NOT* say that the uncertainty interval is a normal distribution of possible values. It doesn’t say that the uncertainty interval is a uniform distribution of possible values.
It just says that the true value of y lies somewhere in the interval.
Stating that the uncertainty interval is a probability distribution means you have some expectation of what the actual true value is. Value 1 has a 50% probability of being the true value, Value 2 has a 25% probability of being the true value. Value 3 has a 1% chance of being the true value.
If you know these probabilities then why do you have an UNCERTAINTY interval?
Look up the definition of the word “if”.
If the distribution is normal, then what it says applies, e.g. 68% confidence that Y lies within 1 SD of the estimated mean. If the distribution is not normal you cannot make that claim, but it does not mean that there is not a standard deviation, and hence a standard error.
“It just says that the true value of y lies somewhere in the interval.”
No. It says the true value has a likelihood of lying between these two values.
“Stating that the uncertainty interval is a probability distribution means you have some expectation of what the actual true value is.”
Of course you have an expectation of what the true value is. there wouldn’t be any point in measuring anything if it didn’t lead to an expected value.
“Value 1 has a 50% probability of being the true value, Value 2 has a 25% probability of being the true value. Value 3 has a 1% chance of being the true value.”
That depends on how you are defining probability.
“If you know these probabilities then why do you have an UNCERTAINTY interval?”
Because they are probabilities (or likelihoods), not certainties.
BZZZZZT—still don’t grok UA yet.
“Look up the definition of the word “if”.
If the distribution is normal”
But you can *NOT* assume normal! The distribution of temperatures from the south pole to the north pole is not Gaussian. The distribution of temperatures from the east coast to the central plains is not normal. The distribution of temperatures from the central plains to the west coast is not normal.
See the attached picture of temperatures in Kansas for 1/4/2023 at noon. Are they normally distributed from north to south? East to west?
You *always* want to assume normal whether it is justified or not.
“If the distribution is not normal you cannot make that claim, but it does not mean that there is not a standard deviation, and hence a standard error.”
If you don’t have a normal distribution then what do you think the standard deviation tells you? And the standard deviation of a population is not the same thing as the standard deviation of the sample means, typically known as the standard error.
Once again you are throwing out crap hoping something will stick to the wall.
“But you can *NOT* assume normal!”
I’m not assuming anything, it’s the description from the GUM that is assuming normal. And to be clear assuming something is not believing it to be true, it’s just saying under that particular assumption then these things are true. If the assumption doesn’t hold then the other things may not hold.
“The distribution of temperatures from the south pole to the north pole is not Gaussian.”
You keep loosing the plot. They are not talking about the distribution of values, they are talking about the distribution of the uncertainty.
“You *always* want to assume normal whether it is justified or not.”
Stop these constant strawman attacks. It’s getting really tedious. I do not assume any population is normal. The whole point of the CLT is it doesn’t matter what the distribution of the population is, if the sample size is large enough the sampling distribution will tend to normal. Didn’t you read Kip’s last article on the subject.
“I’m not assuming anything, it’s the description from the GUM that is assuming normal.”
And you have absolutely no understanding of why the GUM assumes that!
IT’S BECAUSE THE GUM ASSUMES MULTIPLE MEASUREMENTS OF THE SAME THING!
And anyone familiar with measurements in the real world will tell you that you can’t even assume that multiple measurements of the same thing provides a Gaussian distribution. You have to *prove* that assumption is valid.
“You keep loosing the plot. They are not talking about the distribution of values, they are talking about the distribution of the uncertainty.”
But you always assume all measurement uncertainty cancels! So you can then use the stated values to determine everything!
Measurement uncertainty has *NO* probability distribution, only stated values do. You simply cannot look at a measurement uncertainty interval and say *this value* is the most likely true value based on the probability distribution of the possible uncertainty interval values. There is no standard deviation of the uncertainty interval values. There is no average of the uncertainty interval values.
It’s Kip’s closed box. You can’t see inside the box no matter how much you would like to! It’s the best description of measurement uncertainty I’ve seen yet.
And you can’t get out of your own way in order to understand that!
“And you have absolutely no understanding of why the GUM assumes that!”
Let me guess, is it because if you assume that you can say that there’s a 68% of chance of Y being within the standard uncertainty range? Am I close? ”
“IT’S BECAUSE THE GUM ASSUMES MULTIPLE MEASUREMENTS OF THE SAME THING!”
I’d say you are wrong, but the fact you wrote it in all caps makes a persuasive point.
I’m not sure if you understand what “assumption” means in this context. The statement you are are quibbling about is simply of the form if X then Y. It’s saying if you can assume the distribution is near normal (X) you can infer specific things about the confidence intervals (Y).
Of course you would be right to say that in many cases it’s probable that the distribution will be close to normal, especially if it’s the result of adding or averaging multiple things. But that’s not the point of that statement.
“But you always assume all measurement uncertainty cancels! ”
How hot are your pants today?
“Measurement uncertainty has *NO* probability distribution…”
So what do you think the standard measurement uncertainty is?
“It’s Kip’s closed box.”
Or your closed mind.
“You can’t see inside the box no matter how much you would like to! It’s the best description of measurement uncertainty I’ve seen yet.”
But even Kip says there’s a probability distribution with the two dice. You know 7s are more likely than 12s.
bellcurveman is running on empty…
“Let me guess, is it because if you assume that you can say that there’s a 68% of chance of Y being within the standard uncertainty range? Am I close? ””
No, you totally missed the whole point, as usual.
“I’d say you are wrong, but the fact you wrote it in all caps makes a persuasive point.”
Which, once again, only shows that you cherry pick from the GUM. You have absolutely no idea what it actually says.
3.1.1 The objective of a measurement (B.2.5) is to determine the value (B.2.2) of the measurand (B.2.9),
that is, the value of the particular quantity (B.2.1, Note 1) to be measured.
Why do you keep reading “MEASURAND” as “MEASURANDS”? Do you need a new set of reading glasses? Or maybe some other assistive reading apparatus?
“I’m not sure if you understand what “assumption” means in this context. “
I know what “assumption” means. Making an assumption carries with it the responsibility to justify the assumption. Just assuming that all measurement uncertainty is normal is not justifying it.
:It’s saying if you can assume the distribution is near normal (X) you can infer specific things about the confidence intervals (Y).”
What is your justification for that assumption? The word “if ” is not a justification.
“How hot are your pants today?”
Mine are fine. Why do you always keep circling around to all measurement uncertainty cancels – which is the purpose of “assuming” it to be Gaussian!
You keep denying you do this but it is built into EVERY SINGLE THING you assert!
“So what do you think the standard measurement uncertainty is?”
I’ve told you what it is. I’ve given you the GUM defintion of what it is. And you just absolutely refuse to read anything that shows you don’t understand what it is.
2.2.4 The definition of uncertainty of measurement given in 2.2.3 is an operational one that focuses on the
measurement result and its evaluated uncertainty. However, it is not inconsistent with other concepts of
uncertainty of measurement, such as
⎯ a measure of the possible error in the estimated value of the measurand as provided by the result of a
measurement;
⎯ an estimate characterizing the range of values within which the true value of a measurand lies (VIM:1984,
definition 3.09).
Although these two traditional concepts are valid as ideals, they focus on unknowable quantities: the “error” of the result of a measurement and the “true value” of the measurand (in contrast to its estimated value), respectively. Nevertheless, whichever concept of uncertainty is adopted, an uncertainty component is always evaluated using the same data and related information. (bolding mine, tpg)
“they focus on unknowable quantities”
What does *THAT* mean to you? If the quantities are unknowable then that implies, to anyone who understands metrology, that you do *NOT* have a probability distribution. Having a probability distribution implies that you know something about the “UNKNOWABLE”!
“Why do you keep reading “MEASURAND” as “MEASURANDS”?”
Give a context for me saying that? There can be one measurand, there can be many. You can add a number of different measurements, each of a different measurand to produce a new measurand. That’s what the combined uncertainty is all about.
“What is your justification for that assumption? The word “if ” is not a justification.”
I can’t help you if you don;t understand basic logic. The justification is the word “if”. I say that if it rains tomorrow I will get wet. You can say that means that under the assumption that it rains I will get wet. I don’t have to prove it will rain tomorrow for that statement to be correct.
“Why do you always keep circling around to all measurement uncertainty cancels – which is the purpose of “assuming” it to be Gaussian!”
I don’t. You are just incapable of hearing anything that disagrees with your world view. How many times do I have to tell you that “cancelling” is never total, and has nothing to do with assuming a Gaussian distribution. Uncertainties cancel whatever the distribution. I don’t know why I bother to keep typing this – you’ll just unsee it as usual.
“I’ve told you what it is. I’ve given you the GUM defintion of what it is.”
You keep giving me the various GUM definitions, but never explain how they work if there is no probability distribution. I’m sure the bolding means something to you, but to me they are just confirming the point.
I suspect you are just misunderstanding what probability distribution we are talking about. It isn’t the probability of what the real value is, given omniscience. It’s either the probability distribution of errors, or it’s the probability of what the measurement may be when you don’t have perfect knowledge.
““they focus on unknowable quantities”
What does *THAT* mean to you?”
It means there has to be a probability distribution.
“If the quantities are unknowable then that implies, to anyone who understands metrology, that you do *NOT* have a probability distribution.”
Then that person would have to explain all the references to probability distributions in the GUM, and then explain what the standard deviation of the uncertainty means if there is no distribution, and why equation 10 is used to justify adding in quadrature.
“Having a probability distribution implies that you know something about the “UNKNOWABLE”!”
You don’t know what the measurand is, that doesn’t mean you know nothing about it. The probability is associated with the measurement. You take a measurement and now you know something about the measurand. You don’t know what it is, but you do have a proability that it lies with a distribution. It’s more likely to be closer to the measurement you’ve just made, than something far away from the measurement. There would be no point in measuring anything if you could say that.
Really, if you measure a board at it says it’s 6′ long, are you saying you still can say nothing about the actual length of the board?
“””””Give a context for me saying that? There can be one measurand, there can be many. You can add a number of different measurements, each of a different measurand to produce a new measurand. That’s what the combined uncertainty is all about.””””””
Go back and read again. There is only ONE measurand. It may take several physical measurements of various phenomena that make op the measurand to find the value for it.
Volume of a cube – 3 measurements – length, width, and height (LWH)
Pressure of an ideal gas in a cubical container – 5 measurements – length, width, height, temperature, moles of the gas. (P= nRT/V)
Velocity – 2 measurements – distance, time
Why do you think the GUM defines a functional relationship in Section 4? It is not so you can use measurements of different measurands to get an average.
“Go back and read again. There is only ONE measurand”
Not more endless quoting of the GUM. Well it it annoys Kip.
4.1.1
So what are these “other quantities”? Note 1
So through all that legalese, it should be clear that each of these “other quantities” is itself a measurand, or it’s associated random variable.
Sol there are N + 1 measurands here. Y, the one you are trying to determine, and X1 … XN, which are the component measurands that determine Y.
Are these Xs just measuring the same thing. They can be, as Note 2 explains. But they can be completely different things, as shown in the example, or just think of Tim’s volume.
V = 2piR^2H
In that function R and H are two separate measurands, which are combined to find a 3rd measurand V.
Amnd if that’s not clear enough, 4.1.2 starts
(my bolding).
“When it is stated that Xi has a particular probability distribution”
Xi is the stated value of the observation!
“So through all that legalese, it should be clear that each of these “other quantities” is itself a measurand, or it’s associated random variable.”
So what? You haven’t shown that the functional relationship is a STATISTICAL DESCRIPTOR!
“Are these Xs just measuring the same thing.”
If you will actually READ the GUM instead of just cherry picking you will find that the Xi values are many times determined by REPEATED OBSERVATIONS of Xi! The standard deviation of those repeated measurements of the same thing give you a standard deviation of the stated values for the specific Xi!
Again, stop CHERRY PICKING. Actually STUDY the GUM for meaning!
Incredible, absolutely incredible—this is like trying to reason with Moonies or $cientologists. You right again, Tim, this is a brain-washed cult.
Are you agreeing or disagreeing with that quote? Did you notice who said it?
They worship statistics as a god. All bow down to randomness and Gaussian probability distributions!
“Xi is the stated value of the observation!”
Read the whole thing. the Xi’s can be used to mean either the measurand or a random variable representing the probability distribution of all possible measurements. In neither case is it a stated value.
“So what?”
The so what is I responding to being told the statement “You can add a number of different measurements, each of a different measurand to produce a new measurand.” was untrue, and there was only one measurand.
If you didn’t keep trying to move the goalposts you might not need to keep asking these questions.
“If you will actually READ the GUM instead of just cherry picking you will find that the Xi values are many times determined by REPEATED OBSERVATIONS of Xi!”
You can do that to reduce the uncertainty of the measurement, it doesn’t alter the fact that each of the Xi’s may be a different measurand.
“Actually STUDY the GUM for meaning!”
You’re not a good advert for that approach.
“Read the whole thing. the Xi’s can be used to mean either the measurand or a random variable representing the probability distribution of all possible measurements. In neither case is it a stated value.”
Of course X_i is the stated value of a measurement!
Can you read?
“You can do that to reduce the uncertainty of the measurement, it doesn’t alter the fact that each of the Xi’s may be a different measurand.”
You are full of shite! Each X_i is a repeated observation of the same thing.
Using your logic I could measure the height of a Shetland pony and measure the height of an Arabian stallion, average the two and get the average height of a horse!
It’s just pure malarky!
“Of course X_i is the stated value of a measurement!
Can you read?”
This is why I say the GUM is not my favorite document on the subject. It feels very much like it’s written by a committee, with all the lack of clarity that brings.
This thread started with the claim that “There is only ONE measurand. It may take several physical measurements of various phenomena that make op the measurand to find the value for it.”
And me pointing out that section 4.1 “modelling the measurement” explicitly talked about a measurand computed from different measurands.
This is equation 1
Y = f(X1, X2, …, XN)
with section 4.1.1 saying that for economy of notation the symbols X1 etc can be used to mean either the measurand or the associated random variable representing the outcome of an observation of that quantity. Neither of these are stated values, they are either the measurand or the random variable. Note these are all capital X’s
(In any event my point here is proved. The GUM refers to a measurand computed from different measurands. Regardless of how these values are determined the function is using measurands, plural)
Section 4.1.2 continues that each of the Xi’s may themselves be viewed as measurands depending on different quantities. (it’s says a few other things about the nature of f which may be relevant to another discussion.)
Section 4.1.4 then introduces another set of notations, this time using lower case x’s and y, to represent the estimates for these capital letters. This is what I’d regard as using the stated values. Hence x_i is a stated value used as an estimate of X_i, to get a value y which is an estimate of Y. It also says the estimate may be based on an average of multiple estimates of Y.
So it seems to me that x is the stated value and X the measurand, or the associated random variable.
But then we have 4.1.3, which says the set of input quantities X1, X2 … XN may be categorized as
The way I read it, this is refering to what’s described in the following section, estimating the measurand Xi from a single observation xi. But I don’t find the language very clear, and it’s possible it means the Xi can be though of as being a stated value. But if that’s the case I don;t see why they use different symbols for the same thing in the following section.
In any case, I don’t see how this is relevant to the question of whether the GUM allows multiple measurands. It clearly states it does in section 4.1.1.
“You are full of shite! Each X_i is a repeated observation of the same thing.”
Keep calm. It can be, but it doesn’t have to be.
Note 2 to section 4.1.1 tells you it’s possible for each of the Xi, to itself be made by repeated observations, in which case they are labeled Xik. It doesn’t make sense doing that if you think all the different Xi’s are different observations of the same thing.
“Using your logic I could measure the height of a Shetland pony and measure the height of an Arabian stallion, average the two and get the average height of a horse!”
You could, and could use equation 10 to estimate the uncertainty. It wouldn’t make any sense to do that, but it doesn’t invalidate the equation.
It is worth repeating. The measurement model Y accept other measurands as inputs and produces an output that is also a measurand. The inputs into Y can not be of different things, but can have different units as well. It just so happens that most examples in JCGM 100:2008, JCGM 6, 2020, and the NIST uncertainty machine are of measurement models Y that accept as inputs different things.
The wording in the GUM is clear, decisive, and unequivocal on this matter. Claiming that the GUM method only works when measuring the same thing, measurands cannot be dependent on other measurands, etc. almost defies credulity.
More BS.
The GUM is *BASED* on multiple measurements of the same thing!
——————————————————
GUM:
3.1.4 In many cases, the result of a measurement is determined on the basis of series of observations
obtained under repeatability conditions (B.2.15, Note 1).
3.1.5 Variations in repeated observations are assumed to arise because influence quantities (B.2.10) that can affect the measurement result are not held completely constant.
3.1.6 The mathematical model of the measurement that transforms the set of repeated observations into
the measurement result is of critical importance because, in addition to the observations, it generally includes various influence quantities that are inexactly known. This lack of knowledge contributes to the uncertainty of the measurement result, as do the variations of the repeated observations and any uncertainty associated with the mathematical model itself.
(bolding and italics are mine, tpg)
—————————————————–
You simply cannot have repeatability conditions when you are measuring different things at different times in different conditions.
“The wording in the GUM is clear, decisive, and unequivocal on this matter. Claiming that the GUM method only works when measuring the same thing, measurands cannot be dependent on other measurands, etc. almost defies credulity.”
The GUM specifically talks about repeated measurments of the same thing. See above. And no one is saying that measurands can’t be dependent on other measurands.
But the characteristics of measurand_1 cannot be determined from characteristics of measurand_2. You can’t measure the length of a steel tube in a warehouse in Kansas City and use it to calculate the volume of a steel tube in Miami, FL.
You *can* measure the length of a measurand, e.g. a table, *and* measure the width of the same table and calculate the area of the table top through a functional relationship. There is no functional relationship that will let you calculate the volume of one steel tube by measuring its length and the diameter of a different steel tube.
Section 2.2.3: NOTE 3 It is understood that the result of the measurement is the best estimate of the value of the measurand
It doesn’t say the best estimate of the value of the measurands.
You can’t even get your objections straight.
“The justification is the word “if””
The problem is that you then parley that “if” into “always”!
“Uncertainties cancel whatever the distribution.”
No, they don’t always cancel. The quadrature rule requires that all uncertainty be random, NO SYSTEMATIC UNCERTAINTY.
You *always* claim to have studied Taylor’s tome on uncertainty but you just make it obvious with everything you post that you haven’t. You are a CHAMPION CHERRY PICKER.
Taylor, Chapter 3, Page 58:
“Chapter 5 discusses the normal, or Gauss, distribution which describes measurements subject to random uncertainties. It shows that if the measurements of x and y are made independently and are governed by the normal distribution, then the uncertainty in q = x + y is given by ẟq = sqrt[ (ẟx)^2 + (ẟy)^2 ].” (bolding mine, tpg)
“If the measurements of x and y are independent and subject only to random uncertainties, then the uncertainty ẟq in the calculated value of q = x + y is the sum in quadrature or quadratic sum of the uncertainties ẟx and ẟy.” (bolding mine, tpg)
bellman: “ How many times do I have to tell you that “cancelling” is never total, and has nothing to do with assuming a Gaussian distribution.”
You just keep on claiming that you don’t assume all error is random but EVERYTHING, EVERYTHING you post shows that claim is just so much vacuum! Assuming uncertainty is Gaussian is so ingrained in your brain that you can’t evade it. You simply cannot get out of that box you have taped yourself into!
Since every field temperature measurement around the world has some systematic bias then using RSS UNDERSTATES the total uncertainties from adding values in order to calculate an average.
If we both have Vantage View weather stations connected to our computers and you take a single temperature reading from your unit at 0000 UTC and I do the same then what is the most significant uncertainty factor going to be? Is it going to be random error from neither of us being able to read a computer screen? Or is it going to be systematic bias from calibration drift in both units?
Should the uncertainties of the two readings be added in quadrature or added directly?
“No, they don’t always cancel. The quadrature rule requires that all uncertainty be random, NO SYSTEMATIC UNCERTAINTY. ”
Do I have to be spelling that out everytime that I’m talking about random independent uncertainties. That was the context. Your nitpicking is getting as bad as your karlo puppet.
If they are not all independent use equation 11, not 10. It doesn’t alter the point that the distributions do not have to be normal.
“Assuming uncertainty is Gaussian is so ingrained in your brain that you can’t evade it. You simply cannot get out of that box you have taped yourself into!”
These pathetic lies and ad hominems do not help your cause.
“You are a CHAMPION CHERRY PICKER.”
He says before picking a single word from Taylor.
Honestly, I don’t claim to be an expert on Taylor or any other source. They can all be wrong in parts, whilst getting the overall details correct. If Taylor says that only normal distributions can be added using quadrature, he’s wrong. I don’t see anything in your quotes to suggest he does say that.
Saying that normal distributions can be added in quadrature is not the same as saying non-normal distributions can not be added in quadrature.
If you want to know what Taylor says about uncertainties that are not independent or normal, look to chapter 9. Especially equation 9.9. Which is, of course, equation 11 from the GUM. Note that if the all the non-normal uncertainties are independent this still becomes equation 10 – hence adding in quadrature.
This is why you are called bellcurvewhineman!
Get off the curve, man!
You can understand why Kip feels the need to pitch his examples to the level of a 6 year old.
He claims that he doesn’t assume randomness and Gaussian probability distributions but then he states that he does!
He can’t help himself.
You still don’t understand what the word “assume” means do you?
If I say assuming X we can conclude Y, is not the same as saying I think X is true under all circumstances.
“Do I have to be spelling that out everytime that I’m talking about random independent uncertainties.”
I asked you to tell me what the systematic bias is for the temperature measuring station at the Topeka Air Force Base.
You never answered.
And then you come back with “I’m talking about random independent uncertainties”.
You live in another dimension where there is no such thing as systematic uncertainty. That way you can assume everything is random error and it cancels.
And nothing ever intrudes into your warm, little statistical world.
“If you want to know what Taylor says about uncertainties that are not independent or normal, look to chapter 9.”
You are CHERRY PICKING AGAIN! You didn’t even bother to read the text to see the assumptions behind Eq 9.8 and 9.9!
—————————————————
Taylor: “Suppose that to find a value for the function q(x,y), we measure the two quantities x and y several times, obtaining N pairs of data, (x1,y1), …, (xn,yn). From the N measurements x1, …, xn, we can compute the mean ẋ and the standard deviation σ_x in the usual way; similarly from y1, …, yn, we can compute y_bar and σy. Next, using the N pairs of measurements we can compute N values of the quantity of interest
q_i = q(x_i, y_i), i = 1, …, N.
Given q1, …, qn, we can now compute the mean q_bar, which we assume gives our best estimate for q, and their standard deviation σ_q, which is our measure of the random uncertainty inthe values q_i”
(bolding mine, tpg)
——————————————————
Once again, we see you assuming all uncertainty is random and cancels and we can use the stated values to determine the uncertainty of the data.
YOU DON’T EVEN KNOW WHEN YOU DO IT!
It’s because you cherry pick stuff you don’t even bother to understand!
You just continue to circle back to ignoring that real world field measurements *always* have systematic uncertainty and are, therefore, NOT AMENABLE TO STATISTICAL ANALYSIS.
And then you whine that you don’t do assume anything and people are lying when they say you do!
P A T H E T I C
“You live in another dimension where there is no such thing as systematic uncertainty. That way you can assume everything is random error and it cancels.”
Desperate attempt to move the goal posts.
The discussion was about your claim that only normal distributions cancel, now you try to switch to systematic errors.
If I say independent uncertainties cancel using quadrature it does not mean I deny the existence of non-independent uncertainties. If I say random errors cancel, it does not mean I deny the existence of systematic errors.
“You are CHERRY PICKING AGAIN! You didn’t even bother to read the text to see the assumptions behind Eq 9.8 and 9.9!”
What you point out are not assumptions,. they are what equation 9.9 is all about. It’s showing how to handle the covariance between variables.
As always you try to find loopholes to distract from the fact that chapter 9 destroys you claim that Taylor is saying that in order to add in quadrature all distributions must be Gaussian.
“Once again, we see you assuming all uncertainty is random and cancels and we can use the stated values to determine the uncertainty of the data.”
I am literally pointing you to the equation that shows how to handle uncertainty that is not entirely random. If you had an ounce of comprehension you would realise that shows I am not claiming “all uncertainty is random.”.
And what do you mean by assuming we can use the stated values to determine uncertainty? This is all about using the uncertainties of the individual measurements, not the stated values.
“You just continue to circle back to ignoring that real world field measurements *always* have systematic uncertainty and are, therefore, NOT AMENABLE TO STATISTICAL ANALYSIS.”
The logic of what you are saying is that Taylor, Bevington, et al are just wasting their time. All this statistical analysis to show how errors are propagated is completely useless when you make actual measurements.
“As always you try to find loopholes to distract from the fact that chapter 9 destroys you claim that Taylor is saying that in order to add in quadrature all distributions must be Gaussian.”
Do you have short-term memory problems? I gave you the quote from Taylor showing otherwise! Do you need it again?
Someday you should actually sit down and *STUDY* Taylor instead of just cherry picking!
“This is all about using the uncertainties of the individual measurements, not the stated values.”
I asked you to show me where in Tables 9.1 and 9.2 you find measurements given as “stated value +/- uncertainty”. You just ignored the request. My guess is that you will continue to do so!
I’ve given you the exact quote where Taylor says youi are wrong, all you give me are vague quotes where he may have mentioned normal distributions in some different context. Stop pretending you have some deeper understanding of the sacred tome than anyone else.
“I asked you to show me where in Tables 9.1 and 9.2 you find measurements given as “stated value +/- uncertainty”. You just ignored the request. My guess is that you will continue to do so!”
Stop lying. I explained what Taylor is doing there. I’ve no idea why you think he should add uncertainty intervals top the stated values when he is using the stated values to determine the uncertainty.
“Since every field temperature measurement around the world has some systematic bias then using RSS UNDERSTATES the total uncertainties from adding values in order to calculate an average.”
Stop trying to deflect. We are not talking about any specific problem. We are talking about the general case – all random uncertainties adding in quadrature.
But, unless you can assume that all the uncertainties are perfectly correlated in temperature readings, you do not need to use direct addition, the best estimate will be somewhere between the two. Why do you want to keep saying the uncertainty of the average is the average uncertainty?
“If we both have Vantage View weather stations connected to our computers and you take a single temperature reading from your unit at 0000 UTC and I do the same then what is the most significant uncertainty factor going to be?”
Close to zero to do with calculation of global anomalies and uncertainties. Let’s assume that each has a systematic error that adds 1°C to the temperature and zero random uncertainty. Than the measurement uncertainty will be ±1°C. But if we are trying to determine a global temperature from just these two readings, the uncertainty caused by the sample size of 2 will be vastly greater. Say global temperatures have a standard deviation of 10°C, then the SEM from these two readings will be 10 / sqrt(2) ~ = ±7°C, with a 98% confidence interval of ±14°C.
If we had a sample size of 10000 this would go down to ±0.14 but the measurement uncertainty would remain at ±1, but that would require us to be using 10000 thermometers all of which had an identical systematic bias, at which point I might wonder why we keep using these obviously flawed stations.
Of course, nobody is actually interested in the global temperature, only in the change in temperature, and that means those systematic errors cancel. If they are truly systematic. If every station was reading 1°C too hot this year, and it’s a truly systematic error, then every station would have been 1°C last year.
In reality, as I keep trying to say, the uncertainty of any global temperature anomaly is much more complicated than just running the measurement and sampling uncertainties through the propagation equations.
“Stop trying to deflect. We are not talking about any specific problem. We are talking about the general case – all random uncertainties adding in quadrature.”
You are running away. This whole subject, the ENTIRE blog, is about temperature measurements and using them to prove global warming!
“But, unless you can assume that all the uncertainties are perfectly correlated in temperature readings, you do not need to use direct addition, the best estimate will be somewhere between the two. Why do you want to keep saying the uncertainty of the average is the average uncertainty?”
I DON’T want to say the uncertainty of the average is the average uncertainty! THAT IS YOU SAYING THAT!
It’s why I tried to show you that if you have two sets of boards, one of which has an uncertainty 0.08 and the other 0.04 that the uncertainty of the average of the two boards is *NOT* the average uncertainty. The uncertainty has to be either 0.08 or 0.04 — by definition. Yet *YOU* kept trying to say that the average uncertainty was the uncertainty of the average – right in the face of reality!
“Close to zero to do with calculation of global anomalies and uncertainties.”
It has EVERYTHING to do with the calculation of global anomalies and uncertainties.
“But if we are trying to determine a global temperature from just these two readings, the uncertainty caused by the sample size of 2 will be vastly greater.”
Only because you believe all error is random and cancels. With systematic bias that bias ADDS each time you add another measurement! The uncertainty of the average is *NOT* the average uncertainty.
“Say global temperatures have a standard deviation of 10°C, then the SEM from these two readings will be 10 / sqrt(2) ~ = ±7°C, with a 98% confidence interval of ±14°C.”
Once again, THE SEM TELLS YOU HOW CLOSE YOU ARE TO THE POPULATION MEAN, NOT THE UNCERTAINTY OF THE POPULATION MEAN!
How many times does this have to be pounded into your head before you get it? Somehow you have it in your head that the population mean is 100% accurate and the closer you can get to it the more accurate you are!
“Of course, nobody is actually interested in the global temperature, only in the change in temperature, and that means those systematic errors cancel.”
Systematic biases do *NOT* cancel. That is why Taylor, Bevington, Possolo, and the GUM say measurements with systematic biases ARE NOT AMENABLE TO STATISTICAL ANALYSIS.
You have ONE HAMMER, and by Pete EVERYTHING is a nail and you are going to beat everything with that hammer!
Look at all the uncertainty tomes you want, it doesn’t matter whether you add or subtract measurement values (i.e. an anomaly), the uncertainty adds with both. You cannot decrease uncertainty by subtracting one value from another.
Just like it doesn’t matter if you have random variables Z = X + Y or Z = X – Y, the variances of X and Y add for both!
You can’t even get the basics right.
“You are running away. This whole subject, the ENTIRE blog, is about temperature measurements and using them to prove global warming!”
Really? So what was all that nonsense about dice in a box? It’s almost as if you are suggesting that Kip has an ulterior motive in trying to disparage correct statistical analysis.
“I DON’T want to say the uncertainty of the average is the average uncertainty! THAT IS YOU SAYING THAT!”
Apart for the fact iot’s something I never say, don’t believe in, and have repeatedly told you I don’t agree with.
You keep repeating this obvious lie and with so many unnecessary capital letters, yet I still have no idea why you believe it, or what point you think you are making.
It’s really not that difficult. Consider your first example. 100 thermometers, each with a measurement uncertainty of ±0.5°C.
Obviously the average uncertainty is ±0.5°C.
You claim the uncertainty of the sum is ±5.0°C, and this will also be the uncertainty of the average.
Kip claims the uncertainty of the sum is ±50.0°C, and that the uncertainty of the average is 50 / 100 = ±0.5°C.
I claim, under the same assumptions as you that the uncertainty of the sum is ±5.0°C, but say that uncertainty of the average is 5.0 / 100 = ±0.05°C.
So given the average uncertainty is ±0.5°C, which of us is claiming the uncertainty of the average is the average uncertainty?
Not me, I’m saying that it could be only a tenth the size of the average uncertainty. I only say the uncertainty of the average is the same size as the average uncertainty under the assumption that there is a complete correlation between all uncertainties.That is, if one is reading 0.5°C to warm all other readings will also be 0.5°C too warm.
“Really? So what was all that nonsense about dice in a box? It’s almost as if you are suggesting that Kip has an ulterior motive in trying to disparage correct statistical analysis.”
It was Kip’s way of trying to explain that uncertainty is unknown, the closed box,
Correct statistical analysis is *NOT* assuming that all measurement uncertainty is random, Gaussian, and cancels and that no systematic bias exists.
If you do *not* know the distribution then your choices are limited as to what the uncertainty interval is.
“It was Kip’s way of trying to explain that uncertainty is unknown, the closed box,”
But uncertainty is not unknown. Or at least it’s what you are trying to estimate. What’s not known is the true value and the error.
“Correct statistical analysis is *NOT* assuming that all measurement uncertainty is random, Gaussian, and cancels and that no systematic bias exists. ”
But you can assume that with the dice in a box. There’s no reason to suppose the rules of probability cease to exist just because you’ve hidden the dice.
“If you do *not* know the distribution then your choices are limited as to what the uncertainty interval is.”
But you do know the distribution of the dice.
“It’s why I tried to show you that if you have two sets of boards, one of which has an uncertainty 0.08 and the other 0.04 that the uncertainty of the average of the two boards is *NOT* the average uncertainty.”
You keep trying to show different things with this example because you are never clear about the details.
What do you mean by sets of boards? When you say one set has an uncertainty of 0.08, do you mean each board’s measurement had that uncertainty, or do you mean the uncertainty of the average was 0.08? What do you mean by “the uncertainty of the average of the two boards”? Which two boards? Do you mean the average of the two sets, or the average of each set?
“Yet *YOU* kept trying to say that the average uncertainty was the uncertainty of the average – right in the face of reality!”
Oh no I didn’t. But please keep on lying, it shows how everything everything else you say is unreliable.
“It has EVERYTHING to do with the calculation of global anomalies and uncertainties.”
“It” being the following question.
“If we both have Vantage View weather stations connected to our computers and you take a single temperature reading from your unit at 0000 UTC and I do the same then what is the most significant uncertainty factor going to be?”
I stand by assertion that it has little to do with calculating a global monthly anomaly.
“With systematic bias that bias ADDS each time you add another measurement!”
No it doesn’t. It’s defying all sense to say it does. Consider my station has a bias of +1°C, your station has a bias of +1°C. How can these two degrees possibly cause the average to increase by 2°C?
“THE SEM TELLS YOU HOW CLOSE YOU ARE TO THE POPULATION MEAN, NOT THE UNCERTAINTY OF THE POPULATION MEAN!”
Stop shouting, it doesn’t make your point more convincing.
How close I am to the population mean is exactly what I want to know. It’s the population mean I’m trying to find, the closer I am to it the better. There is no uncertainty in the population mean, any more than there is in the true value of a length of wood. The question is how much uncertainty is there in sample mean caused by the measurements and the sampling.
“Somehow you have it in your head that the population mean is 100% accurate and the closer you can get to it the more accurate you are!”
Finally you ascribe something to me that is correct. I’m really not sure why you would think the population mean is not “accurate”.
“Systematic biases do *NOT* cancel. That is why Taylor, Bevington, Possolo, and the GUM say measurements with systematic biases ARE NOT AMENABLE TO STATISTICAL ANALYSIS.”
As Kip would say, this isn’t that nasty statistics, but gold old arithmetic. If this year was 1°C too warm because of statistical bias, then last year was also 1°C too warm – hence they cancel. If you mean there is some statistical bias that changes year to year, then guess what, it isn’t systematic.
“You have ONE HAMMER, and by Pete EVERYTHING is a nail and you are going to beat everything with that hammer!”
You’ve got one cliche and will beat it until it fits any circumstance.
“You cannot decrease uncertainty by subtracting one value from another. ”
But as you said, this is systematic so “NOT AMENABLE TO STATISTICAL ANALYSIS”.
“Just like it doesn’t matter if you have random variables Z = X + Y or Z = X – Y, the variances of X and Y add for both!”
Operative word, “random”. If you add a systematic bias to each, e.g. add 1 to the mean of each, than the mean of X + Y increases by 2, but the mean of X – Y will be the same.
“You can’t even get the basics right. ”
Why do you always have to have these meaningless tags in your comments?
“The problem is that you then parley that “if” into “always”! ”
For anyone following, this is Tim’s standard ploy. Invent a strawman and insist it’s something I’m always saying. He want’s to believe that I say all probability distributions are normal, and so in his fantasy I always do say it. It doesn’t matter how many times I say the opposite it just won;t penetrate his cognitive bias.
I think, the problem is he believes that only normal distributions can be combined and therefore if I say it doesn’t matter what the distribution is when combining them, in his perverse mental state this must mean I believe all distributions are normal.
Oh dear, the irony is now totally overwhelming.
I’ll echo that. He makes an absurd declaration that an average is Σ[x_i^2, 1, N] / N, conflates sums (+) with quotients (/), makes numerous algebra mistakes, and in the same breath gaslights me by saying the algebra is simple.
You’ve not disproved a single assertion that I have made.
You can’t even accept that if q = x + y then q/n = x/n + y/n
The most simple process of dividing both sides of an equation by the same thing.
Sometimes the truth hurts! It is *YOU* that says that uncertianty intervals have probability distributions and that values further away from the stated value have a lower probability of being the true value.
That *IS* assuming the probability distribution is Gaussian.
I have never said that only normal distributions can be combined! Tell me again who is making up strawmen?
If you combine a skewed left and a skewed right distribution what do you get? I know you won’t answer because it would undercut your assertion that values further away from the stated value have a lower probability!
You live in a fantasy world. Utterly and completely. One where uncertainty always cancels. And you quote GUM Eq 10 as proof – without one iota of understanding about the assumptions that must be made so you can use Eq 10!
“Sometimes the truth hurts!”
Not half as much as someone continuously lying about me.
“It is *YOU* that says that uncertianty intervals have probability distributions and that values further away from the stated value have a lower probability of being the true value.”
When did I say that all distributions have probabilities that are smaller the further form the states value?
I’ve already pointed out your misunderstanding, but as usual you ignore that and just repeat the lies.
“I have never said that only normal distributions can be combined!”
Fair enough. I should have said combined using quadrature.
“If you combine a skewed left and a skewed right distribution what do you get?”
I’ve already demonstrated that adding in quadrature of two skewed distributions will give the expected result. What the distribution will be will depend on the exact distributions you are talking about, but will probably by a bi-modal distributions. The more distributions you add, the closer it will get to a normal distribution, just as the CLT says.
“I know you won’t answer because it would undercut your assertion that values further away from the stated value have a lower probability!”
I keep forgetting your psychic powers, which are 100% accurate as long as you ignore anything anyone actually says.
“You live in a fantasy world.”
As opposed to your “real” one where you can keep arguing with claims I’ve never made?
“And you quote GUM Eq 10 as proof – without one iota of understanding about the assumptions that must be made so you can use Eq 10!”
Proof of what? I keep telling you what assumptions are made in that equation. Uncorrelated input quantities, and a linear function. And these assumptions are not absolute – as long as the assumptions are reasonably correct the result will be reasonably correct.
The assumption you keep making, that all the uncertainties have to be Gaussian, is just not correct.
“When did I say that all distributions have probabilities that are smaller the further form the states value?”
So you agree that you can’t just assume a random, Gaussian distribution. Then how do you determine how much cancellation you get?
“Fair enough. I should have said combined using quadrature.”
Quadrature truly only works well when you have normal distributions. You’ve been given authoratative quotes on that.
“I keep telling you what assumptions are made in that equation. Uncorrelated input quantities, and a linear function.”
Those are *NOT* the only restriction. How do you get uncorrelated input quantities if you use the same instrument for all measurements?
“reasonably correct.”
According to who?
“The assumption you keep making, that all the uncertainties have to be Gaussian, is just not correct.”
It’s correct if you want Eq 10 to be accurate!
“So you agree that you can’t just assume a random, Gaussian distribution.”
Your obsession with me is driving you insane, assuming you had any sanity to start with.
I’ve told you more times than you can count, that I don;t assume all distributions are Gaussian.
“Quadrature truly only works well when you have normal distributions.”
Somehow, I see you huddled in the corner of a darkened room, clutching a blanket around you and endlessly repeating “it only works with normal distributions, it only works with normal distributions”
Your wrong. I’ve explained over and over why you are wrong, you somehow think that just repeating it somehow makes it true.
“If you combine a skewed left and a skewed right distribution what do you get? I know you won’t answer because it would undercut your assertion that values further away from the stated value have a lower probability!”
Remember those two distributions I mentioned here.
Here’s the distribution of the sum of those two skewed distributions.
It isn’t normal, but is isn’t the bimodal distribution I expected.
“It isn’t normal, but is isn’t the bimodal distribution I expected.”
So what? It shows that when you combine them you can’t assume cancellation! They aren’t Gaussian!
What do you mean no cancellation. They cancelled in exactly the same way adding in quadrature predicted.
“You keep giving me the various GUM definitions, but never explain how they work if there is no probability distribution”
You just keep on showing that you’ve never actually studied the GUM for meaning. All you ever do is just cherry pick things you think show someone is wrong.
The GUM assumes all random error from multiple measurements of the same thing! Thus adding in quadrature using weightings for each term in the functional relationship is used.
It’s why we keep pointing out to you that while you deny it vehemently all you ever do is assume that uncertainty is totally random and cancels. You always circle back the same thing EVERY SINGLE TIME!
AKA the hamster wheel that squeaks and squeaks but always comes back around to the same spot..
My wheel keeps spinning, whereas your hamster is long dead, and the wheel is stuck permanently on troll mode.
bellcurvewhinerman—always generating noise on his hamster wheel.
Poor baby, do you think I care what you think?
From the way you are compled to respond to my every comment with some predictable put down, and spend the rest of the time speculating on the state of my mind in most other threads, then yes I do think you care what I think. You care very deeply, so deeply it’s turned you into a troll who’s only point in existing is to spend time pollution these comment sections with your inane trolling.
I also suspect you will confirm this hypothesis by replying to it with another witty put down. I suggest something like “irony overload”, or “I know what you are but what am I?”. You could always prove me wrong by just resisting the urge to always have the last word. We shall see.
Here you go, puppy:
More noisy word salad; trying to reason with you lot is like trying to reason Moonies or $cientologists.
And what would be the point? The cost of acknowledging the truth is too high for you, so it will never happen.
The projection is a bit amusing, though.
You may continue now, bellcurvewhinerman…
“It’s either the probability distribution of errors, or it’s the probability of what the measurement may be when you don’t have perfect knowledge.”
And here we are, once again, assuming that all measurement uncertainty is random so we can say it cancels.
It’s what you assume EVERY SINGLE TIME!
From the GUM, Section 3.2.3:
“It is assumed that, after correction, the expectation or expected value of the error
arising from a systematic effect is zero.”
3.2.4 It is assumed that the result of a measurement has been corrected for all recognized significant systematic effects and that every effort has been made to identify such effects.
The entire GUM surrounding Eq 10 assumes ZERO systematic bias.
“why equation 10 is used to justify adding in quadrature.”
It’s only justified if ALL uncertainty is random! It assumes all systematic uncertainty has been corrected.
Do *YOU* know what the systematic bias is for the temperature measuring system at Forbes AFB in Topeka, KS?
If not, then how do you apply Eq. 10?
I keep saying, you, most statisticians, and climate scientists assume all uncertainty is random and cancels. You can therefore use the standard deviation of the stated values as your uncertainty.
You can deny it all you want but your assertions put the lie to those denials.
It’s an inconvenient truth for you, for statisticians that have never learned a single thing about metrology, and for climate scientists who also know zip about metrology! BUT IT IS STILL THE TRUTH NO MATTER HOW INCONVENIENT.
“ You don’t know what it is, but you do have a proability that it lies with a distribution.”
And once again we circle back to your meme that all uncertainty is random and cancels.
“It’s more likely to be closer to the measurement you’ve just made, than something far away from the measurement.”
And, once again, we circle back to your belief that all uncertainty is random, Gaussian, and that it all cancels. If that is the case then the stated value is the true value of the measurement.
You simply do *NOT* know where in the uncertainty interval the actual measurement lies. You simply do *not* know anything about the values in the uncertainty interval.
Remember, we are talking about SINGLE measurements of one thing. One measurement of one thing does not create a probability distribution made up of multiple measurements. Trying to create a probability distribution by combining single measurements of different things mean you have to add the variances (i..e the uncertainty) of each single measurement. No cancellation. V_total = V_1 + V_2
“Really, if you measure a board at it says it’s 6′ long, are you saying you still can say nothing about the actual length of the board?”
Your real world experience, or actually your lack of, is showing again. If you are building a stud wall, for instance, you simply don’t assume that all the 6′ boards you have measured are actually 6′ long. You cut them all to the same length which may or may not be exactly 6′ long. That way you don’t get ripples in your ceiling drywall!
So, NO, you can’t say you know anything about the actual length of the board. You *make* them all the same, you don’t assume they are all the same!
You’ve triggered the noise generator again this fine AM, Tim!
“So, NO, you can’t say you know anything about the actual length of the board.”
Sorry, trying to resist replying to all these near identical comments. But this just seems absurd. You’ve cut all your boards to what you think are 6′ in length, you’ve presumably measured them and found they are 6′ within your level of uncertainty, but you still insist you know absolutely nothing about the actual length. They could all be 6cm they could all be 6km?
“You’ve cut all your boards to what you think are 6′ in length,”
OMG! I cut them to fit in the space where they go! If the rafters are not quite 6′ high or are a little bit higher than 6′ doesn’t matter!
I MAKE THEM ALL THE SAME LENGTH! Their exact length is irrelevant!
Once again you show that you have absolutely *NO* experience in the real world. You live in a non-real dimension created from your delusions.
I’m just trying to figure out what relevance any of this has to measurement uncertainty. And especially to your claim that measuring a board as being 6′, tells you nothing about the actual length of the board.
“I MAKE THEM ALL THE SAME LENGTH! Their exact length is irrelevant!”
But what if the length was relevant? You still have to measure them.
If I didn’t MAKE THE BOARDS ALL THE SAME LENGTH then you would get ripples in the drywall on the ceiling!
It simply doesn’t take much variation to show up when you put paint on the ceiling and mount a ceiling light!
You’ve never once built anything. It’s obvious. How would *YOU* make sure you cut them all to the same length?
I don’t care! This has got nothing to do with measurement uncertainty.
“No. It says the true value has a likelihood of lying between these two values.”
But you *STILL* don’t know where in the interval it is!
“Of course you have an expectation of what the true value is. there wouldn’t be any point in measuring anything if it didn’t lead to an expected value.”
And, as usual, you are conflating two different things hoping to confuse the issue.
“But you *STILL* don’t know where in the interval it is!”
Of course not. That’s why it’s uncertain. If you knew the actual value you would be certain.
“You do *NOT* have an expectation of what the TRUE value is. If you did then why do you have an uncertainty interval?”
Expectation doesn’t mean you know, it means you have the best estimate of where it is.
“You are trying to conflate measuring one thing multiple times with measuring multiple things one time each.”
What are you on about now? We were just discussing the definition of measurement uncertainty from the GUM. It doesn’t matter if it’s a single measurement or a combined value.
“Of course not. That’s why it’s uncertain. If you knew the actual value you would be certain.”
You claim to know the probability associated with each value in the uncertainty interval. So you *must* know which value is the true value. Meaning there is no uncertainty. That’s why you always ignore the uncertainty interval and just use the stated values!
“Expectation doesn’t mean you know, it means you have the best estimate of where it is.”
The measurement itself is a “best estimate”. If you know the probability associated with each point in the uncertainty interval then you also know the best estimate of the true value.
You are turning yourself inside out trying to rationalize your own memes!
“What are you on about now? We were just discussing the definition of measurement uncertainty from the GUM. It doesn’t matter if it’s a single measurement or a combined value.”
Of course it matters! Why do you think the GUM talks about a MEASURAND. There is no “s” in MEASURAND.
3.1.1 The objective of a measurement (B.2.5) is to determine the value (B.2.2) of the measurand (B.2.9),
that is, the value of the particular quantity (B.2.1, Note 1) to be measured. A measurement therefore begins
with an appropriate specification of the measurand, the method of measurement (B.2.7), and the
measurement procedure (B.2.8).
3.1.4 In many cases, the result of a measurement is determined on the basis of series of observations
obtained under repeatability conditions (B.2.15, Note 1).
3.1.6 The mathematical model of the measurement that transforms the set of repeated observations into
the measurement result is of critical importance because, in addition to the observations, it generally includes various influence quantities that are inexactly known.
3.1.7 This Guide treats the measurand as a scalar (a single quantity). Extension to a set of related
measurands determined simultaneously in the same measurement requires replacing the scalar measurand and its variance (C.2.11, C.2.20, C.3.2) by a vector measurand and covariance matrix (C.3.5). Such a replacement is considered in this Guide only in the examples (see H.2, H.3, and H.4).
(all bolding mine, tpg)
Why do you, bdgwx, climate scientists, and statisticians all treat the GUM as if it is addressing single measurements of multiple things? IT DOESN’T, except in a few spots that are never referenced by anyone!
You CONTINUALLY show your ignorance concerning metrology but just keep on motoring along as if you know more than anyone else about the subject!
“You claim to know the probability associated with each value in the uncertainty interval. So you *must* know which value is the true value. ”
Utter nonsense. I’m really not sure if you are genuinely failing to understand this simple concept, or are just arguing for the sake of it.
Lets put two dice in a box. The sum of the dice have a value which I don’t know. But I do know what the probability of any number is. I do know that there is a 1/6 probability of it being 7, or a 1/36 chance it’s a 12, or a 2/3 chance it’s between 5 and 9 inclusive. What I don’t know is what value it actually is. And I can’t say what it’s value actually is unless I was allowed to open the box. But that doesn’t mean I can’t say what values are more likely.
More smokescreen…
It also allows you to predict the next toss. It was a point Steven Mosher tried to make in the CTL post that spiraled so far out of control that it was claimed that 1) science does not make predictions and if you are making predictions then you aren’t doing science, 3) if you use statistical inference then you aren’t doing science, 4) quantum mechanics is completely deterministic and my favorite 5) superstition is a viable alternative to science when the goal is prediction.
Your faith in statistics is misplaced. It may allow you predict the pattern of the next 1,000 throws, but it won’t let you predict with any accuracy what the NEXT throw will be.
Casinos would love to see you come in. You would consistently bet that the mean would come up next on every throw. It is why I don’t gamble very often and only for entertainment.
Even Kip didn’t realize that even on a throw of two dice, each dice is independent and mutually exclusive of the other. What is the probability of dice A being a 6 and independently, dice B being a 1. Or A = 3 and B = 4. It why the house always wins and love to see naive gamblers like you come in.
It is why trying to “predict” what will happen NEXT requires a functional relationship that is deterministic. It is what SCIENCE is all about.
It is why correlation is not causal. There is no guarantee that the NEXT value can be predicted. You can only predict what it might be.
Science is predicting what will occur next with no doubt. Statistics is about predicting what the next PATTERN may be.
It is why you add uncertainties of independent measurements of different things, even with the same device to get an upper bound on uncertainty.
“It is why trying to “predict” what will happen NEXT requires a functional relationship that is deterministic.”
Depends on what you mean by “predict”. For any game of chance you will never have a deterministic formula that will tell you what the next result will be, the best you can have is a better understanding of the probability of what will happen next.
“Even Kip didn’t realize that even on a throw of two dice, each dice is independent and mutually exclusive of the other.”
Really?
“It is why you add uncertainties of independent measurements of different things, even with the same device to get an upper bound on uncertainty.”
Yet every book on metrology say you can add in quadrature, and just adding is likely to give you too large a bound.
Do you realize even quadrature is only an assumption Can you imagine an uncertainty where the square of one uncertainty is added to the actual value of another and the cube root is then taken? Think nonlinear errors with active components like a vacuum tube or transistor in combination with passive components. Why are some calibration correction sheets curved?
There is a reason experimental uncertainty exists. Determining uncertainty of components is impossible and how they combine is unknown.
Can you not think of circumstances where the upper bound is the best choice? How about weight limits for bridge supports? Or rainfall attenuation on microwave signals along with other fading components like humidity or dust?
“Do you realize even quadrature is only an assumption”
It’s not an assumption, it’s a justified statistical technique. It might be an approximation of reality, but that’s true about any maths.
The rules, based used to get the general equation are based on first order Taylor series. It assumes the function is linear and will be more of an approximation when it isn’t. You can use higher order series if it’s very non-linear.
“Can you not think of circumstances where the upper bound is the best choice?”
Yes, when the uncertainties are entirely non-independent. And in cases where you have a small sample, such as 2-dice and you absolutely need a 100% certainty of the result.
But the problem isn’t when might it be appropriate. The problem I have with these essays is Kip’s assertion that anything less than the upper bound is never appropriate.
“How about weight limits for bridge supports?”
I doubt any bridge is built to handle the absolute worst case, because that’s potentially infinite. There has to be a risk assessment. You want to handle all plausible scenarios and then some, but there has to be some cut off. If you are saying you can’t build a bridge until it’s capable if handling a scenario which only has a 1 in 10^100 chance of occurring, you will never build a bridge.
“It’s not an assumption, it’s a justified statistical technique.”
It’s only a justified statistical technique if ALL UNCERTAINTY IS RANDOM AND GAUSSIAN!
And, once again, we circle back to you assuming that all uncertainty is random and Gaussian, EVERY SINGLE TIME!
You just can’t get out of that paper bag you’ve trapped yourself in, can you?
“Yes, when the uncertainties are entirely non-independent.”
AND when they are not random and Gaussian!
“The problem I have with these essays is Kip’s assertion that anything less than the upper bound is never appropriate.”
That’s *NOT* what Kip said. It’s what applies when you don’t know anything about the uncertainty! If the systematic bias is significantly larger than the random error, which is entirely possible in one single measurement of one single thing, then direct addition is perfectly acceptable.
“If you are saying you can’t build a bridge until it’s capable if handling a scenario which only has a 1 in 10^100 chance of occurring, you will never build a bridge.”
If you assume, as you do, that all uncertainty is random and cancels then not only will you be risking large payouts of money and criminal penalties you’ll never design a second bridge! No one will trust your judgement!
“For any game of chance you will never have a deterministic formula that will tell you what the next result will be, the best you can have is a better understanding of the probability of what will happen next.”
Which is why you don’t have a functional relationship.
Volume =πR^2H is a functional relationship. It tells you exactly what the volume will be. A probability distribution can’t do that. An average is a probability descriptor, not a functional relationship.
“Volume =πR^2H is a functional relationship.”
Yes it is, well done. So is mean = (x + y) / 2.
“It tells you exactly what the volume will be.”
Only if you have exact values for height and radius.
“A probability distribution can’t do that.”
But if you are looking at the uncertainty of your measurements you need an uncertainty interval, and that’s a probability distribution.
“An average is a probability descriptor, not a functional relationship.”
An exact average is a functional relationship. The sum of two dice is a functional relationship, the average of two dice is a functional relationship.
“Yes it is, well done. So is mean = (x + y) / 2.”
(x+y)/2 is a STATISTICAL DESCRIPTOR. There is no combination of x and y that will give you (x+y)/2. If you have a 6′ board and an 8′ board, you’ll never find one that is 7′ long. If you have an 8-penny nail and a 10-penny nail you’ll never find a 9-penny nail. If you have a 1lb hammer and a 2lb hammer you’ll never find a 1.5lb hammer.
Why is this so hard for you to understand?
“But if you are looking at the uncertainty of your measurements you need an uncertainty interval, and that’s a probability distribution.”
If all of the uncertainty is systematic bias then where is the probability distribution? What is it?
“The sum of two dice is a functional relationship, the average of two dice is a functional relationship.”
Nope. The average is a statistical descriptor. There is no guarantee that the average exists in the real world. So it can’t be a *functional* relationship.
Because to acknowledge the truth would collapse his entire propaganda narrative. The GAT becomes meaningless, along with all his treasured GAT trends.
Think about all the time bgw has invested in his climate curve fitting “model”—the cost of the truth is way too high for him.
“(x+y)/2 is a STATISTICAL DESCRIPTOR.”
And it’s a functional relationship. Is the point where you demonstrate once again that you don’t understand what functional relation means?
“There is no combination of x and y that will give you (x+y)/2.”
What a weird thing to say. OK, I’ll take that challenge, x = 2, y = 4, gives me (2 + 4)/ 2 = 3. Hay, I’ve just done something Tim thinks is impossible.
“If you have a 6′ board and an 8′ board, you’ll never find one that is 7′ long.”
I’m guessing this is what using imperial measurements does to your brain. Apparently having two boards means you can never find a third board of a different length.
“Why is this so hard for you to understand?”
Because it’s balderdash. I’m sure you meant to say something else, but it still won’t make sense.
And yes, the real point is you still don’t understand that an average does not need to have one physical object that it the same size as it to be an average. An average length of 7 feet does not require there to be a physical board that is 7 feet.
“If all of the uncertainty is systematic bias then where is the probability distribution? What is it?”
Depends on if you know what the bias is or not. If you know it then you know with probability 1 that the measured value is equal to the true value plus the systematic error. And as you know what that is, you can then say with no uncertainty what the true value.
If you know there is absolutely no random uncertainty, but there is an unknown systematic uncertainty, you are going to have to figure out what your state of knowledge is of this unknown systematic error.
One example of this might be Kip’s assertion that all measurement uncertainty is caused by rounding. If you measure something with absolute certainty that there are no errors, but your instrument only gives you an answer to the nearest integer, you know the uncertainty is ±0.5, and it’s reasonable to assume that this will be a rectangular distribution. Now if you keep measuring the same thing with this instrument you have a systematic error. If the true value is 9.8, it will always be rounded to 10 so there will always be the same error of 0.2. So there you have a systematic error with a known rectangular distribution. The consequence is you still only know the true value is between 9.5 and 10.5, and even if you measure it hundreds of times and take the average, you can’t reduce that uncertainty. The average will always be 10.
Of course, if you are measuring lots of different things with different lengths, this systematic uncertainty becomes random.
“And it’s a functional relationship.”
An average is *NOT* a functional relationship. It is a statistical descriptor. There is no guarantee that an average exists in reality. A functional relationship describes reality. Volume, energy, charge, velocity, etc.
Take average velocity as an example. Car A travels the distance at a high speed, stopping just before the end point and then creeping forward to cross at the same time as Car B. Car B travels at a slower speed over the part of the distance and then at a high speed at the end. Both have the same average speed but that average doesn’t actually exist in reality.
Only statisticians living in a dimension separate from reality believes that average velocity exists in reality.
“I’m guessing this is what using imperial measurements does to your brain. Apparently having two boards means you can never find a third board of a different length.”
ROFL! The parameters of the example are that you have TWO BOARDS! Of course *YOU* have to change that by adding a third board in order to justify that the average length board exists!
“An average length of 7 feet does not require there to be a physical board that is 7 feet.”
Like I said, you live in an alternate “statistical” dimension that has no points of congruence with actual reality!
“One example of this might be Kip’s assertion that all measurement uncertainty is caused by rounding. “
That’s not what he said. Your statement is just a reflection of your poor reading skills.
“you know the uncertainty is ±0.5, and it’s reasonable to assume that this will be a rectangular distribution. “
It is only a reasonable assumption to a statistician living in an alternate reality. It’s a CLOSED BOX. You simply have no idea of what the probability distribution is inside that box other than its possible range!
TG said: “A functional relationship describes reality.”
You also said that “Functional relationships are *NOT* non-deterministic.“.
How does your definition handle the case of a function that describes reality but is non-deterministic like would be the case with QM processes or Kip’s two dice scenario.
TG said: “You simply have no idea of what the probability distribution is inside that box other than its possible range!”
We know the probability distribution Kip’s two dice scenario.
We know the probability distributions of many of the quantum mechanics boxes.
And just like we can exploit the QM distribution to make statements about the future hit points of photons/electrons/etc on a backdrop behind two slits we can exploit the distribution to make statements about the future outcomes of Kip’s two dice toss.
I’ve said before that I don’t get too hung up on definitions. I’m more than willing to adopt any vernacular that aids with discussion. So what word or phrase to do think is best for a function that describes a non-deterministic reality?
Toss more stuff into your rock soup of irrelevancies, I know you can.
“An average is *NOT* a functional relationship.”
You obviously have a different definition of “functional” than I do. Could you provide a link to your definition. If you don’t accept (x + y) / 2 as functional, but do accept πR^2H, could you say if x + y is a functional relationship or not?
“Both have the same average speed but that average doesn’t actually exist in reality. ”
Well it does. Both cars must have been traveling at the average speed at least at one point in their journey. In reality, any measurement of speed is going to be based on some sort of average, even if it’s only over a short distance. But none of this is relevant to the question of when you can add in quadrature.
“Only statisticians living in a dimension separate from reality believes that average velocity exists in reality.”
I’m really getting tired of your constant attempts to take ownership of the “real world”. There are lots of ways of understanding the world, many of them exist outside your workshop. To me, statistics is one of the best tools we have to understand the real world in all it’s messiness.
To me, your contention that averages don’t exist because you can’t touch them is also avoiding the real world.
You are conflating “functional relationship” to a statistical” average! They are different things.
An average or mean is not a functional relationship, it is a statistical parameter of a set of numbers. The fact that it might be the same as a data point does not make it a measurement.
“”””””
3.1.1 The objective of a measurement (B.2.5) is to determine the value (B.2.2) of the measurand (B.2.9), that is, the value of the particular quantity (B.2.1, Note 1) to be measured. A measurement therefore begins with an appropriate specification of the measurand, the method of measurement (B.2.7), and the measurement procedure (B.2.8).
“”””””
“They are different things.”
And I say an average is a functional relationship. Just endlessly claiming it isn’t doesn’t get as very far. You need to explain with evidence what your definition of a functional relationship is.
I see nothing in the GUM to suggest they are using functional in anything other than the usual mathematical meaning of the word. Indeed, 4.1.2 notes that the function might be as simple as Y = X1 – X2, modelling a comparison between two measurements of the same thing. Does that agree with your definition of functional?
“A measurement therefore begins with an appropriate specification of the measurand, the method of measurement (B.2.7), and the measurement procedure (B.2.8).”
Non of which suggests to me that the measurand cannot be a mean, with the method of measurement being the average of multiple values.
B.2.9 defines measurand as “particular quantity subject to measurement”. No mention of any exeptions, just a quantity.
B.2.5 defines measurement as “set of operations having the object of determining a value of a quantity”. Note the word set.
B.2.1 defines a measurable quantity as “attribute of a phenomenon, body or substance that may be distinguished qualitatively and determined quantitatively”
Nothing in any of the definitions says that if something is a statistical parameter it cannot be measurand.
But no matter. Assume it cannot and the GUM simply doesn;t deal with the mean of multiple things – I’ll make my usual comment that in that case talking about the measurement uncertainty of a mean is impossible, and we will just ahve to rely on the usual statistical techniques for determining the uncertainty caused by sampling.
And all of this is still a distraction from the question I keep asking, which is not about the mean but about the sum. Do you think the sum of two different values is a measurand? If so do you think that if the uncertainties are random and independent then equation 10 is the correct equation to use to determine the uncertainty of that sum?
JG said: ” but it won’t let you predict with any accuracy what the NEXT throw will be.”
Yes it will. Using inference I know that 7 is the most likely outcome and that values above and below that have diminishing likelihood. Therefore 7 ends up being the best prediction for the next toss as that prediction minimizes the error of the prediction.
This technique is used in weather forecasting. Global circulation models are run numerous times with stochastic, parameterization, and initial condition perturbations. The mean or median is then used as the going forecast. This is how the NWS makes gridded forecasts using the NBM. When you click on your local gridded forecast from weather.gov it is using this technique.
JG said: “Casinos would love to see you come in. You would consistently bet that the mean would come up next on every throw.”
No. They would hate me because I would minimize my loss and their gain. That’s assuming I was forced to play. In reality I would predict that in a casino environment anything I tried would result in a loss over the long run so I would decide based on that prediction that the most optimal choice is to not play.
JG said: “It is why trying to “predict” what will happen NEXT requires a functional relationship that is deterministic. It is what SCIENCE is all about.”
The obvious counter example here is quantum mechanics.
So you are an expert in QM as well CO2 warming pseudoscience?
No one is fooled by this charade.
“””””Yes it will. Using inference I know that 7 is the most likely outcome and that values above and below that have diminishing likelihood. Therefore 7 ends up being the best prediction for the next toss as that prediction minimizes the error of the prediction.””””””
“””””. In reality I would predict that in a casino environment anything I tried would result in a loss …”
These two statements are incongruent. If 7 is “the best prediction” you should make money hand over fist! The fact that you recognize that won’t happen is tacit admission that 7 IS NOT a prediction of the next throw.
JG said: “These two statements are incongruent. If 7 is “the best prediction” you should make money hand over fist!”
That’s not true. To exchange money there have to be rules governing when and how much of a buy-in and payout there is. A smart game administrator would craft these rules such their per toss expected return is positive and yours is negative.
The irony here is that smart game administrators use statistical techniques to make predictions regarding how much money they will gain/lose on future events given a specific set of rules for the game. They then decide on the rules based on those predictions and their gain/loss goals. It would be egregiously negligent if administrators of games of chance like Let’s Make a Deal didn’t make predictions regarding their future gains/losses prior to allowing the game to proceed.
“Therefore 7 ends up being the best prediction”
Best prediction? Not the only prediction?
A functional relationship will give you an exact value.
Volume = πR^2H. It gives you an exact prediction.
Speed = dy/dx. A functional relationship. It gives you an exact prediction.
Y might be a 7? That is not a functional relationship!
It’s the same with quantum mechanics. There are multiple predictions for the hit point of a photon on a backdrop behind two slits. The position of the hit is non-deterministic and represented by a probability density function. Unlike the dice PDF the photon hit point PDF is continuous with an infinite number of predictions. However, the best prediction of the next hit is where the PDF peaks. That is what will minimize the error on predictions. The interesting thing about this scenario is that it is a case where there is a functional relationship derived analytically without making statistical inferences. Yet that functional relationship is non-deterministic (like most relationships in QM).
Nonsense. QM completely describes the wave function densities on the other side of the slits.
karlomonte said: “Nonsense. QM completely describes the wave function densities on the other side of the slits.”
You say it is deterministic. I say you’re grandstanding. If you disagree then I challenge you to publish your findings that you can predict deterministically where each and every photon will decohere on the backdrop in the exact order and exact spot before you observe it. Do this for the countless other QM processes as well. Win a Nobel Prize and rock the foundation of the quantum realm and even the core of science itself. You want to make extraordinary claims? Then you need to step up and present extraordinary evidence.
You have not the first clue about QM, you can’t even get past the fact that a photon is not a particle.
You need to show where I made one, clown.
Strawman. I never said a photon was just a particle.
Challenging the fact that QM makes non-deterministic predictions or claiming that it is completely deterministic is an extraordinary claim.
Remember, this line of conservation is related to all the absurd statements I was told when I said science makes predictions. This includes Pat Frank’s claim that QM is completely deterministic.
In contrast with:
Another lame backpedal attempt by you fraud pseudoscience AGW clowns.
The wave equations used to develop the probability predictions are deterministic. They may give probability predictions, but they are based upon mathematical formulas that have precise inputs.
I can use Maxwell’s equations to predict the “ideal” field strength of an electromagnetic field based upon well-defined inputs. These are deterministic. It is science.
The problem with declaring that you can curve fit and predict the future is that you are using constantly varying coefficients and not constants. That is what makes correlation fail in prediction. It is why climate models fail. They are not deterministic.
Can I infer from today that tomorrow will be the same? I can, but it is not deterministic because there is a large chance that tomorrow won’t be like today.
Did your “model” predict December’s UAH value? If not, why not?
I tried to tell him this, but instead he launched into a bizarre tirade in which it is obvious he thinks photons are bullets.
It’s the viewpoint of lots of climate scientists. The earth “shoots” photon bullets at the CO2 molecules in the atmosphere. Most of them have apparently never even heard of the inverse-square law. Energy/m^2 radiated from a point source on the earth goes down by the inverse square law so that the intensity changes by the square of the distance at which it is received. If that energy is then re-radiated back to the earth it will see a decrease by 2 x (1/distance^2) (path loss out + path loss in) when it reaches the earth again. Even if all the back radiation occurs at 2m in height it will see a decrease of 1/(2^2) going out and another 1/(2^2) coming back. That’s actually a path loss of 1/distance^4.
I’ve really never seen any of this reflected in any analysis of back radiation. Like so much other things it’s just ignored.
TG said: “Most of them have apparently never even heard of the inverse-square law.”
Your hubris shows no boundaries. The concept you are looking for is called view factors. For example, the view factor from surface to TOA is 4π*(6378000 m)^2 / 4π*(6448000 m^2) = 0.9784 and the view factor from TOA to the surface is 4π*(6448000 m^2) / 4π*(6378000 m)^2 = 1.022. Everybody knows about this.
TG said: “ If that energy is then re-radiated back to the earth it will see a decrease by 2 x (1/distance^2) (path loss out + path loss in) when it reaches the earth again. Even if all the back radiation occurs at 2m in height it will see a decrease of 1/(2^2) going out and another 1/(2^2) coming back. That’s actually a path loss of 1/distance^4.”
That is a violation of the 1LOT. If X W/m2 goes from the surface to TOA then TOA would received 0.9784*X W/m2. Then if all of that energy got reradiated back to the surface then the surface would received 1.022*(0.9784*X) W/m2 = X W/m2. A similar calculation can be made for the 2m case as well with the view factors being much closer to 1.
Of course, you don’t think arithmetic can be performed on intensive properties so you won’t accept any of this and continue to go around telling people the law of conservation of energy is bogus. I’m just making it known to the lurkers how things actually work.
TG said: “I’ve really never seen any of this reflected in any analysis of back radiation. Like so much other things it’s just ignored.”
Not only do scientists know about view factors (they created the concept afterall), but climate scientists know about the rectification effect that occurs when spatial averages of temperatures or radiant exitance are used in the SB law.
Of course, you don’t think the SB law is meaningful so it performs arithmetic on an intensive properties and since you only think it works when the body is equilibrium with its surroundings. I’m just making it known to the lurkers how things actually work.
And BTW, because you are the king of making up strawmen it is prudent that I make it clear what I didn’t say. I didn’t say all of the radiation from the surface gets reradiated back to the surface in the real world. It doesn’t. I didn’t say all of the radiation from the surface gets absorbed at 2m. It doesn’t. I didn’t say the radiation from a spot on the surface only spreads out by a factor of 0.9784 at TOA or that it focuses by 1.022 from TOA to the surface. It’s doesn’t. That’s not how view factors work. I didn’t say the real atmosphere can be modeled effectively with only two layers. It can’t. There is an infinite number of strawman you can construct and it’s impossible preempt all of them. I’m just going to nip the strawman thing in bud right now. I’m only addressing your assertion that energy disappears in violation of the 1LOT under the non-real scenario where all of the energy coming from the surface gets reradiated back to the surface. Remember, it’s your scenario. If there are any criticisms of it don’t pin them on me.
Says the King of Hubris…
karlomonte said: “Says the King of Hubris…”
Defending the 1st Law of Thermodynamics is not hubris.
Defending the Stefan-Boltzmann Law is not hubris.
And yes. I will absolutely defend the 1LOT and the SB law. They are unassailable laws of physics whether you, Tim Gorman, Jim Gorman, etc. agree with them or not . You call it hubris. I call it unfalsified science.
Hey clown, you forgot “QM” and “decohere” in this rant.
HTH
You are right, all the 1D energy balance diagrams do this; reality is infinitely more complex. As JCM says, they also ignore convection and lateral transport.
“The interesting thing about this scenario is that it is a case where there is a functional relationship derived analytically without making statistical inferences. Yet that functional relationship is non-deterministic (like most relationships in QM).”
“If you disagree then I challenge you to publish your findings that you can predict deterministically where each and every photon will decohere on the backdrop”
Functional relationships are *NOT* non-deterministic. A functional relationship *would* allow you to determine where each photon would hit.
You keep describing a “statistical function” while calling it “functional relationship”.
It doesn’t matter if that statistical function is determine empirically or by using statistical math. It’s still a statistical relationship.
And if you can’t *see* that backdrop then how to you empirically determine the statistical function?
You keep trying to use examples where you can *see* what is happening as proof that you know what is happening in a closed box.
You can’t see inside an uncertainty interval to know what is happening. It is a CLOSED BOX.
JG said: “The wave equations used to develop the probability predictions are deterministic.”
Probabilistic predictions are not deterministic.
TG said: “Functional relationships are *NOT* non-deterministic.”
It is sounding like you and JG have different definitions of “functional” and “deterministic” than the rest of us.
TG said: “You keep describing a “statistical function” while calling it “functional relationship”.”
Yep.
TG said: “t doesn’t matter if that statistical function is determine empirically or by using statistical math. It’s still a statistical relationship.”
Yep. That’s why QM is inherently probabilistic and non-deterministic.
The question still remains…does it make predictions and is it based on science?
I say yes to both. Pat Frank says no. And based on your and JG’s kneejerk reaction to nuh-uh everything I say I assume you and JG don’t think QM makes predictions or is based on science either.
TG said: “You keep trying to use examples where you can *see* what is happening as proof that you know what is happening in a closed box.”
That’s science. You make a statement about a future state of the box and then you look inside it to see your predictions faired.
It’s the same with Kip’s two dice example. We can make a prediction about what the next toss will result in, but to assess the skill of that prediction we must actually perform the toss.
More religious pseudoscience noise…another demonstration of the truth of Unskilled And Unaware.
Still think photons are bullets?
I never said photons were bullets. You said that. Don’t pin your comments on me.
Liar, and a lame backpedal. Here is what I wrote:
And in response you generated a bunch of nonsense noise about “photon positions” (and deftly demonstrating you have not clue one about wave functions):
Which I simplified into a bullet analogy. Embrace your kook ideas, don’t run away from them.
There’s no backpedaling from me. I stand by what I said. That is, the claims that 1) science does not make predictions, 2) if you are making predictions then you aren’t doing science, 3) if you use statistical inference then you aren’t doing science, 4) quantum mechanics is completely deterministic and my favorite 5) superstition is a viable alternative to science when the goal is prediction are absurd.
And what you wrote is deflection and diversion. The existence of the wave function does not imply that QM is “completely deterministic”. Nor does it mean that QM does not make predictions or that it isn’t science.
And If you think the bullet analogy is a kook idea then you probably shouldn’t have mentioned it. Remember, that was your idea. You and you alone own it. I’m not going to bullied into embracing your “religious pseudoscience noise” ideas.
And my point still stands. You can predict non-deterministic outcomes like what happens in the many QM processes or in Kip’s dice toss scenario. Just because they are non-deterministic and inherently probabilistic does not mean that they cannot be predicted or that doing so is anti-science.
More trendology pseudoscience noise ignored.
Of course he does. He doesn’t know that shot noise is. How a transistor works. He obviously doesn’t know what the inverse-square law is either!
“Pat Frank says no.”
Really? Frank says QM is invalid. Do you have a quote to back that up?
Pat Frank didn’t say QM was invalid. He said it was “completely deterministic”. He also said initially that QM does make predictions, which I agree with. But then contradicts that later by saying “not in any scientific sense” when the context of the prediction is for non-deterministic outcomes. He also said in the context of science that “inference has place” and that “The rest of your comment is about statistical inference. Not science.” So it appears Pat Frank only thinks QM makes predictions and is scientific because he also thinks it is completely deterministic and does not use statistical techniques. I gave him plenty of opportunities to clarify his position. No further clarification was given. [here]
Is your position that any discipline with non-determinism is not scientific as well? Is it your position that QM is “completely deterministic” as well? Is it your position that prediction is not a purpose of science as well? Is it your position that superstition is a viable alternative to science when the goal is prediction as well? Is it your position that statistical inference is not scientific as well?
If no to all of the above then we agree and your comments here are misplaced and should be directed toward Pat Frank, Kip Hansen, JCM, and the others who hold those positions. Everyone already knows I think they’re all absurd.
More word salad, unreadable.
I’ll ask again – have you ever heard the term “shot noise”?
Do you have even the faintest clue as to how a transistor works?
“However, the best prediction of the next hit is where the PDF peaks.”
Then you should be on 7 every time in a casino since it comes up the most often, right?
Don’t punk out and say you wouldn’t gamble in a casino. Yes, they use statistics to determine how to set the rules. But they *DON’T* use statistics to change how often 7 comes up, they can’t do that and remain an honest game.
TG said: “I’ll ask again – have you ever heard the term “shot noise”?”
No.
TG said: “Do you have even the faintest clue as to how a transistor works?”
No.
TG said: “Then you should be on 7 every time in a casino since it comes up the most often, right?”
If the goal is to predict to the value of the next toss then yes.
But that may not be an optimal goal. When you introduce money into the game you have to consider expected return. For example, if 7 has an expected return -$100, 6 and 8 have an expected return of -$50, and all others have an expected return -$75 then to maximize your position and minimize the house’s position (which is still losing proposition BTW) you would select 6 or 8 even though 7 is the most likely result.
TG said: “Don’t punk out and say you wouldn’t gamble in a casino.”
You can call me a punk all you want. I’m still not going to go to casino where the rules of the game necessarily mean I have a negative expected return. And based on my prediction that playing will result in the loss of money I would optimally choose not to play since that’s my right.
TG said: “But they *DON’T* use statistics to change how often 7 comes up, they can’t do that and remain an honest game.”
Strawman. I didn’t say they did. I said they set the buy-in and payout values in such a manner that they have a positive expected return and you have a negative expected return.
“No.”
I didn’t think so or you wouldn’t be making the assertions you are.
“No.”
I didn’t think so or you wouldn’t be making the assertions you are.
You live in a statistical dimension that doesn’t seem to have many congruent points in common with the real world the rest of us live in. You have lots of company, however. Several of them show up here making the same claims as you.
I just love your word salad trying to rationalize why you wouldn’t bet on the most probable outcome. It’s people like you that keep the casinos open.
At the crap table what you win is based on what you and the others bet, not on an adjusted payout based on what you rolled. The house makes its cut on the other fools, not on the winner. The house cut is based on the 7 being the most common roll – exactly as you said.
Again, you are trying to rationalize your position and failing.
TG said: “I didn’t think so or you wouldn’t be making the assertions you are.”
TG said: “I didn’t think so or you wouldn’t be making the assertions you are.”
My assertions are that
1) QM is not “completely deterministic”.
2) QM makes predictions anyway.
3) QM is based on science.
I’m certainly no expert on QM, but I don’t need to be. I’m confident with these assertions nonetheless.
Is it correct that you disagree with all 3?
TG said: “At the crap table”
Nobody said anything about a craps table.
TG said: “Again, you are trying to rationalize your position and failing.”
My position is that if you want to minimize the error of a prediction on the next toss of two dice then your prediction should be 7.
Is it correct that you disagree with this position?
This is just more noise, why in the world would want to a “minimization” like this?
Obviously you’ve never stepped up to a real Craps table and put your money down. I can assure you that neither the players nor the dealer are ever thinking nonsense like this. They are NOT making predictions of the next roll, unless they are tourists throwing money away on single rolls the Big 6 & 8.
And yapping about “error” here is another indication that just like bellcurvewhinerman, you still don’t understand that uncertainty is not error.
You call it noise. The math calls it an RMSE of 2.4 which is the lowest error among the errors of the 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 predictions.
I’ve never played craps. I never intend to. And I don’t care because Kip didn’t say anything about it craps. But I can tell you that any competent casino is making predictions about their expected return prior to implementing any game of chance so your statement is absurd.
I’m “yapping” about error because that’s what inevitably happens when you make a prediction regarding a non-deterministic outcome whether it be predicting the next toss of dice, the spot where a photon/electron/etc will decohere on a background behind two slits, the amount of time it takes before a neutron decays spontaneously, etc.
Either you can’t read or this is just more distraction noise. I can’t tell the difference.
And WTF does “decohere” mean anyway? This isn’t even a word. Just like bellcurvewhinerman, you fan from the hip and hope you hit something.
karlomonte said: “And WTF does “decohere” mean anyway?”
https://en.wikipedia.org/wiki/Quantum_decoherence
Hah! The noise is the nonsense you generate with your keyboard, learn to read, Hubris King.
There is no RMSE on a roll of the dice. Look up probabilities for a roll of a dice. It is just like a coin. Each side has a 1/6 chance of coming up on each and every roll. What you did before and what you do after matters not one iota. When you roll two dice, each one of them has a sum total probability of 1, and each side of each dice 1/6 of coming up.
They are independent and mutually exclusive.
The house rules are what controls the winning/losing. One the first roll 7/11 win and 2/3/12 means everyone loses. Anything else is “point” and the next roll must be that same number. You figure it out. You roll an nine, you lose if you roll a 7. The losing combinations far outnumber the winning combinations.
+100
Get a room.
“Yes it will. Using inference I know that 7 is the most likely outcome and that values above and below that have diminishing likelihood.”
So what? Do you *ever* bother to look at a weather forecast? What does it give for rain “predictions”? What does the “prediction” mean?
You still aren’t addressing the fact that functional relationships are determinative. Weather forecasts are not, not for temperature, not for rain, not for wind, not even for humidity or pressure.
“No. They would hate me because I would minimize my loss and their gain.”
Exactly how would you do that? By betting on an outcome with a SMALLER probability of happening? The excuse that you wouldn’t play is just a tacit admission that you can NOT predict the next outcome. Word it using any word salad you want, it just boils down to your assertion that you can know the next outcome is wrong and you know it.
“The obvious counter example here is quantum mechanics.”
Do you think most of us have no basic understanding of quantum mechanics? Quantum mechanics lay at the base of how the venerable transistor works. It allows the amount of electrons that will tunnel through the energy barrier at the junctions to be estimated. But it can’t tell you WHICH ones will and it can’t even be specific in the number which do. Does the term “shot noise” mean anything to you?
You are as bad as bellman at trying to cherry pick stuff to throw against the wall while not even understanding what you are throwing!
1000% correct.
The pseudoscience clowns won’t let this one stand, Jim, this is my prediction of the future.
They have no interest in truth, all they do is skim for loopholes to support their a priori CO2 control knob temperature rise religion.
“It also allows you to predict the next toss.”
I think you need to define “predict” there. If the toss is random you can’t predict what the next toss will be, but you can say what the probability of a specific result will be. But to do that you need the prediction interval not the confidence interval. What the CLT allows you to do is predict the likely range of the long term average.
To predict is to make a statement about a future event. Using inference we can say that 7 is the most likely result of a toss of two dice. Therefore the best prediction for the next toss would be 7. And sure enough a monte carlo simulation shows an RMSE of about 2.4. The worst prediction would be 2 or 12 which results in an RMSE of about 5.5. Therefore if you were tasked with predicting the next toss the most skillful prediction would be 7.
You have hijacked my statement. SCIENCE PREDICTS the next event through a functional relationship. STATISTICS PREDICTS a pattern based on probability.
Do you know what independent and mutually exclusive means? It means there is a 1/6th probability of any number on each die. There is a 1/6th probability of getting a 1 on one die and a 1/6th probability of getting a 6 on the other die each time they are rolled.. The same happens on the next roll. Your “inference” is not worth spit when you go to actually play. Heck just go to the roulette table and bet on red/black based on the previous spin. If the previous was red, the next has to be black, right?
“But I do know what the probability of any number is.”
How do you know that? Once again, you fall back into the old meme of all uncertainty is random and cancels. You say you don’t but it shows up EVERY SINGLE TIME!
If *any* of the dice has a systematic bias, i.e. a loaded dice, you can’t possibly know what the probability of any number is because you don’t know the loading! Is it loaded to give more one’s? More five’s?
It is a statisticians blind spot that you and they just can’t ever seem to break out of!
All you can know is what the range of possible values is. Period. Exclamation point. It is a CLOSED BOX. The term uncertainty means YOU DON’T KNOW!
You have never, NOT ONCE, read the GUM for meaning. NEVER! All you know is how to cherry pick stuff you think might prove someone wrong – with absolutely no understanding of what you are posting.
From the GUM:
3.2.1 In general, a measurement has imperfections that give rise to an error (B.2.19) in the measurement
result. Traditionally, an error is viewed as having two components, namely, a random (B.2.21) component
and a systematic (B.2.22) component.
NOTE Error is an idealized concept and errors cannot be known exactly.
3.2.2 Random error presumably arises from unpredictable or stochastic temporal and spatial variations of influence quantities. The effects of such variations, hereafter termed random effects, give rise to variations in repeated observations of the measurand. Although it is not possible to compensate for the random error of a measurement result, it can usually be reduced by increasing the number of observations; its expectation or expected value (C.2.9, C.3.1) is zero.
NOTE 1 The experimental standard deviation of the arithmetic mean or average of a series of observations (see 4.2.3) is not the random error of the mean, although it is so designated in some publications. It is instead a measure of the
uncertainty of the mean due to random effects. The exact value of the error in the mean arising from these effects cannot be known.
(all bolding mine, tpg)
CANNOT BE KNOWN!
That’s called UNCERTAINTY! It’s a CLOSED BOX!
An analogy that I think is apt:
A person is convinced that it is possible to travel faster than the speed of light, based on lights-in-the-sky, ufology, whatever, and thinks the physicists who say otherwise are wrong.
The person then decides he is going to prove the physicists are wrong, even though he has no formal training in the subject. He proceeds to get a copy of Halliday & Resnick and starts going through it, looking for loopholes to get around the limits of relativity.
Is this person going to learn physics this way?
No, he won’t take the time to work through the problem sets, which is where the real learning happens.
This is exactly what these CO2-driven global warming people who haunt CMoB are doing with uncertainty—looking for loopholes, anywhere, that will allow them to continue to claim these impossibly tiny “error bars”, and along the way attack anyone who dares to go against their religious pseudoscience wth statistical smokescreens that only serve to confuse and hide the real issues.
This is not an objective search for truth.
Wow. That’s a blast from the past – brings back all sorts of memories.
Still available and in-print!
https://www.amazon.com/Fundamentals-Physics-David-Halliday/dp/0471216437
My copy has two volumes. I think it was set up to be a two semester textbook.
“No, he won’t take the time to work through the problem sets, which is where the real learning happens.”
You nailed it.
Let’s bring this back to temperature measurements. Where are the measurements that define a probability function for a temperature measurement?
There are none! What Kip has tried to do is show a method of determining an uncertainty interval experimentally. Trying to make an argument against it needs refutation using the same basis, not launching into some diatribe about 100’s of dice and millions of experiments.
Here is what the GUM says :
“””2.2.3 The formal definition of the term “uncertainty of measurement” developed for use in this Guide and in the VIM [6] (VIM:1993, definition 3.9) is as follows:
uncertainty (of measurement)
parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand
NOTE 1 The parameter may be, for example, a standard deviation (or a given multiple of it), or the half-width of an interval having a stated level of confidence.
NOTE 2 Uncertainty of measurement comprises, in general, many components. Some of these components may be evaluated from the statistical distribution of the results of series of measurements and can be characterized by experimental standard deviations. The other components, which also can be characterized by standard deviations, are evaluated from assumed probability distributions based on experience or other information.
NOTE 3 It is understood that the result of the measurement is the best estimate of the value of the measurand, and that all components of uncertainty, including those arising from systematic effects, such as components associated with corrections and reference standards, contribute to the dispersion. “”””
Look at Note 1 carefully. “The parameter MAY BE, … , a standard deviation, … , or the half-width of an interval, …, ”
Look at Note 2 carefully. “…, evaluated from ASSUMED probability DISTRIBUTIONS BASED ON experience or OTHER INFORMATION. ”
I have outlined before what NOAA and NWS consider the uncertainty intervals. If you need them I can certainly post them again. Those should be used with temperature measurements. You will not like them.
As to the LLN and CLT, they do not apply to single measurements as there is no distribution to use to determine a measurement uncertainty.
The weak,and strong LLN really do not apply to temp averages since they both assume Independent and Identical Distributions (IID) samples of a population.
“What Kip has tried to do is show a method of determining an uncertainty interval experimentally.”
What experiment? He’s throwing dice inside a box and saying no one is allowed to look at the result.
“launching into some diatribe about 100’s of dice”
But his claim is meant to work for summing or averaging 100s of measurements. How is it misleading to perform an experiment that is closer to that.
“Look at Note 1 carefully. “The parameter MAY BE, … , a standard deviation, … , or the half-width of an interval, …, ””
Which contradicts Kip’s claim that the parameter MUST NOT BE a standard deviation.
“Look at Note 2 carefully. “…, evaluated from ASSUMED probability DISTRIBUTIONS BASED ON experience or OTHER INFORMATION. ””
Which answers your opening question “Where are the measurements that define a probability function for a temperature measurement?” You don’t need measurements, you can just assume the distribution.
“What experiment? He’s throwing dice inside a box and saying no one is allowed to look at the result.”
That’s called an uncertainty interval! You can’t see inside the uncertainty interval. IT’S A CLOSED BOX!
“But his claim is meant to work for summing or averaging 100s of measurements. How is it misleading to perform an experiment that is closer to that.”
Kip’s claim will work for any number of dice. Your experiment was ill-formed, you didn’t make enough rolls. It’s outcome was misleading because the experiment was ill-formed.
This measurement is interesting in a way, because the possible uncertainty range should not have a normal distribution. The experiment is done in a specified way and the only variable should be the instrument measuring uncertainty. However all the instruments are calibrated using the same standard, and unless they all drift around across a normal distribution, should all measure pretty much the central calibrated value. I would expect the measured value for e_m to be a very narrow distribution, but the possible error range to be larger because of the absolute accuracy of the calibration standard. These errors certainly do not add in quadrature.
The problem is that complex measurement such as m_e typically involve lots of separate measurements that are then combined for the final result. For example, manufacturers of digital voltmeters give specifications for “error” bands that must be converted to uncertainty. There is no probability for these, and the JCGM standards for uncertainty expression tells how to combine the individual elements.
davezawadi ==> I think you are referring to the m_e measurement? If so, it is not given with an absolute measurement uncertainty, but rather with a Standard Uncertainty — which is a different animal.
karlomonte ==> m_e is a measured constant, but the “uncertainty” listed in the table is not “absolute measurement uncertainty” is is rather a statistical construct called “Standard uncertainty” for which Bellman supplies the definition in this thread.
Yet I still submit that there is really no probability distribution that can be attached to it. As Tim Gorman says, the probability is 1 at the (unknowable) true value and 0 everywhere else in the interval. NIST does not make 100 billion independent measurements of m_e and average them to get the tiny uncertainty.
karlomonte ==> “no probability distribution that can be attached to it” — and I would agree.
Depends on whether you are using classical or Bayesian statistics.
In classical terms, you are correct the probability of the true value being within a given range is either 1 or 0, and you don’t know which. That’s why you talk in terms of how likely is it you would get the result you did for a specific value.
In Bayesian terms, you can talk about the probability of the true value being within a given range because the probability is based on your state of knowledge.
Not what I (i.e. Tim) wrote, try again…
It is what you wrote, and I can;t help you if you don’t understand the implications.
And now you retreat back to gaslighting.
The only one that doesn’t understand is you. There isn’t any difference that I can see between operational measurement uncertainty and Bayesian theory applied to measurement uncertainty. They both describe how sure you are that the true value is inside the uncertainty interval but neither describe an actual probability distribution for the values inside the interval.
You are cherry picking again. Trying to throw some crap against the wall hoping something will stick so you can prove someone wrong.
You’d be correct that my understanding of Bayesian statistics is quite limited, but that doesn’t mean I’m cherry picking.
There isn’t a difference between the operation of the two, ignoring priors, but I was talking about the differences conceptually regarding probability. It was in response to you saying that the probability of the true value being a specific point was either 1 or 0.
Classical statistics does not describe the probability that the true value lies in a specific range, Bayesian does because they use different concepts of probability. And Bayesian does describe a probability distribution of the value.
As usual, he’s cherry picking stuff he doesn’t understand.
“In Bayesian terms, you can talk about the probability of the true value being within a given range because the probability is based on your state of knowledge.”
It looks like you are cherry picking again without actually understanding what you are posting.
From the NIST:
“The theoretical Bayesian interpretation of π(Θ) is
that it describes the probability that the true
value τ[Θ] lies within the interval (θl, θh).”
It doesn’t give the probability distribution of the values in the interval, it just describes the probability that the true value lies within the interval.
So it really doesn’t matter if you are using the classical definition of measurement uncertainty intervals or Bayesian interpretations.
“It looks like you are cherry picking again without actually understanding what you are posting.”
When in Rome…
“It doesn’t give the probability distribution of the values in the interval, it just describes the probability that the true value lies within the interval.”
Try think about what you are saying. If you know the probability that the true value lies in a specific interval, you can know what the probability is for any given interval. You can make those intervals as small as you like. You can build a probability distribution.
If you attach a probability distribution to the uncertainty interval then that implies you know which value in the uncertainty interval is most likely to be the true value. How does that imply uncertainty?
It means you know the unknowable true value!
Bellman knows!
Hah!
That the mass is a (approximately known) constant is an assumption, no? We (relatively) assume the mass of all electrons are the same, but are they? One might assume there are differences in different frames.
Yes I’m pretty sure m_e is the electron rest mass and the Uniformity Principle states that all electrons are identical.
A statement can be made but that it is true is an assumption. I’m not asserting it isn’t true, only that its truth is just a matter of faith,
It is, but it only takes two unlike electrons to falsify the assumption. So far this hasn’t happened TMK (and the QM/Heisenberg uncertainty principle effects this of course). For example, free electrons in a vacuum all respond the same ways to electric and magnetic fields.
If they have different masses then wouldn’t that falsify Gauss’ Law?
I haven’t read anything that says that.
I think you are correct; there is also the issue of u_0 and e_0 which are intimately connected to the speed of light.
Something that causes me problems is this scenario,
Two cars speedometers specified at +/-10%, following one another.
If the first car has a speed reading of 30mph what is the range readings possible in the following car? (discounting other possible errors)
My calculations is the range of readings are between 24.5 and 36.3 mph.
Over reading 27,2727 by 10% gives 30mph in first car, under reading 27,2727 by 10% gives a reading of 24.54mph in the following car. The high read out in the second car calculated in a similar way
The cars are travelling at the same speed which will be between, 27,2727 and 33.3333mph.
I think those numbers are right but I am confused by this sort of thing
I too am a bit confused, but I understand that Kip is doing his best to put our confusion right.
Anyway, in your example it’s a “certainty” that the supposedly “accurate” (but politically biased) speed camera beside the road will snap both of them for “speeding!”
Ben ==> “specified at +/-10%” if so, we are not talking about absolute measurement uncertainty anymore, but some other kind of uncertainty.
Those kinds of relative uncertainties have be be worked outin arithmatic as you have done. I don’t blame you for being confused — if it was easy, Briggs, a professional statistician, would not have written a 277 page book “Uncertainty: The Soul of Modeling, Probability & Statistics“. He would have just expained it in a 2,000 word blog essay.
We could work out the speeds of your cars with a little more information. Are the cars travelling at exactly the same speed (distance over ground over time)? are you trying to figure out what the speedometer on the following car will show? (If so, why?)
Your calculation of the range of possible true speeds for the first car gives you the same for the following car (if travelling at the same speed).
Kip, maybe you don’t know that since the 1930s, British speed limits have “legally” allowed a “tolerance” of 10%, due to the limitations of speedometers at that time.
And if in Ben’s scenario the first car slows, the second driver (if he is alert) will do so too. So, Ben has actually understated his case. It’s not impossible that the second car will have stopped altogether.
Neil ==> Ah ha! The light is beginning to shine….this has something do to with traffic speeding laws in the rather semi-enlightened UK. It culd befirged out, arithmetically…but don’t get why we would do so. Ya can’t get out of a speeding ticket by saying “I was just going the speed of the guy in front of me.”
My advice — Just pay the fine and leave your brain for better pursuits.
You could get a speedometer calibration certification for some amount of money. Maybe the court would accept it if it showed you were not ‘speeding’. If so, you might save money in the longer run because the insurance company might not raise your insurance rate.
Around 2002 I saw calibration checks being performed in a major factory for making car speedometers. Final acceptance check was to insert an electric drill special bit into the mechanical speedo drive, then run the drill at full speed, then read the speedo dial to see if you get expected value.
The drill motor was driven by mains ac electricity, with enough poles to give 1,440 revs per minute.
Now, in 2023 with so much poor quality electricity from wind and solar, I expect that a less convenient frequency source has been found. Geoff S
Kip Thank you for the reply
This is based on an experience when I was a teenager. During school holidays I used to go out with my dad who was a sales representive for a vetinary company. On one trip we/he was stopped by police for breaking the speed limit going through Kinlochleven. This was pre Gatso and radar speed traps so it was one speedometer against another.
As Neil Lock says in the UK speedometers had, and still have, a +/-10% tolerance. I think most read slightly on the plus side. So my question relates to a police car following and “booking” a driver who is given a ticket for exceeding the 30mph speed limit by 6mph when his legal speedometer did not indicate a breech of the limit. In recent years the authorities are less tolerant of infractions, exceeding by 10%+2mph very likely gets you a fine and points. So 35mph in my scenario.
I have not researched what methods traffic police use to measure speeds of vehicles and if they are more accurate, with all the other variables involved, tyre wear, digital readouts in 1mph units and the rest.
In my dad’s case he escaped with a ticking off, he probably had broken the speed limit as he had to have a new speedometer after using the method of following someone driving at a constant indicated 30mph.
But although I still have problems with this kind of I find it fascinating.
In NSW, the police used chronometric speedometers in their vehicles, with frequent calibration. Monthly, I think. There was still the problem of keeping station with the vehicle in front, and I think there was a requirement for doing this for some minimum distance.
“Your Honor, I ran up the backside of the car in front because I was staring intently at my speedo trying to determine if the needle was on 27.2727 or 33.3333”
See my reply to Kip above for an explanation
The Empirical Rule in statistics says that for a normal distribution, one can expect approximately 68% of the measurements to lie within one SD, while approximately 95% will lie within two SD.
When a speedometer is calibrated, what is returned is a table of indicated speeds versus the true speed. The percentage error usually varies for each speed. In a well-designed speedometer, there will be a speed that is almost exactly right, commonly for a critical speed such as the highest legal speed.
I work in Electrical Engineering, and utilize the variation/uncertainty of component values and performance all of the time. We use Monte Carlo analysis of the various effects of the variation of the components to give a PDF of the actual predicted/simulated performance of the system. Most of the time we assume a uniform distribution of the components to make sure that we are robust to the margins of the individual components modeled, and this method works well in predicting production variation performance. We do get a mean and sd from the output, and by looking at the distribution we can tell if we will meet our specification.
From a practical standpoint we never get less variation in the system, and the individual components are only weighted in their effects to the final values. If a voltage divider has a 5% resistor and a 10% resistor, we use both percentages against their respective values and then calculate the resulting error in respect to the nominal value. These are additive in nature and we will never reduce the variation through truncation, or averaging.
Most of my Monte Carlo analysis runs simulate 1M systems built with each part being a random variable of mean x variation +/- y% and with a uniform distribution. In practical terms this very accurately simulates the resultant real world builds with some design margin.
From an analytical standpoint, I have never understood how we could have a more accurate value of a temperature using averages, than the base measurements used to obtain it. It does not work this way in the real world. If you measure the same thing 10 times and there no changes to the parameter being measured you then know your measurement variation/error. If you measure something 10 times and the parameter changes, given that you know your measurement error, then you know the variation and change over time. That number will never be more accurate than the basic measurement error no matter how many times you repeat it.
rpercifield ==> Yes, this whole presentation rests on understanding what “original absolute measurement uncertainty” is when it gets out of bed in the morning. (I think you are using the term ‘basic measurement error’, by which I believe you mean the same thing).
Hello Kip,
I basically have two types of error/variation in my systems. One is the original in my missive which is the inherent variation of the parts I use in my design, and the measurement error contained within the various instruments we use to determine what the value of a particular parameter.
Being in the manufacturing industry, we always perform a “Measurement System Evaluation” (MSE). This allows us to know how the equipment, and operator affect the measurement variation. Thus, within our processes we know the limits of what we can resolve, and how much that contributes to the total system variation. Most of the time our measurement system variation is significantly less than the system’s inherent variation, however we never take that for granted, and reflect the measurement system error in the total performance of the system evaluated. We also do not allow for greater resolution just because you average something.
One of the strong points of our organization is the everyone who has input into the design, testing, and manufacturing of our business is trained and taught the same way of evaluation of systems. This means that an engineer to management conversation about the analysis of the system at hand has the vernacular, and expectations.
I have enjoyed your pieces greatly and am in agreement with your assessments of the current state of climate and modeling schemes. Thanks for the different way of presenting reality.
rpercifield ==> Thank you, nice to have someone agree!
Thanks for sharing your in-the-field experience.
rpercifield said: “From an analytical standpoint, I have never understood how we could have a more accurate value of a temperature using averages, than the base measurements used to obtain it.”
The analytical explanation comes from the law of propagation of uncertainty which is defined in JCGM 100:2008 equation E.3. Using the idealized non-correlated form as per equation 10 we have u(f) = Σ[(∂f/∂x_i)^2 * u(x_i)^2, 1, N]. In this equation f is the measurement model function and x_i are the inputs into that function.
The partial derivative ∂f/∂x_i is crucial in the analytical explanation. When ∂f/∂x >= 1/sqrt(N) then uncertainty increases as a result of the arithmetic encapsulated by the function f. When ∂f/∂x < 1/sqrt(N) then uncertainty decreases as a result of the arithmetic encapsulated by the function.
When the measurement model is f(x_1, …, x_N) = Σ[x_i, 1, N] / N then ∂f/∂x_i = 1/N for all x_i. And because 1/N < 1/sqrt(N) then it necessarily follows that u(f) < u(x_i) for all x_i.
The meaning is very deep and requires a lot of understanding in multivariant calculus But that is the analytical explanation for why the uncertainty of the average is less than the uncertainty of the individual components that went into the average.
rpercifield said: “It does not work this way in the real world.”
Yes. It does. I encourage you to verify this yourself using a Monte Carlo simulation. Or as a convenience the NIST uncertainty machine will do both the deterministic JGCM 100:2008 (GUM) method and the Monte Carlo method for you.
rpercifield said: “If you measure the same thing 10 times and there no changes to the parameter being measured you then know your measurement variation/error. If you measure something 10 times and the parameter changes, given that you know your measurement error, then you know the variation and change over time. That number will never be more accurate than the basic measurement error no matter how many times you repeat it.”
I’m not sure what you mean exactly. I will say that if your measurement model is f(x_1, …, x_10) = (x_1 + … + x_10) / 10 then it is necessarily the case that u(f) < u(x_i) for all x_i when those x_i are uncorrelated. You said you have experience with Monte Carlo simulations. I encourage you to prove this out using a simulation. Or as I said above the NIST uncertainty machine will do the simulation for you.
The bgw bot is stuck in a loop, again…
“I encourage you to verify this yourself using a Monte Carlo simulation.”
An eminently practical suggestion. The freeware is available, and specialty MCS software is not required. Just some noggining with setting it up. And your laptop, whether bought for purpose or not, is probably more than adequate for the task.
I use OpenCalc, which is old enough that it has probably been superceded by better offerings. But if you want help, and have downloaded other freeware, I will do so as well and we’ll jack a a WUWT thread until the scales drop from your eyes…
Let’s do an experiment..
Round 1:
I have a voltage source that has an output of 10VDC. The variation is +/- 20mVDC with a normal distribution and +/-6sigma. Thus, each sigma is 20mV/6 or 3.3333mVDC. The distribution essentially is an average of 10VDC and the 6sigma tales are at +/-20mVDCfrom the mean (9.98VDC to 10.02VDC) . This could be represented as a random variable if necessary, but for our experiment not a requirement. Our measurement system is only able to resolve to 1V and the standard rounding rules apply, =>9.5VDC 10VDC <10.5VDC. Thus, the variation of the signal is less than the resolution of the measurement system. No matter how many measurements you make and average you will never see anything other than 10V. There is not way to get to this data given the limits of the system. This measurement system will not work.
Round 2:
Same as above except that except that we are now have variation at 200mVDC instead of 20mVDC. Given enough samples there may be an outlier greater than the 500mV to register a different value but this would be rare, and your average will provide no data as to the real variation.
Round 3:
We keep everything as in Round 2 and we add a 1% measurement error at 10VDC. That means that the limits for the result are now between 9.41 to 9.60 on the low end and 10.40 to 10.61 on the high side. this will affect the reading given, however this error is in addition to the variation of the voltage source. Have we learned anything about the source variation? probably not, for it is confounded with the measurement error. the probability of exceeding the trip limits for the changing of the digit are better but still very low.
I cannot tell you how many times I have resolved issues due to poor selection of measurement resolution. Yes, in some situations your method could be applied, I have done so. However, it has to be done in a system there it applies. In the production environment it is basically useless. I have not been convinced that it applies to an increase in resolution in the atmospheric sciences. In my post grad statistics classes any average looses data and without an understanding of variance of random variable means a loss of resolution. I perform digital signal processing all the time and can demonstrate that averages remove important signal data due to the nature of the method. While it can help it also can hurt by eliminating information. No matter how hard you try you always lose data in an average.
As an aside I have used dithering and the theorems you listed in you response. They have their place and applications. In my daily work however, we must know that our system is capable of resolving the variation of interest. This analog to digital conversion is critical in making sure that we see what is happening. As above without this resolution, we do not see anything with the averages. This type of real world situations happen all the time. People assume that the system is doing things without the proper resolution of measurement to make those determinations.
I personally use Mathcad in my Monte Carlo simulations at work. To do this I generate transfer functions that utilize Random Variables with a Mean and Distribution Type. These variables are nothing more than arrays, sometimes multidimensional according to the application, and the model is run with millions of values in the array. Our predictions are normally pretty close to actual performance.
My real point is this, if I can only resolve 1 digit, I do not have confidence that I really know what the true average is with a resolution greater than 1 digit. I am not measuring the same item multiple times, for the climate is a dynamic system. No two measurements are done in identical conditions. The unresolved variation underneath is missing from the measurement. To say that we can detect 0.001C changes on the system without knowing the resolution is not plausible in my mind. Just the measurement error is greater than the difference by orders of magnitude. We have also not included drift, and other errors in the system. Too little is known of the performance of all of the components that make up the system and should place large error bars upon any assessments that are far greater than the changes themselves.
rpercefield said: “No matter how many measurements you make and average you will never see anything other than 10V.”
Yep agreed…for that scenario. But that scenario does not sufficiently embody the discussion at hand. Consider the following scenario instead.
You want to know the average of sample of differing voltages. The measurement model is then Y = Σ[V_n, 1, N] / N where V_n is one of the output voltages being measured. Let’s say the measured voltages are 10, 8, 5, 12, and 13. Because the measurement is to the nearest integer it has ±0.5 V rectangular uncertainty.
Using the procedure in JCGM 100:2008 we need to first convert ±1 (rectangular) into a standard uncertainty. We’ll use the canonical 1/sqrt(3) multiplier so u(V_n) = 0.29 V. We’ll then apply equation 10 to get u(Y) = sqrt[5 * (1/5)^2 * 0.29^2]] = 0.13 V. Note that u(Y) = 0.13 V is less than u(V_n) = 0.29 V and less than ±0.5 V rectangular. Thus the average is 9.6 ± 0.13 V (k=1).
Here is the NIST uncertainty machine configuration you can use to verify this.
It is also important to note that with just 5 measurements of rectangular uncertainty the average has a nearly normal uncertainty inline with expectations from the central limit theorem.
The point…we started with individual measurement uncertainty of ±0.5 V rectangular and ended with a standard uncertainty of the average of 0.13 V.
You have access to the equipment. Do the experiment in real life. Measure 5 different voltages and record the average of the measurement and the average of as computed from the output of the voltage source. Repeat a sufficient number of times (50’ish should do it) so that you have a sufficient number of averages to compare. Compute the RMSE of the individual measurements and the averages. You will see that the RMSE of the averages is lower than the RMSE of the individual measurements.
That result is trying to tell you something.
And what does this number tell you?
Not much at all.
It tells you nothing. The populations are not even the same. It’s like trying to average the heights of goats with the heights of horses!
But sitting in the armchair, anything is possible!
“ Because the measurement is to the nearest integer it has ±0.5 V rectangular uncertainty.”
Because the measurement is to the nearest integer it has a MINIMUM of +/- 0.5V rectangular uncertainty.
Fixed it for you.
Excellent summaries, thanks, hopefully the armchair metrologists here will learn something from your experience.
+100!
You and your compatriots simply do not get that the average uncertainty, which is what you get when you divide the total uncertainty by the number of elements, is not the uncertainty of the average!
You look at the statistical average as being something completely separate from physical reality and yet also consider the average to be something that describes physical reality. It’s called cognitive dissonance.
When you take an average you are doing u(q/n). When you use the GUM formula ∂(q/n)/∂xi = (1/n). When you square 1/n you get 1/n^2. You can then extract the sqrt of (1/n)^2 and get (1/n). When you then divide the total uncertainty, u(q/n), you get the AVERAGE UNCERTAINTY. It is the value that, when multiplied by the number of elements (n), gives you back the total uncertainty. The AVERAGE UNCERTAINTY, however, is not the uncertainty of the average.
Again, if you have 50 boards (+/- 0.08′ uncertainty) and 50 boards (+/- 0.04′ uncertainty whose average length is 6′, THE AVERAGE UNCERTAINTY IS NOT THE UNCERTAINTY OF BOARDS BEING 6′ IN LENGTH. By definition, the physical measurement uncertainty of those 6 boards is either +/- 0.08′ or +/- 0.04′. The average measurement uncertainty is *NOT*, I repeat – IS NOT, the actual physical measurement uncertainty of the average.
If you could find someplace to actually measure the global average temperature it’s uncertainty would be the physical uncertainty of the measuring device. If you can’t physically measure it then its uncertainty becomes a function of the measurements made which you use to calculate it. It is *NOT* the total uncertainty divided by n in order to find an average value. It is a direct propagation of the element uncertainties.
When you add 10C +/- 0.5C with 15C +/- 0.3C the uncertainty of the sum is sqrt[ 0.5^2 + 0.3^2] = 0.58 as a minimum. It could easily be 0.8!. It is *NOT* (0.5 + 0.3)/2 = 0.4!
The big problem you have is shown in Taylor’s Eq, 3.18 and 3.46. You have to UNDERSTAND:
3.18. If q = x/w then u(q)/q = sqrt[ (u(x)/x)^2 + (u(w)/w)^2 ]
It is *NOT* u(q)/q = sqrt[ (u(x)/x)^2 + (u(w)/w)^2 ] / w!
3.46. if q = q(x,w) then u^2(q) = (∂q/∂x)^2(u(x)^2 + (∂q/∂w)^2(u(w)^2
In both cases you find the uncertainty term by term and since u(w) = 0 you get u(q) = u(x). If x is a sum of elements then u(x) = RSS u(x_i).
You want us to believe that
u^2(q) = (∂q/∂(x/w)^2 u(x)^2 + (∂q/∂(x/w)^2 u(w)^2
Since u(w) = 0 when w is a constant this reduces to
u^2(q) = (∂q/^(x/w)^2 u(x)^2
==> u^2(q) = u(x)^2/ w^2 this reduces to u(q) = avg uncertainty = u(x)/w
Once again, the average uncertainty is *NOT* the uncertainty of the average.
You can *NOT* reduce uncertainty through averaging. CLT doesn’t help. All the CLT says is that you can get a better estimate of the population mean by increasing the size of your samples. That population mean tells you NOTHING about the distribution of the population itself. You can have a *very* skewed population distribution and the CLT will still generate a Gaussian distribution for the sample means. The issue is that the mean of a skewed distribution tells you almost nothing about the population distribution. The CLT does not allow you to determine kurtosis of skewness of the population distribution You can estimate the population standard deviation but again, in a skewed distribution the standard deviation is basically useless in describing the population. It’s why all five of the statistics textbooks I have say you should use something like the 5-number description of the skewed population and not the mean/standard deviation.
Bottom line? The global average temperature has to have an uncertainty at least as large as the uncertainty of the elements used to calculate global average temperature. That means that the uncertainty of the global average temperature will be at least +/- 0.5C and you simply cannot differentiate annual temperature differences smaller than that uncertainty interval. No more Year1 is 0.01C hotter than Year2. In reality the uncertainty of the global average temperature will be *much* higher than +/- 0.5C.
P.S. anomalies don’t help. Anomalies inherit the uncertainty of the elements used to calculate the anomaly.
TG said: “3.18. If q = x/w then u(q)/q = sqrt[ (u(x)/x)^2 + (u(w)/w)^2 ]
It is *NOT* u(q)/q = sqrt[ (u(x)/x)^2 + (u(w)/w)^2 ] / w!”
ALGEBRA MISTAKE #1: Neither Taylor nor I ever said u(q)/q = sqrt[ (u(x)/x)^2 + (u(w)/w)^2 ] / w.
TG said: “3.46. if q = q(x,w) then u^2(q) = (∂q/∂x)^2(u(x)^2 + (∂q/∂w)^2(u(w)^2
In both cases you find the uncertainty term by term and since u(w) = 0 you get u(q) = u(x).”
ALGEBRA MISTAKE #2: If q = q(x, w) = x/w and u(w) = 0 then u(q) = u(x) / w per Taylor 3.46.
TG said: “You want us to believe that
u^2(q) = (∂q/∂(x/w)^2 u(x)^2 + (∂q/∂(x/w)^2 u(w)^2″
ALGEBRA MISTAKE #3: ∂q/∂(x/w) != ∂q/∂x
ALGEBRA MISTAKE #4: ∂q/∂(x/w) != ∂q/∂w
TG said: “Since u(w) = 0 when w is a constant this reduces to
u^2(q) = (∂q/^(x/w)^2 u(x)^2 ==> u^2(q) = u(x)^2/ w^2“
ALGEBRA MISTAKE #5: ∂q/∂(x/w) != 1/w, it is actually just 1.
TG said: “reduces to u(q) = avg uncertainty = u(x)/w”
ALGEBRA MISTAKE #6: u(x) / w is not an average
MIRACLE #1: When q = x/w and u(w) = 0 then u(q) = u(x) / w.
Algebra mistakes #3 and #5 combine in just the right way to produce miracle #1.
Anyway, burn your miracle #1 into your brain. When you divide a value with uncertainty by a constant then the uncertainty of that value also gets divided by that same constant.
TG said: “You can *NOT* reduce uncertainty through averaging.”
ALGEBRA MISTAKE #7: That’s not what your miracle #1 says. u(x / w) = u(x) / w.
When x = a+b and w = 2 then u((a+b) / 2) = u(a+b) / 2. And we know from applying RSS that u(a+b) = sqrt(u(a)^2 + u(b)^2). So u((a+b) / 2) = sqrt[ u(a)^2 + u(b)^2 ] / 2. And in the case where u = u(a) = u(b) then u((a+b) / 2) = sqrt[2u] / 2 = u / sqrt(2).
TG said: “Bottom line?”
I want to end with this bottom line. I’m really not trying to be patronizing or offensive here, but this is a repeated problem. You keeping making arithmetic/algebra mistakes. I counted 7 mistakes in this post alone. I strongly advise that you forego doing algebra by hand and start having a computer algebra system do it for you.
Can you be any more snooty? I think not, you are at the acme.
Yeah, don’ja just hate it when those mean ol’ “snooties” find numerous math mistakes in your badly flawed assertions?
“The analytical explanation comes from the law of propagation of uncertainty which is defined in JCGM 100:2008 equation E.3. Using the idealized non-correlated form as per equation 10 we have u(f) = Σ[(∂f/∂x_i)^2 * u(x_i)^2, 1, N]. In this equation f is the measurement model function and x_i are the inputs into that function.”
This does nothing but find the AVERAGE UNCERTAINTY. The average uncertainty is *NOT* the uncertainty of the average!
u(q/n) = sqrt[(1/n)^2 u(x_i)^2] ==> sqrt[ u(x_i)^2] / n
That is the very definition of the average uncertainty!
You keep trying to foist this off as the uncertainty of the average but it isn’t. That’s why you can’t explain why a board of average length has a inherent measurement uncertainty that is *NOT* the average uncertainty!
Taylor 3.18 explains this implicitly. So does Bevington and Possolo. The examples have been give to you multiple times.
You define uncertainty term by term. “n” is just another term.
If q = x/w then u^2(q) = u^2(x) + u^2(w)
If w is a constant then u(w) = 0 and you get u(q) = u(x)
The AVERAGE UNCERTAINTY value is u(q)/n = u(x)/n —-> THE AVERAGE UNCERTAINTY VALUE.
Why do you insist on trying to say that the average uncertainty value is the uncertainty of the average! In fact, you may as well say it is the uncertainty of every element of the data set! Which would mean that every single measurement taken of every different thing has the same uncertainty value – an obvious physical impossiblity!
TG said: “This does nothing but find the AVERAGE UNCERTAINTY.”
ALGEBRA MISTAKE #8. Σ[u(x_i), 1, N] / N != u(Σ[x_i, 1, N] / N
The law of propagation of uncertainty does NOT compute Σ[u(x_i), 1, N] / N when Y = Σ[x_i, 1, N] / N.
Use a compute algebra system to verify this.
TG said: “u(q/n) = sqrt[(1/n)^2 u(x_i)^2] ==> sqrt[ u(x_i)^2] / n”
ALGEBRA MISTAKE #9: That does not follow from Taylor 3.18 or Taylor 3.47.
When u(n) = 0 and using Taylor 3.47 the derivation is as follows.
u^2(q/n) = (∂(q/n)/∂q*u(q))^2 + (∂(q/n)/∂n*u(n))^2
u^2(q/n) = ((1/n)*u(q))^2 + ((-q/n^2)*u(n))^2
u^2(q/n) = ((1/n)*u(q))^2 + 0
u(q/n) = (1/n)*u(q)
u(q/n) = u(q) / n
Use a computer algebra system to verify this.
TG said: “u(q/n) = sqrt[(1/n)^2 u(x_i)^2] ==> sqrt[ u(x_i)^2] / n
That is the very definition of the average uncertainty!”
ALGEBRA MISTAKE #10: That is NOT the definition of an average.
The definition of an average is Σ[u(x_i), 1, N] / N.
Note that sqrt[ u(x_i)^2 ] / N != Σ[u(x_i), 1, N] / N.
Use a computer algebra system to verify this.
TG said: “If q = x/w then u^2(q) = u^2(x) + u^2(w) If w is a constant then u(w) = 0 and you get u(q) = u(x)”
ALGEBRA MISTAKE #11: u(q) = u(x) does not follow from Taylor 3.18 when q = x/w and u(w) = 0.
The correct answer is u(q) = u(x) / w.
Use a computer algebra system to verify this.
BTW…you actually got the correct answer in MIRACLE #1 above after making ALGEBRA MISTAKES #3 and #5 that miraculously combined in just the right way to get the right answer to this problem. Suddenly you changed answer though. Why?
TG said: “The AVERAGE UNCERTAINTY value is u(q)/n = u(x)/n —-> THE AVERAGE UNCERTAINTY VALUE.”
ALGEBRA MISTAKE #12: Neither u(q) / n nor u(x) / n are averages.
The definition of an average is Σ[x_i, 1, N] / N.
Read the wikipedia article discussing arithmetic means.
A dozen claim busting such algebra mistakes and counting? Just more Nick nitpicking bd!
I don’t understand why he won’t just use a computer algebra system. There is no shame in doing so. I use them all of the time to verify my work because I know mistakes happen.
And it’s not the mistakes themselves that I’m concerned with here. I certainly make my fair share of them. It’s making 12 mistakes in only 2 posts that could have easily been caught using a computer algebra system and then completely going silent when they are pointed out as if to make a statement that the poster still thinks his math is correct.
And then accidentally get the right answer of u(q) = u(x) / w in one post and then in the very next say it is u(q) = u(x), confusing a sum (+) with a quotient (/), and incorrectly identifying sums and averages is beyond bizarre.
Apparently you can’t follow simple algebra either, eh?
(u(x_i)/n)^2 = u^2(x_i)/n^2
Factor out the 1/n^2 and you are left with total uncertainty divided by the number of elements.
Voila! An AVERGE!
“Voila! An AVERGE!”
Befor I ah didden even know how ta spell unginar, an now I are won…
I know, too easy. And not fair at all. In fact, spello’s mean nada to me. Too much time reading very informative correspondence from very good EASL engineers, to care about them.
Stop drooling, blob, its unsightly.
TG said: “(u(x_i)/n)^2 = u^2(x_i)/n^2 Voila! An AVERGE!”
Neither of those is an average.
An example of an average is Σ[u(x_i), 1, N] / N.
Dividing a single number by a constant does not make an average unless of course your N = 1.
“ALGEBRA MISTAKE #8. Σ[u(x_i), 1, N] / N != u(Σ[x_i, 1, N] / N”
Average uncertainty is total uncertainty divided by the number of elements. That is *ALWAYS* an average.
[u^2(x_1) + u^2(x_2) + … + u^2(x_n)] /n
IS EQUAL TO:
u^2(x_1)/n + u^2(x_2)/n + … + u^2(x_n)/n
That is simple algebra!
—————————–
“TG said: “u(q/n) = sqrt[(1/n)^2 u(x_i)^2] ==> sqrt[ u(x_i)^2] / n”
ALGEBRA MISTAKE #9: That does not follow from Taylor 3.18 or Taylor 3.47.”
Of course it follows:
if q/n = x1/n + x2/n + … + xn/n
then ∂(q/n)/∂x1 = (1/n)
∂(q/n)/∂x2 = (1/n)
then this leads to (partial derivative * uncertainty)^2
u(q/n) = sqrt{ ( u(x1)/n )^2 + … + ( u(xn)/n )^2 }
which then becomes
u(q/n)/(q/n) = sqrt{ (1/n^2) [u^2(x_1) + … u^2(x_n)] }
Again, simple algebra. You just factor out the common (1/n^2)
Take the (1/n^2) out from under the square root and you have
sqrt[ u^2(x_1) + … + u^2(x_n)] / n
THAT IS AN AVERAGE! Total uncertainty divided by the number of elements.
Check YOUR math before telling me I am wrong.
TG said: “Average uncertainty is total uncertainty divided by the number of elements. That is *ALWAYS* an average.
[u^2(x_1) + u^2(x_2) + … + u^2(x_n)] /n
IS EQUAL TO:
u^2(x_1)/n + u^2(x_2)/n + … + u^2(x_n)/n
That is simple algebra!”
ALGEBRA MISTAKE #13: Neither of those are averages!
Again…the formula for an average is Σ[x_i, 1, N] / N.
It is NOT Σ[x_i^2, 1, N] / N.
NOR is it Σ[x_i^2/N, 1, N]
Use a computer algebra and verify this yourself.
TG said: “Of course it follows:
if q/n = x1/n + x2/n + … + xn/n”
You’ve moved the goal post. In this post you said q = x/w. Now you are saying it is q/n = x1/n + x2/n + … + xn/n.
ALGEBRA MISTAKE #14 q/n = x1/n + x2/n + … + xn/n is NOT an average.
When you multiple both sides by n you get q = x1+x2+…+xN. That is a sum.
TG said: “sqrt[ u^2(x_1) + … + u^2(x_n)] / n”
ALGEBRA MISTAKE #15: That does not follow from Taylor 3.47 when q/n = x1/n + x2/n + … + xn/n
However, it does follow when q = x1/n + x2/n + … + xn/n which is an average. Note that q != q/n which could mean that it was just a typo. Do you want to clarify anything regarding this mistake?
TG said: “sqrt[ u^2(x_1) + … + u^2(x_n)] / n
THAT IS AN AVERAGE! Total uncertainty divided by the number of elements.”
ALGEBRA MISTAKE #16: sqrt[Σ[x_i^2, 1, N]] / N is NOT an average.
Note that sqrt[Σ[x_i^2, 1, N]] / N != Σ[x_i, 1, N] / N
This is getting ridiculous. You have increased your algebra mistake count from 12 to 16 in a comment defending your earlier mistakes.
The silence is deafening. This is what I mean. I don’t care that you made algebra mistakes. I make more than my fair share of them. That’s why I use computer algebra system a lot. My issue is that they are brought to your attention and you either defend them or pretend like they never happened. There does not appear to be impetus whatsoever to correct these mistakes.
Again, don’t take this personally. Like I said, I make mistakes all of the time. I’m mortified when I do. But when it is brought to my attention and especially when it is indisputable like would be the case with an algebra mistake I always correct them.
They are the tactics of Nitpick Nick Stokes.
So now performing arithmetic correctly is a “tactic”?
I didn’t make algebra mistakes. I’ve shown you twice how the algebra works.
if q = x + y the q/n = (x+y)/n = x/n + y/n
u^2(q/n) = u^2(x/n) + u^2(y) ==> (1/n^2) [ u^2(x) + u^2(y)]
That is simple algebra!
[ u^2(x) + u^2(y) ] /n IS the average uncertainty.
You can run away from that all you want but you can’t hide.
And it simply doesn’t matter anyway. Eq 10 in the GUM ONLY APPLIES when all you have is random error as your uncertainty – NO SYSTEMATIC BIAS.
That’s what the GUM uses for an assumption. That may be an inconvenient truth for you to accept but it is the truth nonetheless.
I’m quite certain they will circle back around to same old noise.
TG said: “[ u^2(x) + u^2(y) ] /n IS the average uncertainty.”
ALGEBRA MISTAKE #17: [ u^2(x) + u^2(y) ] /n is not an average.
Think about it. For example consider the sample {1, 2, 3}.
The average is (1+2+3) / 3 = 2.
What your formula says is (1^2 + 2^2 + 3^2) / 3 = 4.67.
Again the formula for an average is Σ[x_i, 1, N] / N.
TG said: “[ u^2(x) + u^2(y) ] /n”
ALGEBRA MISTAKE #18: u(q/n) = [ u^2(x) + u^2(y) ] /n does not follow from q = x + y, q/n = x/n + y/n, and Taylor 3.16/3.18 or Taylor 3.47.
The correct answer is u(q/n) = u(q) / n = sqrt[ u^2(x) + u^2(y) ] / n.
You can use a computer algebra system to verify this.
You *have* to be kidding, right?
q = x + y
q/n = the average value
q/n = (x+y)/n = x/n + y/n
u^2(q/n) is the average uncertanty
the uncertainty factor for x = (1/n)^2 u^2(x/n)
the uncertainty factor for y = (1/n)^2 u^2(y/n)
u^2(q/n) = [ (1/n)^2 u^2(x/n) + (1/n)^2 u^2(y/n) ]
factoring out n leaves
[u^2(x/n) + u^2(y/n) ]n^2
——————————
You say u(q/n) = [ u^2(x) + u^2(y) ] /n does not follow. It doesn’t follow. That’s true.
It should be u(q/n) = SQRT[u^2(x/n) + u^2(y/n) ] /n
*That* is what I posted.You can’t even copy what I wrote correctly!
TG said: “You *have* to be kidding, right?”
No. I’m not and neither are the computer algebra systems that disagree with your algebra.
TG said: “”u^2(q/n) is the average uncertanty”
ALGEBRA MISTAKE #19. u(q/n)^2 is not an average.
u(q/n)^2 is the uncertainty of the average squared.
However, q/n is an average because it complies with the formula Σ[x_i, 1, N] / N.
And remember, Σ[u(x_i), 1, N] / N does not equal u(Σ[x_i, 1, N] / N). The first is the average uncertainty because it is computing the average of u(x_i) for all x_i. The second is the uncertainty of the average because it is computing the uncertainty of a function that computes the average.
Use a computer algebra system to prove this for yourself.
TG said: “[u^2(x/n) + u^2(y/n) ]n^2″
ALGEBRA MISTAKE #20: [u^2(x/n) + u^2(y/n) ]n^2 does not follow from Taylor 3.16/3.18 or 3.47 when q = x + y, q/n = x/n + y/n.
The correct answer is u(q/n) = sqrt[ u(x)^2 + u(y)^2] / n.
Or in variance form u(q/n)^2 = [ u(x)^2 + u(y)^2 ] / n^2
TG said: “You say u(q/n) = [ u^2(x) + u^2(y) ] /n does not follow.”
Patently False. I said and I quote “The correct answer is u(q/n) = u(q) / n = sqrt[ u^2(x) + u^2(y) ] / n.”
TG said: “It should be u(q/n) = SQRT[u^2(x/n) + u^2(y/n) ] /n”
ALGEBRA MISTAKE #21: SQRT[u^2(x/n) + u^2(y/n) ] /n does not follow from Taylor 3.16/3.18 or 3.47 when q = x + y, q/n = x/n + y/n.
In the same post you first say u^2(q/n) = [u^2(x/n) + u^2(y/n) ]n^2 then you say u(q/n) = sqrt[ u^2(x/n) + u^2(y/n) ] / n. Those are different solution even acknowledging the variance form of the first.
I am begging you…please start using a computer algebra system to review your work before you post.
Here is how you solve the problem of u(q/n) where q = x + y and q/n = x/n + y/n using method A (Taylor 3.16 and 3.18) and method B (using Taylor 3.47).
Taylor 3.16 and 3.18 Method
Apply Taylor 3.18 first.
(A1) u(q/n) / (q/n) = sqrt[ (u(q)/q)^2 + (u(n)/n)^2 ]
(A2) u(q/n) / (q/n) = sqrt[ (u(q)/q)^2 + (0/n)^2 ]
(A3) u(q/n) / (q/n) = sqrt[ (u(q)/q)^2 + 0 ]
(A4) u(q/n) / (q/n) = sqrt[ (u(q)/q)^2 ]
(A5) u(q/n) / (q/n) = u(q)/q
(A6) u(q/n) = u(q) / q * (q/n)
(A7) u(q/n) = u(q) / n
Record that result and apply Taylor 3.16 second.
(A8) u(q) = sqrt[ u(x)^2 + u(y)^2 ]
Now substitute the result from (A8) into (A7).
(A7) u(q/n) = u(q) / n
(A9) u(q/n) = sqrt[ u(x)^2 + u(y)^2 ] / n
That is your answer. Each step was verified with a computer algebra system.
Taylor 3.47 Method
Work with q/n first.
Compute partial derivatives.
(B1) ∂(q/n)/∂q = 1/n
(B2) ∂(q/n)∂n = -q/n^2
Apply Taylor 3.47.
(B3) u(q/n) = sqrt[ (1/n * u(q))^2 + (-q/n^2 * u(n))^2 ]
(B4) u(q/n) = sqrt[ (1/n * u(q))^2 + (-q/n^2 * 0)^2 ]
(B5) u(q/n) = sqrt[ (1/n * u(q))^2 + 0 ]
(B6) u(q/n) = sqrt[ (1/n * u(q))^2 ]
(B7) u(q/n) = 1/n * u(q)
(B8) u(q/n) = u(q) / n
Record that result and work with q next.
Compute the partial derivatives.
(B9) ∂q/∂x = 1
(B10) ∂q/∂y = 1
Apply Taylor 3.47
(B11) u(q) = sqrt[ (1 * u(x))^2 + (1 + u(y))^2 ]
(B12) u(q) = sqrt[ u(x)^2 + u(y)^2 ]
Now substitute the result from (B12) into (B8).
(B8) u(q/n) = u(q) / n
(B12) u(q/n) = sqrt[ u(x)^2 + u(y)^2 ] / n
That is your answer. Each step was verified with a computer algebra system.
If you have a question about any of the steps please ask and reference the step by its identifier.
Snooty clown.
“u(q/n)^2 is not an average.”
It *IS* the average uncertainty TERM. The actual uncertainty is sqrt[ u^2(q/n) ]
“And remember, Σ[u(x_i), 1, N] / N does not equal u(Σ[x_i, 1, N] / N)”
It is *NOT* (x_i). it is x_i/n
Again, you just admitted that q/n is the average value.
Since q/n = (x1 + x2 + … xn)/n –> (x1/n + x2/n +… +xn/n)
Simple algebra
The uncertainty of the x1/n term is
[(∂(q/n)/∂(x1/n)]^2 u^2(x/n)
The partial of ∂(q/n)/∂(x1/n) is 1/n so we get
(1/n)^2 = 1/n^2
So the uncertainty terms become (1/n)^2 u^2(x_i/n)
It is SIMPLE algebra. If you web site isn’t coming up with this then you either entered the equation wrong of the site is incorrect.
TG said: “The partial of ∂(q/n)/∂(x1/n) is 1/n so we get”
ALGEBRA MISTAKE #22: 1/n does not follow from ∂(q/n)/∂(x1/n) when q/n = Σ[x_i, 1, n] / n.
It is fairly easy to show that ∂(q/n)/∂x1 = 1/n but to perform ∂(q/n)/∂(x1/n) it gets more complicated. We need to use finite differencing.
Let
q(x:1->n) = Σ[x_i, 1, n]
f(x:1->n) = q(x:1->n) / n = Σ[x_i, 1, n] / n = Σ[x_i/n, 1, n]
So
f(x1, x:2->n) = x1/n + Σ[x_i/n, 2, n]
Start with the finite difference rule upon x1.
(1) df(x1, x:2->n) = f(x1+h, x:2->n) – f(x1, x:2->n)
(2) df(x1, x:2->n) = [(x1+h)/n + Σ[x_i/n, 2, n]] – [x1/n + Σ[x_i/n, 2, n]]
(3) df(x1, x:2->n) = [(x1+h)/n + Σ[x_i/n, 2, n] – x1/n – Σ[x_i/n, 2, n]]
(4) df(x1, x:2->n) = [(x1+h)/n – x1/n]
Now perturb x1 by 1/n by setting h = 1/n.
(5) df(x1, x:2->n) = [(x1+(1/n))/n – x1/n]
(6) df(x1, x:2->n) = [x1+(1/n) – x1] / n
(7) df(x1, x:2->n) = (1/n) / n
(8) df(x1, x:2->n) = 1/n^2
So
(9) ∂(f)/∂(x1/n) = ∂(q/n)/∂(x1/n) = 1/n^2
TG said: “So the uncertainty terms become (1/n)^2 u^2(x_i/n)”
ALGEBRA MISTAKE #23: Taylor 3.47 says u(q/n) = sqrt[ (∂(q/n)∂q * u(q))^2 + (∂(q/n)∂n * u(n))^2.
You plugged the wrong terms into Taylor 3.47.
In other words, not only did you compute ∂(q/n)/∂(x1/n) incorrectly, but it’s not even used in Taylor 3.47.
I think what is confusing here is the symbol q. To make it easier to see how the substitutions work you can define f = x + y, f/n = x/n + y/n, and q = f/n = x/n + y/n.
TG said: “It is SIMPLE algebra.”
Not it isn’t.
TG said: “If you web site isn’t coming up with this then you either entered the equation wrong of the site is incorrect.”
Dunning-Kruger.
Story tip: California Will Pay Students $22 an Hour for Climate Activism – PJ Media
Thanks J Boles.
Reminds me of MDL (Method Detection Limit).
https://www.epa.gov/cwa-methods/method-detection-limit-frequent-questions
Basically, it has to do with determining the minimum value of a measurement in a particular lab that can be trusted with 99% certainty.
Gunga ==> The full descriptionis:
“The method detection limit (MDL) is defined as the minimum concentration of a substance that can be measured and reported with 99% confidence that the analyte concentration is greater than zero and is determined from analysis of a sample in a given matrix containing the analyte.”
I’m retired now but the lab in our water plant ran our own Total Suspended Solids on our sludge lagoon discharge.
Using an older method for our MDL we determined that, even though our scale could weigh a gram out to something like 10 decimal places, our MDL was 4.5 mg/L.
Lots of things in the method we used could effect our lab’s results. (Using a graduated cylinder vs a volumetric flask vs a pipet, etc.)
Reminds me of siting issues in measuring temperatures. The instrument may be able to give a reading out to so many decimal places but, considering the conditions, within what range can a particular site’s values be trusted?
Gunga ==> There is actual an official answer to that that will surprise most readers. It is in the document “Automated Surface Observing System (ASOS) — User’s Guide NOAA March 1998” available from a link on this page.
Parameter Range RMSE Max Error
Ambient Temperature -58°F to +122°F 0.9°F ± 1.8°F
Note that 0.9°F is almost exactly 0.5°C.
The uncertainty here is given as a Root Mean Squared Error (RMSE).
It is also important to point out that ASOS report all temperatures including Tmin and Tmax as 1-minute averages. According to you this means Tmin and Tmax have no physical meaning. Your quotes were “one cannot average temperature” and “A sum over intensive variables carries no physical meaning. Dividing meaningless totals by the number of components cannot reverse this outcome.” and “Dividing meaningless totals by the number of components – in other words, averaging or finding the mean — cannot reverse this outcome, the average or mean is still meaningless.” and “The simple scientific fact (from the physics of thermodynamics) that the Intensive Property we call Temperature cannot be added (or the result of an addition of temperatures divided to create an “average”) is not subject to argument from authority.” and numerous other similar quotes.
How do you reconcile your thesis that averages of intensive properties are meaningless with your citation of the ASOS user manual?
bdgwx ==> As I have explained to you before, it does not matter in CliSci that they are inadmissibly averaging temperatures, they do it because they want to find an “average temperature” — which is not a temperature but just a number.
There are many caveats to all those oddities.
What an ASOS does it to try to get some idea of what the ever-varying temperature might have been at some moment. What they end up with is not a real temperature — but an guess-timate of what the temperature might be considered to have been during that minute at the station. National Oceanic and Atmospheric Administration, Department of Defense, Federal Aviation Administration and United States Navy, who wrote the spec, all seem to settle for that guess-timate though they know it is no better than that.
Tmax and Tmin certaintly have physical meanings, but finding their mean does not. Finding the mean of Tmax and Tmin produces just a number, not a physical temperature.
Again…Tmax and Tmin are themselves means. Most of the ASOS implementations take readings every 10 seconds and do Tn = (Tn0 + Tn10 + Tn20 + Tn30 + Tn40 + Tn50) / 6. How is that any different than Tavg = (Tmin0 + Tmin10 + Tmin20 + Tmin30 + Tmin40 + Tmin50 + Tmax0 + Tmax10 + Tmax20 + Tmax30 + Tmax40 + Tmax50) / 12? If Tavg has no meaning then it would follow that Tmin and Tmax have no meaning either.
bdg ==> Tmax and Tmin are “guess-timates of what the temperature might be considered to have been during that minute (of highest or lowest temperatures) at the station.”
If guesstimates are good enough for you, then you are golden.
The way temperatures are reported (including Tmin and Tmax) are good enough for me.
That’s not related to my point though. My point is that all reported temperatures in the digital age (including Tmin and Tmax) are themselves averages. And according to your thesis that makes all reported temperatures meaningless. And if all reported temperatures are meaningless than that makes any discussion of their uncertainty meaningless as well.
I, of course, challenge that thesis. I think both temperatures are their uncertainties are useful, meaningful, and actionable. And pretty much the entirety of science, sans a few contrarians who masquerade as proponents of science, agrees with me on this.
BTW…is the numeric value of a die an extensive property?
Actually they should be handled as numeric integrations, but these concepts are beyond the ken of climate astrologers such as yourself.
Irony alert—this is religious pseudoscience, not rational thought.
Only you climate pseudoscience practitioners believe you can decrease measurements values recorded as integers down to milli-Kelvin resolution through the magic of averaging.
hypocrisy alert. the virtue signaling bdgwx has still failed to recognize what science is all about. He is still stuck obsessing over descriptive statistics, lost in the weeds. Blind defender of consensus. Doesn’t yet know what the game is; aimlessly wandering on the field.
So says the poster who doesn’t think prediction is something science does and who instead thinks superstition is a viable avenue for prediction. Obviously you and I have a different worldview of what science is and isn’t. It’s been made clear to me that the division is insurmountable so who am I to try and rock that boat again?
detrimental self-deception, distortions, and lies. willful ignorance and narcissism.
Science predicts by developing theories that can be verified by future observation, not by adjusting past observations.
A linear regression line as a prediction is *NOT* forecasting, never has been, never will be.
The original purpose was to graph an independent variable and a dependent variable to see if the hypothesized functional relationship was linear.
Using time as the independent variable, when time is not part of determining the dependent variable, is a trend, not a relationship.
You are conflating different issues. ASOS takes multiple readings as close as possible in order to simulate as well as it can multiple measurements of the same thing. Those readings are impacted by all kinds of things like settling time, buffering, etc. But it is the best that can be done with an ever-changing measurand.
Averaging 10 minute readings, however, is completely different thing. In that case the uncertainty grows significantly as the measurements are combined. You are truly measuring different things each time. The accuracy of the averaged reading is *NOT* the average uncertainty. The accuracy of that averaged value is the RSS of the uncertainties of the individual measurements.
Each and every time he pontificates about measurements he just reinforces how vacuous his real world knowledge is.
So averaging an intensive property for 1 minute is okay, but 10 minutes is unacceptable? What about 5 minutes?
You are trying to make perfect the enemy of good.
Did I stutter in my post? Averaging over 1 minute is the best you can do at trying to take multiple readings of the same measurand. Is it a perfect process? NO!
Averaging over ten minutes works IF, and only IF, the proper measurement uncertainty is applied and propagated!
Did you even bother to read my post or are you just throwing out crap hoping something sticks to the wall?
Over one minute the hope is that the measurement uncertainty will be insignificant compared to the measurements. That does *not* apply to averages made up of 10 minute increments. There the measurement uncertainty associated with the average is *not* insignificant.
You just can’t ignore measurement uncertainty the way you want to, at least not in the real world.
It’s just a sad commentary on the state of education in this country that so many PhD scientists and statisticians think that you can increase measurement resolution if only you can take enough measurements (i.e. use a yardstick to measure down to the thousandths digit), that all distributions are Gaussian, that all measurement uncertainty cancels, that stated values are all 100% accurate with no measurement uncertainty, and that average uncertainty is uncertainty of the average.
Have you changed your position that arithmetic can be done on intensive properties or not? Do you think 5-minute average of an intensive property is meaningless or not?
Let me reiterate what KM said, there is no reason with the newer digital thermometers that 1 second measurements throughout a 24 hour period shouldn’t be integrated to find the appropriate average temperature!
Why isn’t that happening? Trendologists such as yourself would scream your head off over losing the ability to splice the new info to the old info! Soinstead, let’s handicap the new info and make temperatures an average over five minutes to mimic the hysteresis of LIG thermometers. Then use Tmax and Tmin to calculate a mid-range (not a true average) temperature. I suspect there is also trepidation and consternation about what a true average from integration would show!
Read the ASOS user guide at this location.
https://www.weather.gov/asos/
“”””””Once each minute the ACU calculates the 5-minute average ambient temperature and dew point temperature from the 1-minute average observations (provided at least 4 valid 1-minute averages are available). These 5-minute averages are rounded to the nearest degree Fahrenheit, converted to the nearest 0.1 degree Celsius, and reported once each minute as the 5-minute average ambient and dew point temperatures. All mid-point temperature values are rounded up (e.g., +3.5°F rounds up to +4.0°F; -3.5°F rounds up to 3.0°F; while -3.6 °F rounds to -4.0 °F).””””””
“All mid-point temperature values are rounded up …”. ROUNDED UP!!!! THEN CONVERTED TO Celsius. Does that rounding and converting add any uncertainty? Look at the the table on page 12 that has accuracy and error standards. Hmmm ! Some pretty large values here.
I feel unwell just reading that. If you’re going to report in tenths of a degree Celsius, measure in tenths of a degree Celsius or better.
I know. That is a criticism I’ve voiced amongst other weather enthusiasts numerous times over the last 20 years.
Then there is this issue:
The average Tmax for my location in January is something like 40F or 5C—what does this number tell me about what the high T tomorrow might be?
Pretty much nothing because the range of possible values can be -10F to 65F.
You nailed it. The weather forecast is a prediction. The climate 80 years from now is *NOT* a prediction. It is just the extension of a trend line with no physical relationship involved. You may as well trend the postal rates over the past 100 years along with a scaling factor. You’ll get the same result the CGM’s give!
Values matter. And maybe not just questionable values reported due to siting issues.
I had a candid face to face conversation with someone from the NWS who had come out to inspect our NWS precipitation gauge.
(We’d been reporting for 50 years. I was able to fill in some of the missing values because I had access to the paper and electronic records and personal knowledge of the guy who didn’t report the values via their “new” online entry system. If one day was missed then the whole monthly average was marked as “M” for missing for our station. I was able to reduce about 24 “M”‘s to just one because I could not find a record for just one day back around 2005.)
Anyway, he told me that the FAA had taken the responsibility for the reporting stations for many of the country’s airport stations. The 3 in Ohio were almost always the hottest in Ohio. If they only called in the NWS to check out a station if they thought there was a problem with the instruments. But, in the meantime, they still reported what they knew were bogus readings from a faulty sensor.
One of my usual typos:
“If they only called in the NWS to check out a station …”
Should be:
“They only called in the NWS to check out a station …”
(Drop the “If”.)
Perhaps a new level of “siting issues”?)
Sometimes you see funny usage of statistics. I enjoy golf and many times the announcer will tell that X number of players have played a hole with an average of 3.5 shots or under par. Well that is silly knowing that in golf only whole shots can be counted. So the average player did not play 3.5 shots on that hole.
Kip,
“The above is the correct handling of addition of Absolute Measurement Uncertainty.”
You are in a world of your own with this stuff, but it is aggravating when you quote what you claium is authority, when what it says is the correct statistical understanding, whi h you apparently can’t read. You wrote a post on the Central Limit Theorem, when you don’t have a clue what it says, even though you linked to a Wiki post that defined it correctly. And you persist with this misunderstanding of Absolute Measurement Uncertainty, even though your link says correctly what it is.
It isn’t some variant uncertainty. The only reason to describe it as absolute is to distinguish it from relative uncertainty, expressed as a fraction or %. Your link (which is just another blog) says it clearly enough:
“Absolute error or absolute uncertainty is the uncertainty in a measurement, which is expressed using the relevant units.”
It is just uncertainty, and that is correctly expressed as at stats books say, as a standard deviation.
Nick ==> You stubbornly refuse to use the full definition of absolute measurement uncertainty. It is expressed as “absolute” because it is precisely that. It is the exact amount (in relevant units) that the measurement is uncertain by. It is used “to express the inaccuracy in a measurement”
I will make one more to set you right on this: Original Measurement Uncertainty is when a measurement is KNOWN to be uncertain to a specific amount of the relevant units. It is not a probably this or a possible that. It is an exact amount by which we are uncertain of what the true value of the measurement was when taken. It always represents a range of possible values, which is the range of uncertainty.
The simplest example is the officially recorded value for a daily mean temperature rounded to the nearest whole degree. Once the value has been rounded, it then becomes “Recorded-Whole-Degree +/- 0.5 degrees”. The true value is somewhere within that range but we do not and cannot know what it was (with just that recorded value) any closer than to state the range.
There are many many other kinds of uncertainty, and many other names and definitions for those various types. But I have given understandable definitions and clear examples. Quoting partial definitions and “other definitions” of “similar concepts” is not useful.
I am not talking about statistical definitions — just physics and the art of measurement. (Plus a little arithmetic.)
At least say you understand the “rounded temperature” issue as “absolute measurement uncertainty…that would be step one.
Kip,
“You stubbornly refuse to use the full definition of absolute measurement uncertainty.”
You stubbornly refuse to quote the definition. Who uses it your way? And what exactly do they say? All you give is people making the standard observation about units, to distinguish absolute from relative.
Nick ==> What words would you use to describe the measurement uncertainty in the rounded temperature reading example? Or a measurement made by guessing using a ruler only marked in whole inches?
Because in both cases the uncertainty stems from the known uncertainty of the actual length or actual temperature, due to measurement device or measurement process. While the uncertainty is know precisely the true value is not. Thus we get 78° +/- 0.5° or 10 in +/- 0.5 inches.
That is the proper definition of original absolute measurement uncertainty. The uncertainty in such cases, as in all my examples, is known and absolute.
Nick ==> You neither answer the question nor Take the Challenge.
Nor do you attempt to explain why absolute measurement uncertainty forced by the measuring device and/or process, which has a known uncertainty range but no known true value, is a probability problem.
I also bet that you have not taken the two or three minutes necessary to look at the physics YouTube that makes this quite clear.
Kip,
“That is the proper definition of original absolute measurement uncertainty.”
For Heaven’s sake, quote that definition! The actual words. And give a source, if you have one. Who are they and what do they actually say?
Nick ==> Ah, come on….even you can understand this. Absolute Measurement Uncertainty is a sub-set of “absolute uncertainty” wherein the uncertainty derives from the uncertainty inherent in the measurement device or process — AND is a know quantity. It is a known quantity to an absolute value. A ruler than can measure no more precisely than the nearest cm — a scale that can weigh no more accurately than the nearest kilogram. etc etc etc. These all have known absolute measurement uncertainty deriving from the least count.
There are some other instances that create practical absolute measurement uncertainties.
How about this?
From “Operational Measurement Uncertainty and Bayesian Probability Distribution”
“Suppose Θ is a random variable with a
probability distribution π(Θ) which expresses
the state of knowledge about a quantity. The
domain of π(Θ) is the range of possible value
for that quantity. Suppose (θl, θh) is a result of
measurement expressed as an interval for that
quantity where θl and θh are any two possible
values of Θ and θl < θh. Now suppose[Θ] is a
conceptual true value of that quantity. The
theoretical Bayesian interpretation of π(Θ) is
that it describes the probability that the true
value τ[Θ] lies within the interval (θl, θh).”
“Thus, π(Θ) describes the probability associated
with a result of measurement expressed as the
interval (θl, θh). The operational interpretation
agrees with the essential GUM and aligns with
Bayesian thinking.”
This is *NOT* describing the probability distribution for the values within the interval but only the probability that the true value lies within the interval.
That is how measurement uncertainty has *always* worked, at least in the physical world.
The trendologists can’t handle the truth, but all they can manage in reply is a downvote.
To paraphrase – well – you, Quit whining.
Hi blob! How’s the TDS going these days, getting any professional help for it?
And thanks for demonstrating my point to a T.
Stop skimming to find something to nitpick, and read the whole article.
Yet practitioners of climate science routinely ignore and drop all of the standard deviations that arise from the averaging of average averages while traveling to the Holy Trends. Why is this?
Because it would put the lie to the ability of discerning differences in the hundredths digit.
Kip: Sorry, but I have to take issue with your premise that a throw of dice is analogous to a physical measurement of some property. The result of a throw of dice is determined by simply counting the spots on the top faces. There is no uncertainty to this result. Indeed, measurement uncertainty does not apply to numerical values obtained by counting objects. One assumes that the result of counting things has no uncertainty (US election results excepted).
Measurement uncertainty applies to measurements on a continuous scale where a true value exists to a near infinite level of precision. E.g. I could determine the exact value of Pi if I could measure both the diameter and circumference of a circle with infinite precision.
The primary reason statistical methods apply to real world physical measurements of things like length, weight, temperature, pressure, force, etc. is that the calibration of our instruments is based on comparisons to standard references. In essence we make a series of measurements of “known” references over the instrument’s range multiple times and then analyze the deviations statistically. In most cases this results in a normal distribution of errors. The standard deviation of this distribution becomes a primary element of the instrument MU. Another element of the instrument MU is the uncertainty of the calibration reference itself which is also generally determined statistically based on normally distributed data. In some cases calibration may rely on methods that produce rectangular or triangular distributions, but even those cases result in using an estimate of the equivalent standard deviation for use in defining an uncertainty budget.
You asked for diagrams and examples. There are plenty of both in the ISO G.U.M.
I do agree with you that in the field of climate science, measurement uncertainty is rarely handled properly and is often intentionally obscured or ignored. It is generally large and invalidates many of the claims made by alarmists. See Dr. Pat Frank’s paper on Uncertainty Propagation in climate computer models for an example.
“Kip: Sorry, but I have to take issue with your premise that a throw of dice is analogous to a physical measurement of some property…”
Mr. Layman here.
Kip can correct me if I’m wrong, but I think he was using the roll of the dice to illustrate and communicate the point for such readers as me who are not statisticians.
“It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.” – Somebody Famous
And my personal favorite secular quote:
“Everybody is ignorant, only on different subjects.”
Will Rogers
Just how would you go about determining if a specific die was “fair”? You’d do an experiment – eg roll the die100 times – and see if any particular result turns up more frequently than could be explained by simple chance. That is, you’d apply statistical methods. You might determine, for example that 3 turns up with greater frequency than chance alone could account for with a P-Value of 0.001 using a Pearson Chi Squared test. That would mean there’s only a 1% chance that the die is fair. Not much uncertainty there.
Rick ==> As you must know, even rolling a pair of dice a million times won’t give you an exactly even distribution… but using the actual frequency distribution will reveal a loaded pair of dice.
Rolling a single die or pair of dice is a random process, but tiny difference in manufacture or internal densities or unevenness of the playing surface introduce chaotic responses — thus never an exactly even distribution.
We don’t need any statistics, other than to make a frequency distribution, which only involves counting.
Kip: Yes, a histogram can tell you to be suspicious of something being off. But if you want to quantify the confidence you have in such a determination you need math and probability calculations – that’s statistics. Uncertainty is a quantification of the potential error AT A GIVEN CONFIDENCE level. The entire subject of MU is essentially an exercise in mathematical statistics. Read the GUM – it’s all “normal distribution” statistics.
Rick C ==> Statistics is an interesting topic and has its place in the world.
“But if you want to quantify the confidence you have in” …. that is a statistical concept, not a real world concept — at least for dicing.
I could look at the frequency distribution and know how confident I was in my evaulation…your use of statistics to to be able to convince others that your level of confidence is “correct.
How do you know what was rolled in a closed box? You can’t determine anything on which to base a statistical analysis.
In this case you have just a few options. Assume all dice are fair, assume some dice are unfair, assume all dice are unfair. The second two don’t really lend themselves to much analysis from outside the box.
Rick ==> I am using the dice as a simulation….simulating the act of measuring (counting how many dots, in this case) an instance. But in a special way — we are not allowed to know the actual number of spots resulting from each roll, but only to be told that they are 100% certain to be in a given range.
This simulates the common practice of taking a measurement but then rounding it to some value with a range. Daily Mean Temperatures in CliSci are handled this way. The actual value of T(mean) is recorded and used as a whole number AFTER ROUNDING. This is the “box” in which the T(mean) has been placed, a box 1 whole degree big. This leaves us with a recorded value for T(mean) such as 76°F +/- 0.5°F. We know the uncertainty exactly (absolute uncertainty) but we don’t/can’t know the true value of the measurement — it has been rounded (thrown away), but we know the central value of the range.
In the dicing example, we hide the actual true value of the roll (in the box) but we know the true value is somewhere within the range represented by the central value (the mean of possible values) and the uncertainty range (from the reality of possible dice rolls). A sneaky way of simulating “least count” of a measuring instrument. In our case, the least count of the hidden dice roll is the entire range.
“Measurement uncertainty applies to measurements on a continuous scale where a true value exists to a near infinite level of precision.” Shift back to arithmetic — consider the house carpenter. Consider measuring temperature or wind speed. Consider measuring variable processes….like a dice roll.
We are not taking here of measuring the precise length of a single stainless steel rod a thousand times to get a true value.
We are simply investigating what happens in this particular kind of absolute measurement uncertainty when we must ADD two values (with their absolute measurement uncertainties).
Why? Why concern ourselves with this question? Because adding two or more measurements that have absolute measurement uncertainty is nearly always followed by dividing the sum by the number of measurements.
If you cannot t\do the addition correctly, the next step, division, will be wrong.
I was taught that the real test for the validity of value (usually an average of a bunch of measures) was to repeat the experiment and get an answer at least similar to the first one. The function of statistics was to get an estimate of how likely that was.
Fran ==> getting back to you, don’t know if you are following….
1) Throwing dice is always a one-time-only event. No dice roll depends on the previous dice roll. So, we can’t do multiple experiments on the results of a dice roll. (I know you already know that bit….but must be stated out loud.) Nor can we measure the temperature at a specific place and a specific time more than once (unless we have multiple thermometers mounted very close to one another). Both the time and the temperature will have changed.
2) A single measurement that has been rounded from its measured value to to some level of precision (think rounding to the nearest whole degree or to nearest centimeter) loses the ability to be known except as a range of possibilities….it lies within that range. But with rounding, we know with a certainty that the original measurement is in that range somewhere.
3) Like with throwing dice, reality trumps statistics. I have seen a man using fair dice throw 4 double-sixes in six rolls. It doesn’t matter how likely that is (and it is very unlikely) because it actually occurred in the real world. This is why pragmatists and not big fans of overly strict statisticians.
Wow! You just said a mouthful for sure!
“and it is very unlikely”
About 1 in 100000. Unlikely but not outside the bounds of probability.
“This is why pragmatists and not big fans of overly strict statisticians.”
So are you saying that having seen this happen once, you will ignore all probabilistic arguments? If someone put a bet on you being able to reproduce it, what odds would you accept?
Uncertainty has no probability distribution. If it did there would be no reason for the use of “stated value +/- uncertainty”. You’d just use the best estimate of the true value based on the probability distribution of the values within the interval.
He’s now trying to backpedal away from this claim, while at the same time bob is doubling-down on it.
“You’d just use the best estimate of the true value based on the probability distribution of the values within the interval.”
This is just insane. It’s because there is a probability distribution you have uncertainty. I might know the most likely value is the stated value, but I also know there’s a high probability that the value is not the stated value but lies within a range of values, the uncertainty interval.
Kip,
you are confusing two fundamentally different things. Measuring the temperature using a thermometer graduated in degrees will give you a measurement of X degrees +/- 0.5 degrees. However there is a single precise temperature that we do not know. Throwing a die and asking what the expected value is corresponds to a very different sort of measurement. For starters that is no signal correct answer — the first time might result in 2 while the second throw might result in 6. Saying that the answer is 3 +/- 2.5 means something completely different to saying that the temperature is X degrees +/- 0.5.
Different symbols and words mean completely different things in different subject areas. Your very first example in a previous essay talked about the meaning of +/- X in for example the quadratic formula. There there are two answers -b +/- X and the +/- symbol does not mean an error or a range etc but something different and precise. You are getting yourself confused by trying to apply the meaning of words and symbols from one branch of mathematics to another.
You didn’t read it, either, eh?
Izaak ==> “will give you a measurement of X degrees +/- 0.5 degrees. However there is a single precise temperature that we do not know” Exactly, but we know the uncertainty “absolutely”.
“Saying that the answer is 3 +/- 2.5 means something completely different to saying that the temperature is X degrees +/- 0.5.” Normally that would be true. But I have created a circumstance that simulates the rounded temperature (in which the original value has been discarded or ignored, in either case we do not have it and can not retrieve it). We don’t need to resort to “different subject areas” when I have clearly defined my terms and circumstances. You may have different “meanings” but in my examples, you must use my definitions — there are so many. I give explicit examples and even graphics explaining this — so there can be no whining about “I learned a different defintion of the +/- symbol”.
When we are dealing with “absolute measurement uncertainty”, we mean that while we don’t know the real true value, we know it is invariably found within the range given by the X +/- uncertainty — somewhere.
bdgwx’s ensemble of curve fitting + autocorrelation for next month’s UAH inferred an anomaly value of 0.16C for December 2022, down from November’s 0.17C.
Spencer posted 0.05C for December 2022.
Has bdgwx missed the target, or hit the target, with his projection of 0.16C with his statistic RMSE 0.107C?
Persistence (lag1) alone would have inferred a value of 0.17C with an RMSE statistic of 0.12C for the record. Has this missed the target, or hit the target?
What does any of this mean? Is it meaningful, or meaningless? What have we learned about the nature of the system? I suspect not very much, perhaps even fooling ourselves into thinking we know anything at all. This can sometimes (often) be worse than conscious ignorance.
It is the power of not knowing that drives progress. It is false certainty which stymies the evolution of knowledge.
JCM said: “Has bdgwx missed the target, or hit the target, with his projection of 0.16C with his statistic RMSE 0.107C?”
It missed by 0.11 C.
JCM said: “What does any of this mean?”
It means the number of > 0.11 (1σ) prediction errors is now 196 out of 516 and the number of >0.22 (2σ) prediction errors remains at 38 out of 516. Not only are the 515 preceding predictions within expectation, but the observed value for 2022/12 of 0.05 was with the expectations of the 0.16 ± 0.22 C prediction as well.
Given Christy et al. 2003 assessed uncertainty of 0.10 C (1σ) we expect a perfect model to deviate more than 0.20 C about 23 times out of 516 predictions.
JCM said: “What have we learned about the nature of the system?”
1) That persistence is an effective model.
2) That the non-autocorrelation model using CO2, ENSO, AMO, PDO, and volcanic AOD cannot be falsified at even the minimal 2σ significance level.
BTW…I’m curious…how did you’re superstition model that you said was at least as good as my non-autocorrelation model fair? Would you mind sharing details regarding how you use superstition to make predictions so that we can try to replicate it and assess its skill?
Yes, the crow, raven and jackdaw were calling late again this year.
The list of different uncertainties reminded me of the various types of infinity. Infinity minus infinity could be anything from -infinity to infinity, and seldom is zero, but a statistician would always think it is.
Actual data and likely data are wildly different things. It is likely 40 deg F in my fridge. Most of the time it is, give or take a couple degrees. But when I’m hungry and don’t know what I want, it is entirely possible that it will be 50 in there because I’ve stood there with the door open. Maybe only engineers have this problem.
whatlanguageisthis ==> Don’t forget the modern defrost cycle in your frost-free fridge.
A pair of dice rolled simultaneously have a value distribution as illustrated above that is entirely logical. There are 6 times the combinations for rolling a 7 as there are for a 2 or a 12 and the graph reflects that.
kalsel3294 ==> Yes, exactly that. I link a chart of all the possible rolls and their frequencies, possible combinations, etc in the essay (somewhere.)
Under “rolling a pair of dice”, there are a couple of mentions of a result of “1” which is not possible.
Capt Dave ==> (I am reviewing comments, thought I had seen this one)
Thank You! Sharp eyes and my typo. I have corrected that.
Kip is confused about the term absolute uncertainties, which simply means uncertainties in the same units as the measurand, as opposed to percentage uncertainties. It is nothing more complicated than that.
However the fundamental problem is that rolling two dice is not an appropriate analogue to rounding thermometer readings to the nearest whole degree. The result can only be one of a small number of integral values.
It is a combination of only two rectangular distributions, and although we can see the result is beginning to look related to Gaussian, it is still a long way from it.
Whereas the interval from 19.5 to 20.5 degrees contains an infinite number of real temperatures.
If Kip repeated his experiment with a billion dice, he would come up with something that was close to Gaussian. We could calculate the mean and 95% confidence range of the distribution. This would be a better analogy.
Kip also says there are many references to support his position, and then proceeds to give us a youtube as a so-called reference. That says it all really.
KB ==> I am just making an example of absolute measurement uncertainty that is easy to see without a lot of complication.
infinite number of real temperatures. And, yes, that is the point isn’t it? The fact that the simple “thought experiment” version with dice has a limited number of possible values is what makes it easier for most readers to understand.
In your attempt at a correction of the term “absolute measurement uncertainty”, you drop the “measurement”. When there is an absolute measurement uncertainty, caused by the “least count” of the measuring instrument or measurement process, and that uncertainty is known, then the rules are as I give.
If you can demonstrate that the examples are not correct — meaning Take the Challenge — then we will be corrected.
I point out that still, after three essays, there has not been a single even attempt to Take the Challenge and show that the method used to add two absolute measurement uncertain values is other than I show.
Did you view the physics class YouTube? Too proud to learn? Its only a few minutes, and there are many many that explain this simple point about adding absolute uncertainties.
The Youtube you linked was about simple basic definitions. It said nothing about combining uncertainties. As far as it went, it agreed with what I said.
The fact that the simple “thought experiment” version with dice has a limited number of possible values is what makes it easier for most readers to understand.
No it is misleading. It has a small number of discrete values. To make it more representative of the actual situation you need a lot more dice.
If you can demonstrate that the examples are not correct — meaning Take the Challenge — then we will be corrected.
It’s not up to me to prove they are not correct. You are the one espousing bizarre unusual ideas and therefore the burden of proof rests with you.
I have however tried numerous times to show you and others where you have fundamental misconceptions. I’m trying again now.
It looks like one of the misconceptions you have is you think a range which contains 100% of the possible values is called an absolute uncertainty. It isn’t, and even your Youtube tells you that. You also seem to think it can’t be dealt with using the accepted methods of uncertainty propagation. Well yes it can.
In a temperature range of 19.5 to 20.5 degrees, it is not the case that there are only 6 (or any finite number) of possible temperatures within that range. There are an infinite number. Your example with the dice is misleading.
Repeat this dice thought experiment with more dice. You will see, as the number of dice is increased it will start to follow the accepted methods of dealing with uncertainties.
KB,
Sure, the mind can accept that there are infinite numbers between 19.5 and 20.5. To help understanding, people use the concept of “are these numbers different? In mathematics. 20.00001 is not the same as 20.00002. In practical measurement, how can we tell if that difference is measurable and/or repeatable? In practical measurement, does it matter, because it is so small and the tiny difference between two large numbers has forever been recognised as troublesome because of error potential. See the TOA radiative flux balance as an example, with energy in minus energy out measured at top of atmosphere. Subtract two numbers around 1340 and try to retain meaning of the difference, because for some research a difference of 0.1 or 0.5 matters.
It is evident from reading comments on this and other similar articles that people have rather different ideas about what uncertainty is.
My own approach goes like this. You have made some observations and you are thinking about uncertainty. For example, you have observed values of 49.1, 50.2, 49.1, 49.3, 50.0, 50.1. You make another observation whose value is 58.7. Do you dismiss it as an outlier, or do you accept it, then widen your uncertainty bounds?
In my book, you can reject it ONY if you are quite certain that you KNOW the physical reason why it appears so much different.
In this “modern era” of “cancel culture” it is common for researchers to cancal values that do not fit their preconceptions. In my old work of geochemical exploration, we treasured exceptions because they could lead to new knowledge, new mechanisms.
Someone needs to write a new, definitive text to encourage mandatory adherence to a set of concepts that are useful in the practical world where high uncertainty is really very common. That explains why, for example, we have so many deaths from car crashes. People do not believe it happens to them, so they take less care than they should.
Making a statistical estimate that gives a tiny uncertainty does not prevent crashes.
Geoff S
OK Geoff but what to do about outliers is not really germane to the discussion.
We are a long way from that topic yet, we have to get the basics sorted out first.
+100
KB ==> What you mean is you think I need more possible VALUES not more dice.
You are absolutely right that a temperature reading that has been rounded and recorded only as 76° +/- 0.5° represents an infinite number of possible true temperatures.
However, you are wrong about terminology. I use the term absolute measurement uncertainty and not the more generalized term absolute uncertainty. That is your misunderstanding.
Absolute measurement uncertainty stems from the uncertainty of the measuring device (least count, etc) or measurement process (which should have a stated known uncertainty to be considered scientific).
The Challenge is to show that when adding two values that have absolute measurement uncertainty the procedure of adding the values and adding the uncertainties is incorrect or that adding the “uncertainties in quadrature” produces a correct real world result.
Take the Challenge (or not…) But if you can’t do it, just admit it.
I think I am beginning to see what has been going on here. I’ve now seen several Youtubes which say that you add the uncertainties directly. I can only think these are aimed at students at a basic stage of study. This is perhaps where these ideas come from.
OK I will address your challenge.
Subtract one temperature from another, where both have been rounded to the nearest degree.
76 – 66 = 10 degrees.
Both temperature measurements have a possible range of +/- 0.5 degrees.
So the maximum possible uncertainty on the result is indeed 0.5+0.5 = 1.0 degrees. The result would be written 10.0 +/- 1.0 degrees and you could say the range of the result is 9.0 to 11.0 degrees.
However this is a naive view. This is because it is statistically unlikely that both measurements would vary by exactly 0.5 degrees and in the opposite direction to each other, i.e 76.5 – 65.5 = 11.0 degrees.
The other extreme, 75.5 – 66.5 = 9.0 degrees is similarly unlikely to occur by chance.
Where we have an infinity of possible values in the ranges, as we do here, the probability of both values being at the extremes of their possible values, happening by chance, is infinitely small.
By following this method, you are almost certainly overestimating the uncertainty.
All other values are equally likely to occur:
76.1- 65.8 = 10.3
75.6 – 66.3 = 9.3
etc. The list of possible combinations is endless, and not limited to one decimal place either.
This is the common situation in scientific measurements. They are generally continuous distributions, not discrete values like your dice.
The rounded temperature measurements are examples of rectangular distributions. They are not Normal distributions. It would indeed be incorrect to add in quadrature at the first stage. So the sum SQRT(0.5^2 + 0.5^2) = 0.707 is not the correct way of calculating this uncertainty.
The way round this is to know that the standard deviation of a rectangular distribution is the semi-range divided by SQRT(3). This can be proven mathematically but don’t ask me to do it.
So first of all, we divide 0.5 by 1.732 (the square root of 3).
0.5/1.732 = 0.2887
We then add in quadrature, i.e SQRT(0.2887^2 + 0.2887^2) = 0.408
The “standard uncertainty” of the result is 0.408.
However there is another complication, and that is the result of combining two rectangular distributions of the same size is a triangular distribution.
UKAS M3003 gives the multipliers for 95% confidence (Table C.15) for this situation. I’ve looked it up and it is 1.93.
So the 95% confidence uncertainty is 1.93 x 0.408 = 0.787.
We would probably round this to one significant figure, i.e 0.8.
The result is then quoted as:
10.0 +/- 0.8 degrees (95% confidence)
Instinctively you can see this looks a lot better than the linear addition result. It is more than 0.5 but less than 1.0.
So there you go. I’ve told you why you are wrong and I’ve shown you how it should be done. So that is your Challenge addressed in full.
Do I win a bottle of something good ?
Reference (notice not a Youtube !!!)
https://www.ukas.com/wp-content/uploads/schedule_uploads/759162/M3003-The-Expression-of-Uncertainty-and-Confidence-in-Measurement.pdf
KB ==> Ah, but you acknowledge that the correct answer is 10 +/- 1 — but you “don’t like it” — why?
If you want and are willing to accept the full uncertainty — then that is the correct answer.
It is no refutation to say “I don’t like the true answer — I don’t want the full uncertainty — I prefer a truncated, smaller uncertainty range that is not so uncertain.”
You have not shown that the full uncertainty is not “+/- 1”. You have just shown your preference for a (one of many) possible statistical answers. One that does not show the full uncertainty but rather some portion of it.
It is no proof that your preferred statistical answer is in agreement with various generally accepted statistical formulas.
The challenge is to show, in the real world, that 1) the real uncertainty when adding two values with absolute uncertainty IS NOT the two values summed and the two uncertainties summed. or 2) In the real physical world, when adding two values with absolute uncertainty, the correct uncertainty is “0.787” (rounded up or not).
Can you show this graphically? Can you stack poker chips, roll dice, lay out counting bears, cut strips of paper and line them up, all to show how the truncated, smaller uncertainty is physically correct?
No, you can’t and haven’t.
Any high-schooler can plop known, given values into formulas provided by their teacher or text book and work out the math (at least, with the aid of the calculators or laptops).
If they understand the principles, they can demonstrate that it is true and correct in the real world (think Newtonian physics or chemical processes) but the rub is that the principle has to be real-world true to start with.
So why can’t you? It is because the statistical approach that you have used above does not agree with the physical world.
My advice, don’t play the stock markets….that truncated uncertainty can (and will, eventually) wipe you out.
I can’t do pictures on here I can only do text. Yes this makes it more difficult but I can only do what I can do.
I didn’t agree that the uncertainty = 1.0 degrees is the correct result. I said it was the maximum possible uncertainty.
With the result of 10.0 +/- 0.8, we have 95% of possible results within only 80% of the maximum possible uncertainty range.
Only 2.5% of results lie between 9.0 and 9.2, and only 2.5% of results lie between 10.8 and 11.0.
The probability of the correct result being 9.000 or 11.000 is negligibly small. If you bet on either you would stand hardly any chance of winning.
Have you ever built a stud wall in a house? Using your statistical analysis you’ll wind up with ripples in the drywall on the ceiling.
Have you ever mike’d the crankshaft journals on an engine in order to order the properly sized bearings? Using your statistical analysis you could easily wind up with a turned bearing that was sized too large from not considering the entire measurement uncertainty.
Have you ever wondered why they build expensive, digital torque wrenches when the old spring loaded ones would work fine based your statistical analysis?
As I said on the previous discussion, there are risks for both underestimating and overestimating an uncertainty.
If you routinely take the largest possible uncertainty as your guide, you will cost your company money.
Where that balance lies would be the subject of a cost/benefit analysis. It would be rare that the best cost/benefit would be at the largest possible uncertainty level.
Ergo one of the prime purposes of the GUM, a standard guide for the expression of uncertainty.
Hehehehehe, I quote the title of the GUM, and blob pushes the downvote button. What a clown.
Tell that to an engineer designing a bridge where criminal and civil liabilities are attached.
Tell that to a mechanic overhauling the engine in a customers 1972 Corvette with an irreplaceable original engine.
Tell that to the carpenter who would have to redo the entire room in a house because of ripples in the drywall.
Damage to your reputation for providing a quality product can be far worse than just the cost/benefit analysis for a single project.
“KB ==> Ah, but you acknowledge that the correct answer is 10 +/- 1 — but you “don’t like it” — why?”
You nailed it!
“However this is a naive view. This is because it is statistically unlikely that both measurements would vary by exactly 0.5 degrees and in the opposite direction to each other, i.e 76.5 – 65.5 = 11.0 degrees.”
You missed the entire concept behind the closed box! How do you know what is going on? If you don’t know then how do you know the distribution?
You ignore the fact that all measurement uncertainty consists of two factors, random error and systematic bias. u_total = u_systematic + u_random. If you don’t know either of the factors then how do you account for them in a probability distribution? If systematic bias is more significant than random error then you can certainly get values close to the edges of the uncertainty interval.
“The other extreme, 75.5 – 66.5 = 9.0 degrees is similarly unlikely to occur by chance.”
In essence, you are making the same *assumption* that climate scientists and statisticians make – all error is random and will cancel out leaving the stated values as 100% accurate.
It just doesn’t work that way in the real world.
I do know the distribution.
In this example we are isolating one single uncertainty component, i.e the rounding to the nearest degree. We need to take this one stage at a time.
I do know the distribution for this one uncertainty component is rectangular, or close to it.
I also know that the resulting distribution from combining two rectangular distributions is triangular. That is is the probability density of the result.
It does mean that there is a certain amount of “errors cancelling each other out”. That is exactly what is happening.
So it is not black box. I know these things.
When we move onto the other uncertainty components you mention, combining those with the triangular distribution will result in a distribution that looks more Gaussian. So that puts me on even firmer ground, not less.
Why do you think you know the distribution?
Please list for us what the systematic bias is for each NWS temperature measuring station near Kansas City, KS. If you can’t then you don’t know the distribution of the uncertainty for each of the stations.
You say you know the distribution of the uncertainty interval but that is actually a physical impossibility.
You just *think* you know the probability distribution for the uncertainty interval. But the uncertainty interval is a closed box. You don’t know what is going on in the box!
I know the distribution, because at this stage we are only considering the rounding error. So I do know it is rectangular distribution.
I’m not yet including other uncertainty components. But when I do, that will make the final distribution more Normal. The more uncertainty components I include, the more Normal the outcome.
Of course I am not going to list the systematic bias for each station. I don’t know them and I don’t need to for the purposes of this discussion.
If I needed to, I would find out the bias and its uncertainty. I would then correct the figures for the known biases and propagate the uncertainties of the biases into the final result.
If I can’t find the figures, I would try and estimate the uncertainties caused by bias from whatever knowledge I could obtain about the process.
Quite simple in principle but a bit tiresome in practice.
Yep, once again, just sweep instrumental uncertainty under the rug and ignore it.
Of course, you don’t understand that uncertainty is not error, just like the rest of the AGW cultic zoo.
“The more uncertainty components I include, the more Normal the outcome.”
Which only tells me you do *NOT* have a good handle on uncertainty. Systematic bias does *NOT* push anything toward being normal. If anything it creates a skewed uncertainty!
Uncertainty is unknowable. Even in the GUM they assume that the uncertainty of individual measurements are random, Gaussian, and cancel so they can use the mean and standard deviation of the stated values to determine uncertainty.
You have been brainwashed into believing what bellman believes, all uncertainty is random and Gaussian so you can assume it cancels. You can then do statistical analysis on the stated values.
Every book I have on uncertainty states that the presence of systematic uncertainty makes statistical analysis incorrect. That may be an inconvenient truth for you to accept but it *is* the truth nonetheless.
“Of course I am not going to list the systematic bias for each station. I don’t know them and I don’t need to for the purposes of this discussion.”
Of course you need to know them! This is just part and parcel of assuming all uncertainty cancels so you can use statistical analysis on the stated values!
———————————————
Bevington: “The accuracy of an experiment, as we have defined it, is generally dependent on how well we can control and compensate for systematic errors, errors that will make our results different from the “true” values with reproducible discrepancies. Errors of this type are not easy to detect and not easily studied by statistical analysis”
Taylor: “As noted before, not all types of experimental uncertainty can be assessed by statistical analysis based on repeated measurements. For this reason, uncertainties are classified into two groups: the random uncertainties, which can be treated statistically, and the systematic uncertainties, which cannot.”
————————————————
I know this is anathema to statisticians who believe everything can be analyzed statistically and are, therefore, prone to making and believing in assumptions that *force* the data into forms that are amenable to statistical analysis.
But this just isn’t reality, it’s not the reality most of us who work in the real world have to contend with!
Amen! It is as if the more stats they learn the less connected with reality they become.
I cringe whenever I see opinion polling data reported, with statements like “margin of error +/- 3 percentage points”. These numbers are just pure stats, they never include biases inserted by the way questions are worded (biases that are in general unknown!).
When one looks at resolution error for LIG, it is ±0.5, yet the NWS says ±1. Why? Systematic error due to any number of things like drift, environmental, etc. Take wind for an example. Only if it is constant can one say a statistical analysis can remove it. Same with dew point, is it constant?
It is why experimental intervals are used.
To be fair, that’s the basis of comparators or using the same instrument to take both readings. The systematic error should be the same in both cases, so that cancels.
Tim was quoting KB, hence the quote marks.
Mea culpa. I probably should have added more context.
“The systematic error should be the same in both cases, so that cancels.”
Uh, no. If my instrument reads 1 unit high, then any calculation using the values being read will be 1 unit high.
Let’s use voltages of 5 and 10. I read 6 and 11 so the average is 8.5 instead of 7.5. On another meter I read 4 and 9 so the average is 6.5, again instead of 7.5. Any cancelation from using different meters is purely by luck and is unpredictable. No lab could operate in those conditions.
It is one reason systematic error is also called a bias. Every reading from a given device is biased away from the correct reading and will be inaccurate.
Yep, that’s why I specified comparators or using the same instrument to take the measurement.
I very carefully avoided the can of worms where the instrument is within calibration spec but has different biases in different ranges. The usual example given is the use of gage blocks to ensure that micrometer readings are taken at 90 degree intervals to check for thimble or thread wear.
Even a micrometer calibrated against a gauge block can have systematic error. Take, for instance, gear wear in the mechanism. You may calibrate against a gauge block of 10 but due to gear wear, measuring something at 9 can be off – with every measurement. That’s not a random error, it is a systematic bias.
You’ve gone and opened that can of worms…
mea culpa, mea culpa, mea maxima culpa
I think we’re talking about different things here.
I was pointing out that the systematic bias should (can of worms aside) cancel when finding the difference between 2 measurements.
In both cases (6 & 11, and 4 & 9), the difference is 5, even though the absolute readings are off.
The difference doesn’t matter when you are trying to find A VALUE.
By definition, systematic bias can’t cancel. especially when you are using different measurement devices like when measuring temperature at different locations using different devices.
Yeah, but I’m only trying to find the difference, not the absolute values. The offsets cancel.
111 – 105 is 6, and so is 11 – 5.
If the systematic bias is constant through that part of the measurement range, the difference is independent of the systematic bias.
Part of the reason for taking repeated measurements of the same thing with the same instrument is to minimise the effects of systematic bias
D’oh! Ignore the last bit – it’s too early in the morning to be trying to make sense 🙁
Systematic error can’t cancel. It’s in-built, even with comparators.
If you get two different readings that are flagged by the comparator then how do you tell which one is correct? Does one of them have a systematic bias? Do they both have a systematic bias but different in value?
If you get the *same* reading from two different instruments and they both have the same systematic bias then how do you discern that?
There is a reason why it is called *uncertainty*. As the GUM discusses, there are always unknown factors affecting a measurement. “Unknown” means “uncertainty”.
I should have phrased it as as minimising the effect of systematic bias rather than canceling.
In the absence of the worms, the systematic bias of a single instrument should be constant through its range. Repeated readings of 1.0000″ – repeated readings of 0.9750″ using the same micrometer should be closer to 0.0250″ than using 2 different micrometers.
And, yes, there are lots of worms of various lengths, thicknesses and squirminess in that can, any of which can cause the systematic bias to vary within the measurement range.
I still can’t say that it will minimize systematic uncertainty because you neve know which one is correct or even if one is correct. If they are different then it is an indicator that something isn’t right. If they are both the same you just don’t know.
Kip is not dealing with uncertainty. He is showing a probability and a unique description similar to a 5 number description. Look it up. It displays the min/max values so you can see the range.
You are stuck in a box like many folks are.
Well if he is not dealing with uncertainty perhaps he should stop saying he is.
A very interesting group of posts, however, I don’t think loaded dice are good examples of the problem of measurement uncertainty. A dice face has no error, and we can’t know if particular dice are loaded until we undertake multiple experiments, an analogue Monte-Carlo for instance. In theory, drilling six holes on 1-side of a dice vs. drilling 1-hole on another should bias or ‘weight’ the outcome of 100 throws, that is, if it is an ‘honest’ dice …. Which is not your question.
You said at the end “Provide a childishly simple example, such as I have used, two measurements with absolute measurement uncertainties given added to one another.”
Linear errors are additive. Thus, adding two numbers having separate errors together, the errors also add.
So, for example, for two temperatures 62.3 degF and 31.2 degF (each having the same measurement uncertainty of +/- 0.5 deg, which is related to the precision of the instrument [a Fahrenheit meteorological thermometer has a single index marking whole-degrees, whereas a Celsius thermometer has a ½-degree index)] the answer is 93.5 (+/- 1.0 degF).
This is different to the question as to whether two numbers with error, are different.
For example, is 62.3 degF, different to 63.1 degF? Here as the measurement error is 0.5 degF for each measurement and they add together, the error of the difference is 1.0 degF. As they lie within the total error envelope, they could not be regarded as different.
Observing a thermometer to the next decimal place – estimating a value internal to the index, does not increase the precision, which is determined by the scale. Otherwise, why not estimate to two or three decimal places? The error of the best internal guess is half the index (0.5 degF). However, if we used a laboratory thermometer graduated in 0.5 or 0.1 degF intervals (over a shorter range or a longer thermometer), we could claim temperature was say 62.3 +/- 0.25 or +/- 0.05 degF, but such thermometers are
considerablyprohibitively more expensive than run-of-the-mill meteorological thermometers. They also measure far more ‘noise’.Unlike temperature or other continuous scales (km or g for instance), dice have only 1 to 6 dots for which there is no measurement error.
If a scale is not linear, as is the case for rapid-sampling resistance probes that use quadratic response calibrations, measurement errors become more complex to resolve. In such cases, where estimates are frequent (e.g., 1-Hz), internally linearised, instantaneous and (virtually) error-free, uncertainty is calculated relative to the medium being monitored, by re-sampling. Ice for example at one end and boiling water at the other.
Cheers,
Bill Johnston
http://www.Bomwatch.com.au
Bill ==> Well, a fair pair of dice is strictly speaking one that has not been intentionally tampered to be “loaded”. All are welcome to buy pair of casino-grade dice and roll them a million times, recording the values, and place them in a frequency distribution diagram. As we should know, even at a million rolls, the distribution will not be perfect 1-in-6 for each numbered face. But, the more rolls the closer to that perfection, if the dice are “fair”. Those tiny imperfections would add a bit of “chaos” (Chaos Theory definition) to the results (Buttrfly Effect, in a sense.)
On this one: “adding two numbers having separate errors together, the errors also add.” Tell Stokes and a few others here — they don’t want to add them except “in quadrature”. Subtracting the values also requires adding the uncertainties.
In my “thought-experiment” type example, I create uncertainty by hiding the dice — simulating hiding the real value (as in a rounded temperature, your example) — thus requiring us to fall back on the mean (central value) with the (simulated) absolute measurement uncertainty. We verify the correctness by adding two such values with absolute uncertainty (two individual dies) and show that rolling a pair of dice (in the lidded box) had the predicted values.
.
I think we agree about the adding of these measurements with absolute (measurement) uncertainties. Greatly appreciated.
You do good work at BOMWatch, sir.
I’m beginning to realise what is going on here.
Quote:
Linear errors are additive. Thus, adding two numbers having separate errors together, the errors also add.
After posting yesterday, I did indeed find several Youtubes telling us this. I am thinking that it might be part of a basic education syllabus to teach this simplistic view, as a precursor to the more correct method. Hence all the erroneous Youtubes out there.
This is the explanation: in linear addition/subtraction, if you add the stated uncertainties you will indeed obtain the maximum possible error of the result.
However, this takes no account of the probability distribution. The uncertainty derived in this way has an very small chance of actually occurring. (In the limit of an infinite number of uncertainties being combined so that a Normal distribution results, the linear addition of uncertainties will give a total uncertainty bound that is infinitely unlikely.)
The linear addition of uncertainties idea is intended as a simple idea for kids to understand. It’s not the way to deal with temperature records, but it is better than nothing. What has happened here is this basic idea for kids has been elevated to the “correct” method, which it is not.
Dear KB,
The original question posed by Kip Hansen was explicit:
“Provide a childishly simple example, such as I have used, two measurements with absolute measurement uncertainties given added to one another.”
He is not comparing means of populations of measurements; he is comparing two explicit values.
But there was a problem also.
The example to which he referred was: “two numbers absolute measurement uncertainty stated (as in 10 cm ± 1 cm plus 20 cm ± 5 cm).
An absolute measurement uncertainty of 10cm +/- 1 cm implies a tape measure having measurement intervals or indexes 2 cm apart – measurement uncertainty being ½ the interval = 1 cm. The second value 20cm +/- 5cm was measured using a different tape measure – one with intervals every 10 cm apart – the error, ½ the interval being 5 cm. We clearly cannot measure to the same level of uncertainty using different tape measures. And we should be suspicious if uncertainty values of two measurements of the same scale are different.
If we were measuring a distance of 1 metre, using a piece of string 10m long and no internal indexes, we would fold the string in half (= 5m), half again (2.5m), half again (1.25m) and we could measure 1m with an error of 0.125 m), otherwise the error of measuring using the unfolded string would exceed the estimated length 5-fold (1m +/- 5m). As most would know what 1m looks like, we would guess it more accurately than we could measure it using a 10m length of string.
Dice, or counts of individuals (x … n) or counts of “Yes” vs. “No”, have different distributions to continuous variables – each count or result is absolute, is measured without error and embeds no other information. (Rolling a die has to produce an outcome, the probably of a single face showing up is 1 in 6. HOWEVER, as there are only 1 to 6 possibilities, outcomes are not continuous. (See also https://math.stackexchange.com/questions/1204396/why-is-the-sum-of-the-rolls-of-two-dices-a-binomial-distribution-what-is-define)). So, there is no measurement uncertainty in “Yes” or counting 3 fish, or seeing 4 dots on a die, which is why I did not think “loaded dice are good examples of the problem of measurement uncertainty”.
Interestingly, the measurement scale of a Fahrenheit thermometer between boiling and freezing (212 – 32) = 180 whole-degree units, is 180/100 = 1.8 times that of a whole-degree Celsius thermometer, so it is more precise (measurement error is smaller). However, Celsius meteorological thermometers have a ½ degree index, which doubles the number of indices to 200, making them marginally more accurate (180/200) = 0.9 times that of a Fahrenheit one, which only has whole-degree indexes) – equivalent in practice.
So, to work all this out, one needs to know the instrument, the observing protocol and have some idea of the habits of various observers.
While it’s a pretty complex business analysing met-data, analysis can detect if observers recorded rainfall every day, or allowed small falls to accumulate in the gauge until it was ‘worthwhile’ to measure, for instance. Also, due to rounding up or down, for rainfall measured in points (1/100 inch), none occur in classes of 0.2mm, 0.4mm, 0.6mm, 0.7mm and 0.9mm per day.
These are some of the ‘tells’ I use to assess the quality of Australian meteorology data in analysis presented on http://www.bomwatch.com.au, and the effect of homogenisation on data attributes.
All the best,
Bill Johnston
http://www.bomwatch.com.au
Bill most of what you have written here is correct, but it is not relevant to the question. I think we have to address the fundamental misconceptions before we can progress to these other issues.
If you scroll up to my latest reply to Kip you will see I have addressed his Challenge in full.
Briefly, you are combining two rectangular distributions to arrive at a triangular distribution.
You are stuck in a box of using traditional statistics with normal distributions and using statistical parameters of mean and variance.
Describe a normal distribution with a 5 number description (box plot), such as what Kip has shown. Post it here.
“If we were measuring a distance of 1 metre, using a piece of string 10m long and no internal indexes, we would fold the string in half (= 5m), half again (2.5m), half again (1.25m) and we could measure 1m with an error of 0.125 m), otherwise the error of measuring using the unfolded string would exceed the estimated length 5-fold (1m +/- 5m). As most would know what 1m looks like, we would guess it more accurately than we could measure it using a 10m length of string. “
Small nitpick. Every time you fold the string you add to the uncertainty (think the curve at the fold) of the length. Uncertainty adds whether you add or subtract the stated values.
I agree with most everything you’ve posted. The only item I would add is that in a field instrument you *will* get calibration drift, i.e. systematic error, no matter what the resolution of the sensor is. Any estimate of measurement uncertainty must consider that the total uncertainty is a combination of both random error and systematic error. Since you don’t know either factor (closed box), assuming all measurement error is Gaussian and will cancel is not justified.
If you don’t know the value of some of your uncertainties, you need to do some further research to put a figure on them.
If you can’t do that, then you should try and make a realistic estimate.
It’s not a closed box. When you do an uncertainty budget you need figures to work with.
You do not assume that all measurement error is Gaussian. You assign a probability distribution shape to each individual uncertainty component.
When you have several (four or more) individual uncertainties of comparable size to combine, it so happens that the distribution of the result will be close to Gaussian, at least within the usual 2-sigma or 3-sigma limits that are usual.
However it does not need to be close to Gaussian, it just so happens it commonly is.
“If you can’t do that, then you should try and make a realistic estimate.”
THAT is exactly what Kip has done!
“It’s not a closed box. When you do an uncertainty budget you need figures to work with.”
The uncertainty is a figure all on its own. And it *is* a closed box. Unless you can quantify both components of uncertainty, random error and systematic bias, you have a CLOSED BOX!
You may not like that. But an inconvenient truth is still the truth!
“You do not assume that all measurement error is Gaussian. You assign a probability distribution shape to each individual uncertainty component.”
What distribution does systematic error in field measurement devices have? It’s certainly not Gaussian, uniform, rectangular, or triangular!
So what is it?
“When you have several (four or more) individual uncertainties of comparable size to combine, it so happens that the distribution of the result will be close to Gaussian, at least within the usual 2-sigma or 3-sigma limits that are usual.”
Nope! If the uncertainties happen to be skewed, something that has hysterisis for example, nothing you can do will make the distributions close to Gaussian.
I have laughed for two days over the fact that you and your buddies missed the obvious in my example of boards with two different measurement errors. Those measurement uncertainties form a bi-modal distribution! The average of a bi-modal distribution is meaningless! The CLT may cause multiple samples to cluster around the population mean but the population mean itself is meaningless in a bi-modal distribution. The mean tells you NOTHING about the population itself.
This is why statisticians and climate scientists get uncertainty in the real world SO WRONG.
The distribution of temperatures along a line of longitude is not Gaussian. You can’t make it Gaussian no matter what you do. Temperatures at the south pole and the north pole will have different ranges and different variances. So will temperatures at every plus/minus latitude line crossing that longitude line because winter temps have a wider variance than summer temps.. You wind up with a multi-modal distribution of different random variables, all with different variances.
The mean of samples of that data may cluster around a population mean but that population mean is truly meaningless. You simply cannot just average the data elements of a multi-modal distribution and come up with anything that tells you about that population. Especially when you ignore the standard deviation of the stated values!
This doesn’t even begin to get into the fact that temperature is a piss poor proxy for enthalpy!
That is one reason why the satellite record is not a very good index. It’s better than the surface record but is it fit for purpose? I have my doubts. It is *NOT* a global mean temperature. It may be a global mean temperature proxy, but it is a poor one. Even the satellite record does not consider the differences in variances generated from samples drawn from different latitudes or that the samples form a multi-modal distribution. It just lumps them all together and tries to account for instrument uncertainties, nothing more.
And before you start with the “anomaly” argument, anomalies inherit the parent variances. Taking an anomaly does *NOT* cancel out variance. In fact, because you are subtracting two random variable distributions the variances of the parent distribution ADDS making the variance of the anomaly LARGER, not smaller!
What distribution does systematic error in field measurement devices have? It’s certainly not Gaussian, uniform, rectangular, or triangular!
If the systematic errors of the field measurement devices are independent of each other, it’d be fair enough to treat them as individual uncertainties. The uncertainty of the average of the device readings would have close to a Gaussian distribution as long as there were enough devices in the list.
If you mean that you don’t know what is the systematic error, well that problem is common to both “absolute” uncertainty and to the usual method.
If you mean that you don’t know the shape of the distribution of the systematic errors, in that case I’d give them a rectangular distribution. It actually makes little difference to the overall uncertainty, bearing in mind we are going to round it to one or at most two digits.
“If the systematic errors of the field measurement devices are independent of each other, it’d be fair enough to treat them as individual uncertainties. The uncertainty of the average of the device readings would have close to a Gaussian distribution as long as there were enough devices in the list.”
You just don’t get it! Systematic biases in field instruments are not knowable. They are not amenable to statistical analysis.
The uncertainty of the individual device readings are simply not guaranteed to be close to a Gaussian distribution. Northern and southern temperature readings *are* guaranteed to be bi-modal at the least and are guaranteed to have different variances. And that is on top of each of the measuring devices having unknowable systematic biases of varying amounts.
“If you mean that you don’t know what is the systematic error, well that problem is common to both “absolute” uncertainty and to the usual method.”
You already admitted that *YOU* don’t know the systematic biases of *any* field temperature measuring device. How many of the temperatures included in the global average temperature have their systematic biases known or even accounted for? You can’t find systematic biases by statistical analysis of the stated values!
“If you mean that you don’t know the shape of the distribution of the systematic errors, in that case I’d give them a rectangular distribution.”
Why do you assume the uncertainty intervals have a uniform distribution? If there is systematic bias this *can’t* be the case! The bias will create a skewed set of probabilities at the very least!
All you do by assuming a uniform distribution is set things up so you can assume that all uncertainty cancels and you can use a statistical analysis of stated values to come up with an uncertainty figure.
” It actually makes little difference to the overall uncertainty, bearing in mind we are going to round it to one or at most two digits.”
It makes a HUGE difference. Your stated values should have no more significant figures than your estimate of uncertainty. Otherwise you are saying you know the stated value to a precision that is not justified by the possible uncertainty.
If your values are rounded to one or two figures then how do you get uncertainties in the hundredths digit?
bgwxyz thinks he can reach into the past, discern all “biases” in historic air temperature data, and correct them out.
These people ARE gods!
karlomonte said: “bgwxyz thinks he can reach into the past, discern all “biases” in historic air temperature data, and correct them out.”
I never said that. I never said most of the things you claim I have said. I can even predict with reasonable confidence that you will make claim of something I said in the future that will turn out to be untrue.
And now The Backpedal—don’t run away from your pseudoscience, embrace it!
Hubbard and Lin.
Kip,
In your essay, under the graph labeled
Pair of Fair Dice — 1,000 Rolls
is the statement
very close to the same number for 3s and for 11s and the same numbers for 1s as for the 12s
I don’t read the values of 3 and 11 as being “very close to the same number” but I’ll leave that as a difference of opinion. It is the second clause that looks to me to be totally wrong “the same numbers for 1s as for the 12s” as 1 is impossible and the value for 12 shows as being the same as, or very close to, the value for 2.
Am I misunderstanding something or is the statement a simple error?
Andy ==> You are the second “sharp eyed reader” to catch that typo. Thank you. I have corrected it.
The values for 3s and 11s are comparable (very close), and if we incresed the number of rolls, they would be closer to even.
> Again, if we had rolled the pair of dice a million times, the distribution would be closer to perfectly normal – very close to the same number for 3s and for 11s and the same numbers for 1s as for the 12s.
Two issues which cast doubt on the author’s credibility:
Since the graph he is referering to shows the actual number of occurrences of each possible value rather than the percentage of occurrences, this statement is incorrect and implies a common misconception (in one form, known as “the Gamblers Fallacy)”
The number for 3s and 11s and 1s and 12s are very unlikely to approach the same values in this situation.
As the number of trials increases, the relative difference will tend to the theoretical, but the
actual differences with increase.
And calling that distribution normal reveals another fundamental misconception by the author. A “normal” distribution is not triangular, it is something very different.
StuM ==> To create a frequency distribution chart, one uses the number of occurrences of each value in a data set. Excel will happily do it for you, or Matlab, or Panoply.
(Note: The “1s and 12s” is a typo in the essay, should be “2s and 12s”).
There is no wrong implication in the frequency charts….one can make percentage bar charts if one wishes, but frequencies is the usual. Since the charts are 1000 rolls, the number of occurrences of each number over 1000 give the %. The chart will look the same. For a single fair die, about 16.6 % for each number. For the summed values of a pair of dice, each has a different % — figured the same way — or you can use the chart linked which gives the percentages .
As for “normal distribution” — I clearly stated that it “looks like” (appearance) and then overlay them to show the difference. This is a non-issue (except for die-hard statisticians).
In a closed box how do you know what is happening?
Measurement uncertainty consists of two factors, random error and systematic bias.
For most measurements you simply don’t know the size of either factor. That is why u_total = u_systematic + u_random describes an interval whose probability distribution is unknown.
If even one of the dice in the closed box has a systematic bias you’ll get a shifted set of roll totals. But you’ll never know what it is because the box is CLOSED!
You are basically repeating the typical output of a statistician and climate scientist that lives in a statistical dimension where there is no such thing as measurement uncertainty.
Here is an example of the conflicts that we face right now.
…………………….
This example is about measurement of sea surface temperatures by Argo floats. Here is a link:
https://www.sciencedirect.com/science/article/pii/S0078323422000975
It claims that “The ARGO float can measure temperature in the range from –2.5°C to 35°C with an accuracy of 0.001°C.”
I have contacted bodies like the National Standards Laboratories of several countries to ask what the best performance of their controlled temperature water baths is. This UK reply was from the National Physical Laboratory, Teddington.
“NPL has a water bath in which the temperature is controlled to ~0.001 °C, and our measurement capability for calibrations in the bath in the range up to 100 °C is 0.005 °C.”
Without more dives into the terminological jungle, readers would possibly infer that Argo in the open ocean was doing as well as the NPL whose sophisticated, world class conditions are controlled to get the best they can from their budget.
It would be logical to conclude that the above Argo authors were delusional. \
A controlled water bath in a lab should do better than a float in the wild open seas.
…………………………….
It is quite easy to find conflict after conflict like this in climate research. Seems to me that many people who try to do science these days have little idea of what uncertainty is and are seeking promotion by quoting performance better than any others have quoted.
Thanbk you, Kip, for continuing to highlight the important, basic problem of optimistic guesswork versus measurable reality.
Geoff S
Geoff ==> “We” do try — but it is an uphill battle — I struggle to find examples that are simple enough to communicate to most readers, without offending the unbending “only my way” statisticians. None have taken my challenge to provide an example that shows “adding in quadrature” works with real things in the real world in a case such as a rounded temperature measurement. till, I wait for the miracle —
The Argo float claim is a ridiculous travesty of science.
Followed closely by idiotic nonsense like these impossibly small milli-Kelvin GISS air temperature “error bars”:
karlomonte,
Note how the “95% confidence limits hug the shape of the smoothed numbers. They should hug the shape of the raw numbers. Smoothing is not supposed to improve confidence limits.
Geoff—only in climate pseudoscience is nonsense like this rewarded.
JCGM 6:2020 says that smoothing can be used to abate the extent of uncertainty and that moving averages and the like are widely used for that purpose.
How can smoothing not improve the confidence limits? Most smoothing techniques use a function where the partial derivative ∂f/∂x < 1/sqrt(N). See JCGM 100:2008 equation E.3 and 10.
bdgwx ==> “Smoothing” reduces the understanding of the actual data — intentionally. Some want reduced data (some want everything reduced to a single number they can report to the President).
Smoothing, regardless of the book definition, does not and cannot abate the extent of uncertainty. Except in a psychological way — back in the real world, if there was uncertainty in measurement or processing or data evaluation, the uncertainty is only obscured or disregarded.
I will admit that “statistical uncertainty” is not the same as “real world uncertainty” and has its own rules.
KH said: “Smoothing, regardless of the book definition, does not and cannot abate the extent of uncertainty.”
Those are the words from the GUM (JCGM 6:2020) and backed by the law of propagation of uncertainty (JCGM 100:2008). And coincidentally that statement follows an example of a temperature time series. The irony there is palpable.
KH said: “I will admit that “statistical uncertainty” is not the same as “real world uncertainty” and has its own rules.”
That is absurd. Real world uncertainty is statistical.
Your statements are strikingly inconsistent and at odds with those from JCGM, NIST, UKAS, ISO, and others.
Let me get you thinking about the issue by asking a question. Do you really think those countless certificates that NIST issues, which are based on the law of propagation of uncertainty, are meaningless?
bdgwx ==> “Real world uncertainty is statistical.” This belief is why you don’t get the point of the essay.
Exactly! Since when does a unique description of a distribution automatically get declared wrong just because it offends someone’s ingrained paradym of what statistics should be.
I will guarantee you he has never mike’d a crankshaft journal and tried to get the same force applied by the instrument heads at each measurement point.
“Do you really think those countless certificates that NIST issues, which are based on the law of propagation of uncertainty, are meaningless?”
Once that NIST calibrated instrument gets installed in the field the NIST calibration drifts which results in a measurement uncertainty.
You continue to demonstrate that you have no real world experience in the field of metrology. NIST calibration is *guaranteed* to not last over a period of time. Mud daubers build nests in measurement box air intakes. Ants can leave detritus affecting the environment around the sensor. Ice can affect air flow. Heating and cooling can affect the sensor itself changing its characteristics.
Sidebar. Downloaded free R yesterday. Since g’kids are now on a plane to San Francisco Bay Nerdland, I will install and acclimate on rainy days. Goal will be to replicate some of your offerings.
For one, because it throws information away and hides the actual variation. The only purpose of smoothing is for visual presentation.
Looks like I struck another nerve with this one…
Smoothing might work well in a noisy environment to increase confidence limits. Just how much noise is there in a temperature measurement? Is the noise level significant compared to the other uncertainty factors?
But hang on. These are not error bars, they are error ribbons, and because they are calculated vertical to the line (i.e. vertical in y) they appear to more closely hug the line when it is rising (or falling).
The other point is that the precision of an electronic instrument does not indicate to what extent averages calculated say every 30-seconds, from 30 noisy samples, ‘accurately’ measure the medium. Maximum, or average SST for a day may be considerably impacted by spikes due to the instrument, rather than truly reflect the water column.
All the best,
Bill Johnston
from the document: “The ARGO float can measure temperature in the range from –2.5°C to 35°C with an accuracy of 0.001°C.”
This is the resolution of the sensor itself! It is not the measurement uncertainty of the entire float! It is no longer available on the internet but there was an estimate made of the float uncertainty by the manufacturer and it resulted in the typically seen uncertainty of +/- 0.5C. This was based on the variances of possible water flow by the sensor, the instantaneous salinity of the water itself, and several other factors I don’t remember.
Think about it! A SINGLE barnacle attaching itself to the water inlet can affect the water flow by the sensor! That’s a systematic bias that simply can’t be accounted for in laboratory calibration processes. But it *is* part of the total float measurement uncertainty!
All too often in conventional climatology, there are unstated assumptions.
Clyde ==> Far more than we can name or list. One only needs to read the actual process for creating the single-number Global Satellite-“measured” Sea Level to see just how crazy this can be.
Uncertainty can be calculated for measurements, aka data.
But the global average temperature includes little or no data.
There are adjusted raw temperature numbers
After adjustment what is left is no longer data
The numbers are what a person(s) thinks the data would have been if measured correctly in the first place
And then there is infilling, aka guessing.
How can there be an uncertainty calculation for made up numbers?
So, the bottom line, in my opinion, is uncertainty claims for the global average temperature statistic are meaningless.
.And that means official claims of the GAST margins of error can not be taken seriously. Especially for pre-1920 numbers, that include a LOT of infilling and far too few measurements outside the Northern Hemisphere.
“Hey, I would love to be proved wrong on this point”
I would like to claim I have proved you wrong, but that would require me knowing exactly what main point you were making.
The first step in estimating uncertainty was missed:
Should we trust the people collecting and compiling the data?
If the answer is “no”, and it is for me, the data are irrelevant.
Tne surface temperature statistics are not fit for purpose, IMHO.
Exactly right, there is no way to quantify what these pseudoscience persons do to data.
One of the problems demonstrated here is the easy availability of stats functions which you can run any dataset through. I have ‘R’ on my computer which comes builtin with a dozen or so stats functions that I can use on a variety of data. If so inclined I can download scores of stats libraries containing hundred of stats functions.
Now, each stats functions is great. They are powerful and contain worked examples with each one. AND, instructions on how to use them.
Just because we can average a few hundred readings does not mean we should do so.
The same applied to complex stats functions. Just because I have them does not mean I should use them in thoughtlessly!
Steve ==> Ah, yes — quite right, Slim!
I use R on occasion, wonderful array of possible functions — they come with instructions, but the instructions do not contain all the caveats! When and under what circumstances the function produces valid usable results, what the results of this function mean, when NOT to use this function……
Kip,
You managed once again to produce a lively thread with many very good comments. I fetched the electronic version of Brigg’s book yesterday and am about four chapters in — just finishing what he calls the introductory material. Interesting, but lengthy so far…
Here is one more interesting tale about uncertainty. When I was consulting in the Vancouver (WA) area long ago, and teaching metrology at WSU, I had a student who worked at hp’s printer manufacturing operation. He told me that one printer model had 100+ critical dimensions of subcomponents required to make sure it functioned as it should. This would be a lot of measurements to make if one were doing it by brute force. These parts were produced with plastic injection molding. The Phillips Company (Netherlands) was doing the injection molding. They had shown conclusively that just weighing the produced parts was sufficient to demonstrate that a printer would meet functional requirements.
Pretty darned good engineering.
Regards to you…
Kevin ==> Interesting, a post-hoc shortcut. Bet it took them a while to be confident in their method.
But, it is obvious that with injection molding there are a limited way in which parts can come out wrong. The external dimensions are set by the mold. So any overweight means a casting gate or something failed to be cut off. If underweight, then a void exists.
Thanks for the story! Enlightening.
Kip—As none of the trendologists dared take up the challenge, I figured I would take a swing. When I first read the article, the difference between 5 and 3.5 intrigued me, but I skipped past thinking about the implications. Having re-read, here are my ideas.
Both problems, 1 die in a box and 2 dice in a box, can be easily modeled using the BIPM GUM Type B uncertainty evaluations (the statisticians don’t like these, even though they are rooted in probability and statistics!). The ranges of possible values and the frequency distributions are known exactly, so these are really simple calculations. Figure 2 in the GUM (below) just happens to show both the rectangular and triangular Type B cases along with the equations. The Type B equations give the variances as the dark shaded areas:
1 die
range: 1-6
distribution: rectangular
half-width a: 3.5
u^2 = a^2/3.5 = 4.1
u = sqrt(4.1) = ~2.0
The square root of 3 shows up a lot in Type B calculations. Converting this to an expanded uncertainty with the standard coverage factor k = 2 gives:
U = 4.0
This is larger than the 2.5 geometric value!
2 dice
range: 2-12
distribution: triangular
half-width a: 5
u^2 = a^2/6 = 25/6 = ~4.2
u = sqrt(25/6) = ~2.0
The expanded uncertainty with k = 2 is:
U = 4.0
I’m quite a bit closer to the geometric absolute uncertainty of 5, but not exact. More importantly, both the 1- and 2-die calculations give nearly the same value.
What is the difference here? First, the geometric absolute uncertainties are not standard deviations, so blindly combining them with RSS to get U = 3.5 I don’t think is valid. Second, the Type B variances are designed/intended to be combined with other uncertainties which are all quantified as variances.
“More importantly, both the 1- and 2-die calculations give nearly the same value.”
But the range of the two dice is twice as big, so the same absolute uncertainty covers a smaller fraction of the range. This means that 2s and 12s are now outside the expanded uncertainty interval.
Demonstrating again you have no idea about which you pontificate, just toss garbage up and hope it sticks.
What now troll? Are you saying the range from 2-12 is the same as the range from 1-6?
Heheheheh please continue…
Usual refusal to answer.
“WHAAAAAA ANSWER ME PULEEEESE I DESERVE IT!!”
The neutronium-dense bellcurveman head still can’t sort out that I’m done trying to educate him.
You’re not fooling anyone with this tantrum.
I’m not demanding you answer me, I certainly don’t want you to “educate” me. I know you won’t answer me, you know you won’t answer me. You won’t because if you did you you’d have to accept the contradiction in your claims.
HAHAHAHAHAHA
He wrote “tantrum”…
Still not fooling anyone.
More delusions, I don’t care what the mythical “anyone” thinks.
That’s self evident.
Idiot.
bellcurveman needs a reset, it is stuck on its hamster wheel.
Your calculations for the rectangular distribution are a little off. The half width of the distribution is 3, and you divide it by root 3, not 3.5.
u = sqrt(3^2 / 3) ~= 1.73
For a descrete distribution the more exact formula is
u = sqrt((6^2 -1)/12) ~= 1.71
Nickpick Nick has taught you well, young padowan.
Just trying to be helpful. Feel free to correct me the next time I make a mistake.
Karlo ==> Appreciate the effort….you show clearly that the statistical approaches produce results that are not true physically — in the real world.
For a roll of two dice, starting with an unknown value and its actual real world uncertainty, the correct result for “U” (uncertainty) is 5.
One can prove this easily by rolling two dice just three or four dozen times (or playing out a game of backgammon) — you will get all of the 11 possible combined values, each will satisfy the condition : 7 +/- 5
“7 +/- 5” states the full uncertainty.
So now we have to ignore the GUM?
This is all very odd, considering how many times I’ve told I have to understand metrology.
Clown.
I thank you for your informative response, and will give it all the attention it deserves.
“So now we have to ignore the GUM?”—bellcurveman
Hypocrite.
Still deflecting from the point I see.
Should we reject the use of standard uncertainty as advocated by the GUM in favor of only looking at a measure of uncertainty that always contains all possible values, as Kip advocates.
Thanks for hijacking yet another subthread with your sigma/root(N) nonsense.
You’re welcome. But in this case I meade no mention of the SEM. I’m simply trying to determine if you consider the definition of standard uncertainty as defined in the GUM to be wrong, and the only true uncertainty range should be one covering all possible values?
I follow up question might be if you consider equation 10 in the GUM to be valid? Or do agree with Kip Hansen that this is probabilistic nonsense and only correct way to handle measurement uncertainties is to add them?
You are mixing things together and driving a discussion of the GUM when it doesn’t apply.
An average of temperatures IS NOT A MEASUREMENT or a MEASURAND.
You and bdgwx need to give up the claim that an Average is a functional relationship. It is not, it is a statistical calculation used to describe a parameter of a distribution.
The GUM defines an average in only one circumstance, multiple measurements of a single measurand with the same device. With the proper assumptions being met, random errors can be canceled.
If you want to refer to the GUM, use it for describing single temperature measurements, not for averages of independent measurements of different things.
“You are mixing things together and driving a discussion of the GUM when it doesn’t apply.”
It was Karlo who brought up the GUM at the start of this thread. And despite the rest of your comment, I said nothing about averaging. We’re talking about the definition of standard uncertainty, and how it relates to the the sum of two dice.
“An average of temperatures IS NOT A MEASUREMENT or a MEASURAND.”
Then explain how it can have a measurement uncertainty.
“You and bdgwx need to give up the claim that an Average is a functional relationship.”
Why? It is a functional relation, and you’ve never given any explanation as to why you think it isn’t. I suspect you just don’t understand what functional relation means.
“The GUM “defines an average in only one circumstance, multiple measurements of a single measurand with the same device.”
The beauty of maths is that an equation can be used without having to have an example to demonstrate it. Equation 10 is a general purpose equation which can be used for combining any set of independent uncertainties, regardless of whether they have given you an example or not.
Also, they define average in C.2.19. Note 2 says “The average of a simple random sample taken from a population is an unbiased estimator of the mean of this population”.
“. Equation 10 is a general purpose equation which can be used for combining any set of independent uncertainties, regardless of whether they have given you an example or not.”
NO! It can’t. I’ve given you the quote on that.
Here it is again!
3.1.7 This Guide treats the measurand as a scalar (a single quantity). Extension to a set of related measurands determined simultaneously in the same measurement requires replacing the scalar measurand
and its variance (C.2.11, C.2.20, C.3.2) by a vector measurand and covariance matrix (C.3.5). Such a
replacement is considered in this Guide only in the examples (see H.2, H.3, and H.4). (bolding mine, tpg)
This applies to Eq. 10. It is for multiple measurement of the SAME THING – i.e. measurand, Measurand in the singular, not in the plural!
YOU NEED TO STOP CHERRY PICKING and actually read and study up on the subject.
This has been pointed out to you MULTIPLE times. When are you going to do it?
Right there in B&W, and he still denies reality.
Where in the description does it say they all have to be the same thing? Your swiveling is a joy to behold. A little while ago you were trying to use it to calculate the volume of a cylinder, using the radius and height. Two different things.
Solid neutronium, inpenetrable.
“””””The beauty of maths is that an equation can be used without having to have an example to demonstrate it. Equation 10 is a general purpose equation which can be used for combining any set of independent uncertainties, regardless of whether they have given you an example or not.””””””
A perfect example of a cherry-picker. Every equation has assumptions behind it that must be met.
“”””Also, they define average in C.2.19. Note 2 says “The average of a simple random sample taken from a population is an unbiased estimator of the mean of this population”.”””””
You just won’t take the time to learn what measurements are really about will you? Section C provides standard definitions of statistical concepts. The use of these concepts all have certain assumptions that must be met. If you had searched the document, you would have found that C.2.19 is only mentioned ONCE in the entire GUM.
Section 4.2.1 says,
“””””Thus, for an input quantity Xi estimated from n independent repeated observations Xi,k, the arithmetic mean iX obtained from Equation (3) is used as the input estimate xi in Equation (2) to determine the measurement result y; that is, iixX=. Those input estimates not evaluated from repeated observations must be obtained by other methods, such as those indicated in the second category of 4.1.3. “””””
Note the reference, REPEATED OBSERVATIONS! This appears numerous times. D.6.1/E.3.3/E.3.4/E.4.2/E.4.3 (especially where n=1 for a single measurement/F.1.1.2/G.1.2/G.3.2/G.5.2/G6.3/G.6.5. You should take some time to read these and study them. They are not something you can skim and understand, i.e., cherrypick!
“Every equation has assumptions behind it that must be met.”
Indeed there are. But not of the assumptions in equation 10 fail if you are using the mean as your function, and all your uncertainties are independent.
“If you had searched the document, you would have found that C.2.19 is only mentioned ONCE in the entire GUM.”
I was just pointing out that when you said
You weren’t entirely accurate.
“Note the reference, REPEATED OBSERVATIONS!”
Which has nothing to do with the point I’m talking about. Namely using the standard equation for propagating independent uncertainties and applying it to a mean, or a sum.
Once again, you cannot reduce uncertainty with your mindless averaging.
True words!
Again avoiding the question. I’m not talking about averaging, I’m asking about the definition of uncertainty and the correct use of equation 10. Forget averaging, what do you think it says about the uncertainty of the sum of two values with independent uncertainties?
Poor baby, doesn’t get the respect he thinks he deserves.
You’ve been given multiple quotes on what the GUM says.
But you won’t read any of them for meaning.
Why is that?
The GUM’s use of standard uncertainty is *NOT* what you think it is!
Bellman doesn’t understand the difference between singular and plural.
measurand – singlular
measurands – plural
GUM:
The objective of a measurement (B.2.5) is to determine the value (B.2.2) of the measurand
bellman reads that and sees “measurands”.
More lying behind my back. When have I ever suggested that a single measurement will give you the value of more than one measurand?
When you claim that uncertainty has a normal distribution!
I don’t, and that has nothing to do with my question. I sometimes think I might be arguing with a not very sophisticated AI.
Bellman ==> “The GUM” does not trump reality.
Roll the dice, count the dots, do it dozens of time….get a ten year old to do it for you.
““The GUM” does not trump reality“
No, and it’s my favorite document on the subject, but it is the document I’ve been asked repeatedly to follow in order to properly understand measurement uncertainty. There are plenty of other books that explain how to properly combine uncertainties using adding in quadrature.
“Roll the dice, count the dots, do it dozens of time….get a ten year old to do it for you.”
I’ve already done that, you rejected the exercise. If you mean I have to limit myself to just pairs of dice, I’ve already explained why that’s a bad example. But in any event, counting dots isn’t going to resolve the issue, because it’s one of definition.
If you say “absolute uncertainty” means a range that has to cover all possible values, then your dot counting exercise will give you that. If you define “absolute uncertainty” in terms of what’s reasonable to expect, then the dot counting exercise can show that also.
It’s really a question of whether you want absolute certainty in your uncertainty values, or want something less certain but more useful.
“No, and it’s my favorite document on the subject, but it is the document I’ve been asked repeatedly to follow in order to properly understand measurement uncertainty.”
It is your favorite document to cherry pick from – not to understand.
3.1.7 This Guide treats the measurand as a scalar (a single quantity). Extension to a set of related
measurands determined simultaneously in the same measurement requires replacing the scalar measurand
and its variance (C.2.11, C.2.20, C.3.2) by a vector measurand and covariance matrix (C.3.5). Such a
replacement is considered in this Guide only in the examples (see H.2, H.3, and H.4).
Operative word: Measurand – singlular!
Bellman ==> And now you’ve finally gotten to the point: “If you define “absolute uncertainty” in terms of what’s reasonable to expect,” all this time you’ve been going on and on, mutliple essays here you comment over and over, basically saying nothing more significant than “No, you’ are wrong, my favorite book says so.”
You simply substitute a fallacious definition for the true definition.
Absolute Uncertainty never ever did or will be the same as “what’s reasonable to expect“.
For heaven’s sake, what kind of science do you practice?
Hint: it starts with the letters P and S…
These books that define uncertainty are all books on metrology. If you are saying all their definitions are wrong, so be it. But don’t expect the rest of the world to agree with you just because that’s the way you want it.
None of these are “my favorite books”. My only interest in all this was pointing out an obvious error in Tim Gorman’s assertion that uncertainty of the mean grew with sample size. He insisted I read Taylor, Bevington and the GUM. Now you are saying we should ignore everything they say, and for some reason the same people who insisted I read these books are now happy to go along with you saying they are all wrong. I just think that’s strange. But al.so amusing seeing some tie themselves in knots trying to avoid admitting there is a contradiction here.
“No, and it’s my favorite document on the subject”
Argh, major typo. That should have been
“No, and it’s not my favorite document on the subject“
Kip,
You are dealing with folks whose main purpose is to castigate folks who threaten the status quo of Global Temperature.
Your example is not uncertainty as defined by the GUM. Your example is not a measurement of a measurand.
It doesn’t provide a standard deviation.
It IS an example of a TYPE of uncertainty. You are showing a type of mean plus/minus a value that covers the range of possible, discreet values.
I am disappointed that some folks can not think outside the box of their limited statistical training and see what you are trying illustrate!
“It IS an example of a TYPE of uncertainty.”
Which would be reasonable if he didn’t imply that any other type of uncertainty is wrong.
His adding all uncertainties is exactly what Taylor does in his provisional rules, demonstrating a worse case. But Taylor then goes on to explain in certain cases (ie independent uncertainties) this will be unnecessarily large, and shows why adding in quadrature will give a better result.
I’m simply baffled why people who have spent the last two years bashing me over the head with Taylor, now turn round and applaud someone saying you should never add uncertainties in quadrature.
Oh my gosh, cry me a river Herr Doktor Vhiner…sheesh.
Get over yourself.
Squirm all you like – anyone reading these threads can see why you won’t explain what you believe is correct. I’ll ask again, so you can ignore it and whine some more – if you add two measurements, each with independent uncertainties, do you believe it’s better to use the GUM’s approach, adding in quadrature, or Kip’#s approach, plain adding?
Once again, underneath the bellcurveman smokescreen, the real issue:
—TG
And the deflection continues. Again, I am not asking about averages here, I’m asking about adding two things.
Back to gaslighting—maybe your good pally blob will be along to bail you out with a stupid video meme.
And DYOFHW.
Appreciate being mentioned in a response to Bellman. It’s the same kind of left handed compliment that Nick Stokes gets by being mentioned over and over in posts he did not even contribute to.
But Bellman’s posts are much more erudite and informative than mine. All I do is point out the most obvious boners in some of the responses he gets. Eyes burning?
Have you figured out DVM uncertainty yet, blob?
Thanks, but I can’t agree about being more erudite and informative than you.
I’m not asking you to do my homework, I’m trying to get you to say what you believe.
Can’t understand that I won’t play in your three-ring clown show?
What is wrong with you?
A reasonable question for which I don’t have an answer.
But in this case it’s just the enjoyment of watching you squirm. You know why you won’t answer the question, but your pathology won’t let you not get the last word. So you’ll just have to keep answering my question with an insult which just makes you look more petulant.
bellcurvewhinerman now just spams…
Irony overload!
Then why do you “average” uncertainties to get an average uncertainty? The uncertainty of one measurement can’t tell you the uncertainty of the next. You have to ADD the two in order to find an average.
“Then why do you “average” uncertainties to get an average uncertainty?”
If you want an average uncertainty you average the uncertainties. But I don’t want the average uncertainty, I want the uncertainty of the average.
It’s pointless me keep telling you this because you just can focus on the distinction.
Then why do you keep using the average uncertainty as the uncertainty of the average?
Why do you average .08 and .04 to get an average uncertainty when you *know* that the only two valid uncertainties are .08 and .04? That average uncertainty neither describes the uncertainty of the average or the real world!
I am reminded of the old tale of the mule that wouldn’t listen.
Now he seems to think he’s on some kind of public expose crusade. The guy is whacko.
Perfect!
The CLOSED BOX is the best example of an uncertainty interval I’ve ever seen. Since u_total = u_random + u_systematic you don’t have a chance of defining a probability distribution for the uncertainty if you don’t know both the systematic bias and the random error. The uncertainty interval box is CLOSED.
Yes, if temps are rounded to the nearest integer, you have a closed box with no idea what the “mean” of a distribution is. Was the reading +0.3 or +0.45 or -0.39 or -0.1. All you know is that the reading was somewhere around the integer recording.
It is like averaging all the wrong climate models and having someone say, “Jumping Josephat we got the correct answer!” The probability of different single readings totally canceling must be astronomical.
For those who think it is possible, there is enough data out there to the tenths to allow determining what the distributions are when rounding. It would be interesting to see what distribution that would come up with.
Kip—I’m not arguing you are wrong (this is why I called it the “geometric uncertainty” for lack of a better term). I’m just pointing out that the 5 is not a standard deviation/variance, which means that doing an RSS combination from two single die results isn’t valid.
At the outset of this exercise I had no idea how it would turn out, it was gratifying that it came out closer to U=5.
What a lot of people skip past with regard to the GUM is right up front in the title: it is a standard way of expressing uncertainty that allows uncertainty values to be reused by others. Before the GUM came around there were multiple approaches to uncertainty which tended to be incompatible.
These trendologists seem to think they can skim through the GUM and cherry pick something that gives the answer they want, which turns out to be sigma/root(N). It is what they eat and breathe.
karlo ==> As “residual sum of squares (RSS), also known as the sum of squared estimate of errors (SSE),” I don’t think I have done that, have I?
“the 5 is not a standard deviation/variance” Correct, it certainly is not and not intended to be — it is the full uncertainty of the sum of two values with given absolute uncertainties.
“It is what they eat and breathe.” They need a reality-based diet.
Thanks for participating, appreciate it.
Thanks, Kip:
re RSS—no you haven’t, but someone else did for the previous article, and you compared the result in this one. The difference in the two results piqued my interest, so I wanted to see what a Type B analysis would give.
You are exactly correct. The GUM is about ONE thing, the uncertainty of a single measurand. It addresses the situation where multiple measurements are made to calculate a measurand. A volume of an object is a good example. Bellman and bdgwx try to extend this to averages of individual measurements of different things. It simply doesn’t fit.
Tavg is a distribution. It is not a measurand covered by the GUM. Averages of Tavg’s is again a statistical calculation, which is not a measurand.
Armchair metrologists, mindlessly plugging stuff into equations they don’t understand. And when they are called out for this nonsense, they throw up smoke screens of unintelligible unreadable word salads.
Same question I asked Carlo, forget averaging, what does the GUM say about the uncertainty of the sum of two values? Do you accept that could be a measurand? Would equation 10 agree with Kip’s claim that you can only add uncertainties.
No, it is not a measurand. From the GUM.
The GUM deals with physical MEASUREMENTS. Things you do to an object or phenomenon to obtain physical quantities.
Averages are not measurements even though you would like them to be. They are statistical calculations used to obtain one statistical parameter of a distribution of data. Even if an average happens to coincide with a measurement, the average IS NOT a measurement of a measurand.
The GUM only applies to the action of measuring a temperature, i.e., obtaining a single reading at a given time. In today’s world, that is a singular data point that has no distribution of repeated measurements of the same thing associated with it. Therefore, a Type A evaluation of uncertainty is impossible to perform. That leaves only Type B uncertainty to be determined.
The GUM says:
In other words, use the standard uncertainty developed by experts like the NWS or NOAA.
I suggest that you start to learn what a random variable is and how it applies to calculations used to determine “average” temperatures. Then learn about the variance calculations used to describe a random variable and how those variances are treated when you add random variables to obtain an average.
Also from the GUM:
You can *NOT* assume or extract a probability distribution from a GUM standard uncertainty!
“You can *NOT* assume or extract a probability distribution from a GUM standard uncertainty!”
I never said you could. I’m just rejecting the claim that it’s possible to have a standard uncertainty without a probability distribution. You may not know what it is, but it has to exist.
Liar, of course you did. (and unlike yourself, I don’t have a huge database of enemies’ quotes to prove it).
Translation: “I’m going to make an accusation that Bellman said something wrong, but please don’t ask me for any evidence.”
I mean, I’m sure it’s possible I’ve misspoken or mistypes something to that effect, but it would be helpful if you could point me to it so I can make a correction.
Your buddy blob sure maintains up-and-down that it is possible, and I haven’t seen you take him to task for it.
I have no idea what “blob” whoever they are has said. I try to speak for myself. It’s hard enough keeping up with all the nonsense from you and the gorms, without having to read everyone else’s statements.
What is the probability distribution when one value in the set has a probability of ONE and all the rest have a probability of ZERO?
You’ve just described it.
What is it named? What do statisticians title it?
Don’t run away. Answer the question.
Does everything need to have a name and do I need to know it to see what you’ve described.
It’s fully defined. P(X = V) = 1, P(X != V) = 0
However, looking it this a bit more, it might be correct to say that in the mathematical sense this isn’t actually a probability density function. You are claiming one specific point in a continuous space has a probability of 1, whereas all points have a probability of 0.
None of this has anything to do with the actual probability distribution of an uncertainty interval. This is based on what you don’t know. If you knew the exact value of the measurand you would have no uncertainty, and if you don’t what it is you can’t say it has a probability of 1.
I thought you might be seeing a glimmer of light but your last sentence ruined that hope.
Let’s do an example. I get a block from NIST that is 1.00001 ± 0.000005 grams.
I center my triple beam balance and get a reading between 0.9 and 1.0. I do the whole procedure again and I get a reading between 1.0 and 1.1. Rinse and repeat 50 more times.
I end up with a value of 1.1 ± 0.1. is there really a probability distribution between 1.0 and 1.2? I know the actual weight and it has a probability of 1 and everything else is zero. There is no distribution. There is an interval where the true value may lay.
Don’t compare this to random error when measuring one measurand multiple times and assuming a Gaussian distribution where ± values can cancel and give you a “true value”. In the above example the true value might be calculated as 1.1. We know that is incorrect and that there is a Type B error involved.
The real issue is Type B (systematic) and resolution of the instrument. Even after fixing the Type B error your interval will still be +0.1 and -0.1. the real value will lay within that interval but it is not defined by a distribution curve! It is only ONE value within the interval.
“I thought you might be seeing a glimmer of light but your last sentence ruined that hope.”
I’m glad I confirmed your expectations. But I really think Tim is mixing up two concepts here. The uncertainty of a measurand and the uncertainty of the measurement.
“I know the actual weight and it has a probability of 1 and everything else is zero.”
But you don’t. You said the mass was 1.00001 ± 0.000005g. There’s an uncertainty value, you just know it’s between 1.000005 and 1.000015g. You would need to know what that uncertainty interval represents. It might just be cause by the rounding to 0.00001g, or it might be a 95% or other, confidence interval. In either case, you no longer have a single point with P = 1, so you have a probability distribution, wither continuous or discreet.
“There is an interval where the true value may lay.”
Yes, but you haven’t based that on your measurements, you are just taking on trust the values given to you.
What I think you are doing here is introducing a prior. You have a believe of what the mass is and can then using the measurements to shift the probability distribution. Normally I expect you don’t do this as you are measuring something to determine the measurand, and are starting with a neutral assumption of what the true value is.
But, if you are using a prior, you really shouldn’t start with it being 1. It’s nonphysical and makes any measurement irrelevant. You are just assuming you know what it is and will ignore any actual measurement. What if you measured it 50 times with 50 different instruments and they all say the mass it 2.0 ± 0.1g. You can’t accept that because the probability of it being close to 1.000 was 1.
It makes no sense, because nothing is ever certain.
Again you miss the whole point. The NIST weighed block is very accurate but the device being used does not have the resolution to see that. The point is that you can determine an INTERVAL where the true value lays. You don’t know what a distribution is because there isn’t one. THERE IS AN INTERVAL. That is all there is, an interval where the true value is. It is like finding a needle in a haystack. You know it is in the stack, but you don’t know where.
As you buy better and better measuring devices, you can narrow that interval. You could even narrow it past what NIST gave you. But, you would still have an interval with a higher resolution, and there would still be only one true value in that smaller interval.
“The NIST weighed block is very accurate but the device being used does not have the resolution to see that.”
So what was the point of all those measurements? If you are just going to assume you know with 100% certainty that you know the mass to ± 0.000005g, you are just going to rely on that certainty without the need to measure it.
But it still has an uncertainty distribution, even if you don;t know what it is. Even if you know for certain (which Is not possible), that the true value lies somewhere in that interval, the fact you don’t know where in the distribution is what should show you there is a distribution. And if you intend to use the weight in some combined uncertainty you have to make a best guess as to what it is as you have top know the standard uncertainty, if NIST haven’t specified what they mean.
If you only know this is an upper and lower bound, you would presumably use 4.3.7 from the GUM and assume a rectangular distribution, given a standard uncertainty of sqrt(0.000005^2 / 3) = 0.0000029g.
But it would be better to get the information from the manufacturer.
“You know it is in the stack, but you don’t know where.”
Almost as if there’s uncertainty.
“But it still has an uncertainty distribution, even if you don;t know what it is.”
WHAT uncertainty distribution?
You’ve never answered the question of “If you know the distribution of the values in the uncertainty interval then why isn’t the best estimate (read the GUM) the value with the highest probability?” In that case why even try to state an uncertainty interval?
From the GUM: “Although these two traditional concepts are valid as ideals, they focus on unknowable quantities: the “error” of the result of a measurement and the “true value” of the measurand (in contrast to its estimated value), respectively.” (bolding mine, tpg)
You just can’t accept what even the GUM says, can you?
You are a religious fanatic. Your dogma states that everything has a probability distribution and that *YOU*, being omnipotent, know that.
Once again, from the GUM,
They refuse to look at the truth.
“You’ve never answered the question of “If you know the distribution of the values in the uncertainty interval then why isn’t the best estimate (read the GUM) the value with the highest probability?” In that case why even try to state an uncertainty interval?”
Best estimate does not mean certain. I don’t see your problem here. If I knew what the true value was I’d tell you and ther’d be no uncertainty. I don’t so I give you my best estimate and an indication of how much uncertainty there is in that estimate.
“You just can’t accept what even the GUM says, can you?”
I’m not the one disagreeing with the GUM. That would be Kip.
I really don’t know what you think you are highlighting in that quote. The measurand and hence the error are unknowable. That’s why there is uncertainty.
“Your dogma states that everything has a probability distribution and that *YOU*, being omnipotent, know that.”
Knowing what a probability distribution is hardly makes me a fanatic. I’ve still no idea what you are trying to get to wioth this line of argument.
“Best estimate does not mean certain. “
The GUM says you should use the BEST ESTIMATE.
“I’m not the one disagreeing with the GUM. That would be Kip.”
No, that’s *YOU*. I gave you what the GUM says about “best estimate”.
“The measurand and hence the error are unknowable.”
So is the probability distribution of values in the uncertainty interval.
I’m not sure if either of us know what point you are trying to make.
You may or may not know what the uncertainty probability distribution is. But you can make educated guesses as described in the GUM regarding type B uncertainty.
But for the most part it doesn’t matter, because the standard equations are not based on any specific distribution, just on the standard deviation. And you you have to know what the standard uncertainty of the individual measurements (even if they are just educated guesses) because that’s the whole point of the exercise.
You aren’t going to like what you get when you begin computing the combined variance of Tmax and Tmin. It gets worse as you begin averaging days.
Variance of what? The temperatures or the measurement uncertainty?
“You may or may not know what the uncertainty probability distribution is. But you can make educated guesses as described in the GUM regarding type B uncertainty.”
Malarky!
GUM: “4.3.1 For an estimate xi of an input quantity Xi that has not been obtained from repeated observations, the associated estimated variance u2(xi) or the standard uncertainty u(xi) is evaluated by scientific judgement based on all of the available information on the possible variability of Xi .”
You can only assume a probability distribution for a Type B uncertainty if it is based on manufacturers specifications created in a calibration lab – which is itself based on multiple observations!
And that manufacturer specification only applies for the calibration lab. Once that instrument is in use in the field and subject to systematic bias you can no longer *ASSUME* what the probability distribution is for the uncertainty interval. You know what they say about “ASSume”.
“But for the most part it doesn’t matter, because the standard equations are not based on any specific distribution, just on the standard deviation. And you you have to know what the standard uncertainty of the individual measurements (even if they are just educated guesses) because that’s the whole point of the exercise.”
And, once again, you didn’t even stop to think before you posted this!
Standard deviation is not a good statistical descriptor unless you have a Gaussian distribution. Neither is the mean. You keep saying you don’t assume Gaussian distributions for everything but you circle back to doing so EVERY SINGLE TIME!
“You know what they say about “ASSume”.”
Is it that it’s the bedrock of all maths and science. Everything depends on assumptions, the importance is to understand what they are.
I think bellman is right in this case. It does have a distribution, but we don’t know what it is. What you can do about it is a different matter.
Even discrete values (counting people, sheep, spots on the face of a die, whatever) have a probability distribution where the probability of a non-integer value is 0.
“I think bellman is right in this case. It does have a distribution, but we don’t know what it is.”
It *does* have a probability distribution. I’ve never said anything else. The true value has a probability of 1 of being the true value. All the other values have a 0 probability of being the true value. That *is* a probability distribution.
If you don’t know the probability distribution you simply can’t just assume that it is Gaussian. That just ignores the fact that in the real world the distribution will be some kind of a skewed distribution because of systematic bias.
In the statistical world and the climate science world, it is just assumed that the uncertainty interval has a Gaussian distribution and the mean (i.e. the stated value) is the best estimate of the measurement.
As bellman has said, he assumes that values further away from the mean has less of a probability of being the true value.
He denies that he assumes uncertainty is Gaussian and cancels in every case but then he turns around and says that it *is*!
That’s a distinction which is quite difficult to grasp. a priori, the probability is that of the event occurring, post facto, but before the result is known, the probability is of having chosen that outcome.
Before a coin toss, the probability is that of the coin landing heads or tails. After the toss, the coin has landed either heads or tails. There is a true value, but we don’t know what it is until the covering hand comes off.
The planks fall into the same category. They have a length, but we don’t know what it is until they’ve been measured.
And you don’t know the degree of skewness unless you can measure to greater precision.
Tim was referencing uncertainty intervals, not coin tosses, which are attached to single measurements.
Measure the length of one board, and UA will provide the uncertainty: L ± U(L)
The board has a true length (but is unknowable).
Thus, inside the interval, the true value has a probability of 1, but is 0 everywhere else.
This is also the trap many statisticians fall into: confusing uncertainty with error. Error is defined as the difference between a measured value and the true value.
Because true values are unknowable in general, error is of little real utility except for estimating bounds of uncertainty limits, i.e. GUM Type B intervals. The GUM even deprecates using true values.
It’s the same principle, if you squint just right. Well, apart from the coin toss being discrete and the length being on a continuum. The true vale exists and has a probability of 1, but we don’t know what it is.
It’s knowable within an interval. and more precise measuring instruments/methods can narrow that interval.
This gets back to my post facto point, which I can’t have made as well as I’d like. The true value of anything, be it a dimension of an object or an event, has a probability of 1 and any other value has a probability of 0 of being that true value. The probability distribution we are then interested in is that of finding that true value.
If we’re all going to play the QM game, there is a limit to how far the precision can be improved, but that’s above my pay grade.
That seems a bit like the use of “value” in Economics. It doesn’t really have any use, and is now only used by certain ideologies to try to make a point.
I would recommend reading the terminology section of the GUM.
Thank you, mosh 🙂
It was not meant as a put-down, the GUM can explain them better than I.
Actually, I thought you had intended that reply for the line about every field having slightly different meanings of the terms and just replied to the wrong comment.
It’s an interesting field, and learning something new keeps the Alzheimer’s away.
What were we talking about?
Indeed sir!
What day of the week is this?
Forget about tossing coins, throwing dice, or multiple samples of anything. These are all distractions from the real issue.
You have a measurement to make, and a measurement instrument; it can be anything, speed, intensity, whatever. You have exactly one chance to measure it before it is gone forever. Call it X.
The real issue is this: the sample size is exactly one. Put this firmly in mind There is no population from which you are making random samples all Stats 101.
After the measurement is finished, what do you then have? You have one numeric value, that is all.
Uncertainty theory tells you that this number is not the true value of the anything you measured; in fact the true value is unknown and cannot be known, even after doing your best to remove problems in your measurement that you do know about.
Error is the difference between your value and the true value, and it too is basically unknowable.
What else can you know then? Using Uncertainty Analysis you can estimate an interval around your number as ±U(X), which is supposed to be your best engineering estimate of the internal within which you expect the true value to lay. (The GUM provides a standard way of expressing these numbers in a way that allows portability.)
So now you know X±U(X); U can be thought of as a number that gives quantitative information about the quality of your X measurement, how well you know it.
A good analogy is an optical microscope—if the lens is focussed correctly, the image is clear. But if the focus is off, the image is blurred. Exactly like uncertainty, a small uncertainty value is clearer than a higher one. But it will never be perfect, despite your best efforts, there will always be measurement uncertainty..
The only probability distribution you have is that of an impulse function—one at the true value and zero everywhere else. But this distribution doesn’t really tell you anything except that the true value will forever elude you.
And note that multiple blurred microscope images cannot be averaged to get a clearer image!
There’s too much going on in too many threads here. Did I get off track?
One-offs have their own special worms in the can, so you guys are welcome to them.
Static “things” have the advantage that they can be measured with successively more precise and accurate instruments and techniques to reduce the uncertainty interval. It’s very much a case of diminishing returns, though.
Something which can be done with critical one-off (or time series) measurements is to take synchronised measurements with an odd number of instruments sourced from different manufacturers and calibrated at different labs. You can either use all the readings, or let them vote on the reading. Redundancy is your friend.
Ready when you are, Mr De Mille
That is exactly what experimental uncertainty is for. From the MULTIPLE measurements you can calculate a standard deviation which is a stand-in for uncertainty.
I have to disagree, especially with regard to things like air temperature time series. Placing multiple sensors together in the same location is actually quite difficult because of unknown temperature gradients, plus this can alter the local thermal and air flow environment. It will likely end up adding more uncertainty than it reduces.
The tragedy of science. A beautiful theory slain by an ugly fact.
Yep. It’s where the real world meets statistical world.
You have hit the head of the nail. All you can do is give a GUM Type B uncertainty. This has been done by the NWS/NOAA providing the estimated uncertainty. However, some folks won’t like that and will continue to argue.
The real next chapter is how to combine the Type B uncertainty with the measurements from other stations. To do that one must examine the variability issue in sampling. A random variable is a random variable and it won’t change. Climate science needs to start from the beginning and work through the sampling issues and the variance they introduce.
Well put. But it doesn’t address the issue of what you do when you are combining two or more different measurements to get a combined measurement. How do you combine these estimates of uncertainty? Using Kip’s method, or assuming the the uncertainty is independent, using the standard equations?
“And note that multiple blurred microscope images cannot be averaged to get a clearer image!”
I think that depends on why they are blurred. If your lens is out of focus you have a systematic error and averaging probably wonl;t help, but if it’s caused by random noise averaging could help. Certainly that’s the case in astronomical photography where blurriness is caused by random turbulence in the atmosphere. Multiple images can be averaged to get a much sharper picture.
Multiple images of the same thing! Back to multiple measurements.
I was addressing karlomonte’s point about averaging blurred images of the same thing. What’s your point?
“ Multiple images can be averaged to get a much sharper picture.”
Correct.
You said MULTIPLE images. Does that not set off a bell in your head about multiple measurements OF THE SAME THING. independent images of different things, with or without noise, will not do you any good in making one of them more accurate.
BTW, there is no “noise” in temperature measurements. Noise is a unique signal that appears to be the same as the signal you are measuring but, it has a different source. I don’t know of another source that would mimic temperature.
Do you guys ever actually follow what’s being said.
Really? Then how do the uncertainties of temperature readings at different times, different places, and different devices ever cancel at one or more of them?
They don’t cancel at one or more of them. It’s in the average where the cancellation occurs. The individual reading are not more accurate because you’ve averaged a load of different readings, it’s the average (over whatever domain) that is less uncertain.
Here are two random variables, temperatures that have a mean μ and a standard deviation σ. Show us how you add two random variables AND obtain a smaller variance!
Tmax = 81 ± 1°F
Tmin = 65 ± 1 °F
Be sure to explain what probabilities you used where and why.
You don’t reduce the uncertainty by adding two numbers. Why would you want the sum of two temperature readings? See Kip’s essay about why summing intensive properties is meaningless.
Still if you insist, the sum of your two temperatures are 146 ± √2°F, whatever an F is.
That assumes the errors are random and independent. If as is likely they are not independent, given you are using the same antiquated thermometer, thn the uncertainty is ±2°F.
Now if for some reason you wanted the more meaningful mean if the two you have
73±1/√2°F, or 73±1°F depending on how random the errors are.
What is the formula for calculating the mean and variance of the addition of two random variable? Could these be it?
μ(x) = x1p1 + x2p2 + … +xnpn
σ^2 = Σ(xi – μx)^2pi
“What is the formula for calculating the mean and variance of the addition of two random variable? Could these be it?”
Nope. For a start you seem to be adding n variables, not two. I’m also not sure why you keep using p’s for your weighting constants. It’s confusing as p would normally indicate a prime number.
The formula for adding to random variables, X and Y, withe constant weights a and b, as I’m sure has been explained before is
E(aX + bY) = aE(X) + bE(Y)
and
Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)
This is getting old. I don’t disagree with the formulas you have copied. However, you have not shown that you understand what the mean. You need to show the derivation of these instead of just copying formulas. That is the only way you can demonstrate that you know why you are wanting to “scale” these values.
You still haven’t said why are you multiplying “X” by a constant. What is the purpose of multiplying? Do you know?
Do you understand what this is doing. “X” is a random variable. A random variable is a “container” for values. Those values of X are (x1, x2, …, x3, x(n)).
Variance is defined as (x – μ)^2. The variance of a random variable is the sum of the individual “x” values minus mean value “μ” of all the “x’s”.
So that Var(X) = Σ (x(i) – μ)^2 •p(i)
Don’t forget, part of the variance calculation is using the probability of each value in the random variable “X”. The probabilities must add to 1.
When you are multiplying Var(ax) you end up with
Var(aX) = Σ (ax(i) – aμ)^2 •p(i)
Remember, the mean μ doesn’t stay the same when you “scale” the “x” values because mean μ changes also.
The examples on the reference show you that. When they change the payouts (xi values), they recalculate the mean μ that are associated with the new values.
The question is still on the table!
Why do you want to scale the values of “x” components of the “X” random variable? You must answer this before any more simple copying of formulas takes place.
“This is getting old.”
Understatement.
“However, you have not shown that you understand what the mean.”
Whereas you’ve demonstrated you don’t understand this simple point, no matter how many times I explain where you are are going wrong.
“That is the only way you can demonstrate that you know why you are wanting to “scale” these values.”
I’ve explained multiple times why you are scaling the random variables, it’s pathetically obvious. But you just won’t understand because you are confusing two different means.
Once again you scale the variables because that’s what you have to do to get an average. It’s really simple. When you divide the sum by N to get the average you are scaling the variables by 1/N. I don’t know how to make it any simpler.
Skip the patronizing attempt to explain it. It means nothing when you don’t understand we are are not talking about the average of the random variable, but the averaging of multiple variables.
“Well put. But it doesn’t address the issue of what you do when you are combining two or more different measurements to get a combined measurement. How do you combine these estimates of uncertainty? Using Kip’s method, or assuming the the uncertainty is independent, using the standard equations?”
The standard equations are a tool. You have to know what tool is appropriate for the task at hand. As Bevington so aptly points out, statistical analysis is the face of systematic bias is difficult at best.
You can *always* just directly add all the uncertainties. That will give you an upper bound. The standard quadrature equations will give you a lower bound assuming no systematic bias and all random uncertainty. The truth will lie somewhere in between. *THAT* is where informed judgement will be the only tool you have available. Is the systematic bias very much larger than the random error? Is the random error larger? Shift the uncertainty one way or the other, what do you do?
“but if it’s caused by random noise averaging could help”
A glimmer of hope, perhaps. What if it is a temperature measuring device?
“The standard equations are a tool. You have to know what tool is appropriate for the task at hand.”
But according to Kip you don’t. You just have to use arithmetic and if you try to use any statistical tool you are an arithmetic denier. That’s why in his view the only allowed technique is direct adding of uncertainties.
+100
+100!
“That’s a distinction which is quite difficult to grasp. a priori, the probability is that of the event occurring, post facto, but before the result is known, the probability is of having chosen that outcome.”
I understand and I totally agree. But the impulse function is used a lot by EE’s. It has a probability of 1 but it really isn’t a probability distribution. Hard to get one’s head around that one!
It’s kind of the essence of uncertainty. You know that impulse function exists somewhere in the interval but you just don’t know where. Not knowing where doesn’t mean it doesn’t exist!
“There is a true value, but we don’t know what it is until the covering hand comes off.”
Wow! The very essence of uncertainty! It’s a closed box. With measurement uncertainty there is no lid on the box and there is no covering hand!
“And you don’t know the degree of skewness unless you can measure to greater precision.”
Yep. Closed box! Kip really hit paydirt with that one!
“ If you knew the exact value of the measurand you would have no uncertainty, and if you don’t what it is you can’t say it has a probability of 1.”
Thank you. The Gormans should thank you as well.
So blob, as a religious pseudoscientist, how do you divine true values?
Entrails?
I’m religio free. Only The Imaginary Guy In The Sky knows “true values”.
But when I have to, I do what every one else in superterranea does, to get close enough for purpose. I recalibrate, as appropriate, using a standard device/method impractical for every measurement, but useful for this purpose. That device is often, in turn calibrated on an even more awkward device, and so on.
I took 10 seconds to find a relevant example from my world.
https://inis.iaea.org/collection/NCLCollectionStore/_Public/27/029/27029598.pdf
Oh look, blob fires up his web thingie for more cherry picking.
And this latest word salad contradicts what you were previously claiming.
And you conveniently fail to point to what you claim I was “previously claiming”. Wonder why….
Free clue, blob, being religious has pretty much nothing to do a belief in God. You are in bondage to the AGW propaganda.
“Does everything need to have a name and do I need to know it to see what you’ve described.”
YES, it does! The fact that it doesn’t have a name should be a clue to you as to whether it is a probability DISTRIBUTION.
“None of this has anything to do with the actual probability distribution of an uncertainty interval. “
Of course it does – unless you believe there can be multiple true values! Which wouldn’t surprise me at all!
“If you knew the exact value of the measurand you would have no uncertainty, and if you don’t what it is you can’t say it has a probability of 1.”
Did you read this before you posted it?
You just basically said “the true value doesn’t have a probability of 1 of being the true value”.
ROFL!! You just can’t help yourself, can you?
“Shoot first from the hip and ask questions later”
Incredible.
The fact you think everything needs a name before it’s real, just illustrates your lack of understanding of how maths works.
But after a little internet searching I think the distribution you are describing is called the Dirac measure.
ht to
https://math.stackexchange.com/questions/756785/is-there-a-name-for-the-trivial-probability-distribution-px-x-1-for-a-unique
But I really don’t think it makes sense to claim this is a probability function of a measurand in any real world sense. Nothing can have a measure that is a fixed point. The real world is just to fuzzy. How can one of your boards have a length that is exactly 6′? Its length will depend on exactly how you define it’s start and end points, at a sub atomic level.
“Of course it does – unless you believe there can be multiple true values! Which wouldn’t surprise me at all!”
You still don’t get this concept do you. We are talking about the distribution of the uncertainty, not the true value. Either you have to think of this in terms of the distribution of the error when you take the measurement, or as the distribution of all possible values of the measurand. It isn’t that there are multiple true values, it’s just you don’t know which value is the true value.
“You just basically said “the true value doesn’t have a probability of 1 of being the true value”.”
No. I’m saying if you don’t know what the true value is you cannot say that any specific has a probability of 1 of being the true value.
(Or in classical statistical terms you;d have to convert this to a likelihood description.)
“But after a little internet searching I think the distribution you are describing is called the Dirac measure.”
What’s the area under a Dirac function?
“Nothing can have a measure that is a fixed point.”
EVERYTHING has a value that is a fixed point. The issue is whether or not we can accurately determine what that fixed point *is*!
“You still don’t get this concept do you. We are talking about the distribution of the uncertainty, not the true value.”
If you don’t know the distribution then how do you know there is one? The Dirac function *is* a probability distribution, btw!
The “true value” should be part of the uncertainty interval, by definition. Otherwise the uncertainty interval is meaningless.
“No. I’m saying if you don’t know what the true value is you cannot say that any specific has a probability of 1 of being the true value.”
ROFL!! Thanks for agreeing with me! 1. You can’t know the true value, that’s what the uncertainty interval *is*. 2. That does not mean that there is no value that has a probability of 1, it just means you don’t know what that value *is*! 3. you can’t just arbitrarily assign probability distributions to an uncertainty interval the way you want to – you don’t *KNOW* if it is a Gaussian spread or not.
You still don;t understand how much you are barking up the wrong tree. The probability distributions used in all the equations are about the uncertainty, not what the actual value is.
The probability that a measurand is the value it is is 1, and the probability that it isn’t is 0. But you don’t know what that value is, hence the century of more devoted to understanding how to handle uncertainty. The probability distributions used in all these equations are the distributions of “uncertainty” however it’s defined.
“The “true value” should be part of the uncertainty interval, by definition. Otherwise the uncertainty interval is meaningless.”
No. The interval is just describing a range of possible values the true value may or may not lie in. The interval in this case is that give by the standard uncertainty, hence it’s the standard deviation of the probability distribution (however you define it).
Again, you seem to now want to ignore all definitions of uncertainty intervals described in all your sources, in favor of Kip’s impossible range that absolutely captures all possible values.
“You can’t know the true value, that’s what the uncertainty interval *is*.”
So why do you keep asking about the probability distribution of the true value?
“you can’t just arbitrarily assign probability distributions to an uncertainty interval the way you want to”
I can. Every book on metrology and statistics do. You can;t talk about uncertainty if you don’t have a probability distribution.
“You still don;t understand how much you are barking up the wrong tree. The probability distributions used in all the equations are about the uncertainty, not what the actual value is.”
You simply do not know what you are talking about. No matter how many quotes you are provided from the GUM you just blow them off.
Here is another one.
“E.3.4 Consider the following example: z depends on only one input quantity w, z = f (w), where w is estimated by averaging n values wk of w; these n values are obtained from n independent repeated observations qk of a random variable q; and wk and qk are related by” (bolding mine, tpg)
Almost 100% of the GUM is oriented toward statistical analysis of observed values as representing the uncertainty in a quantity instead of stated uncertainty intervals.
“But you don’t know what that value is”
Thanks for repeating back to me what I’ve been telling you for two solid years.
” The interval is just describing a range of possible values the true value may or may not lie in.”
The uncertainty interval is describing an interval the true value may *not* lie in? As usual, you didn’t even bother to think about what you post!
“So why do you keep asking about the probability distribution of the true value?”
I don’t. The true VALUE doesn’t have a probability distribution. The uncertainty interval is an impulse function. The true value has a probability of 1 of being the true value and all other values have a probability of 0 of being the true value. Is an impulse function a probability distribution? I’ve never considered it to be so since its area is undefined – i.e. width = 0. With a width of 0 it’s kind of hard to integrate the impulse function yet it’s considered to have an integral of 1. To me the impulse function appears to be a distribution but not a probability distribution.
“I can. Every book on metrology and statistics do. You can;t talk about uncertainty if you don’t have a probability distribution.”
No, they do *NOT*. If you would actually study your cherry tree for meaning they consider the stated values to have a probability distribution while ignoring the actual measurement uncertainty interval. It’s a ubiquitous approach foisted on everyone by statisticians that don’t have to function in reality.
For example, look at GUM 4.4.3. The Figure 1 graphs of standard uncertainty is labeled “Figure 1 — Graphical illustration of evaluating the standard uncertainty of an input quantity
from repeated observations”.
The very first sentence in the section says “Figure 1 b) shows a histogram of n = 20 repeated observations tk of the temperature t that are assumed to have been taken randomly from the distribution of Figure 1 a).”
It should be obvious to anyone who reads the GUM how it is handling uncertainty. It’s only opaque to those who live in statistical world and refuse to join the rest of us in the real world.
Another big round of hand-waving and fanning from the hip, I can’t even read his tomes.
I’m pretty sure an impulse function can be considered to have a sigma of zero, which is exactly correct for a true value: the uncertainty is zero. A true value then has no uncertainty interval, which agrees with the treatment in the GUM.
The concept of a true value of a measurement seems to escape some. I grew up with a father who was a master mechanic and did not allow sloppy measurements on diesel engines or fuel pumps. The phrase “DO IT RIGHT THE FIRST TIME” was not foreign when, years later, I received quality training.
And you are still missing the point. The probability of the true value being the true value is irrelevant, it’s the probability of the uncertainty in the measurement that matters. It doesn’t matter if the uncertainty is determined from multiple measurements or is a type B uncertainty, neither have anything to do with the probability of the true value being a true value.
“The uncertainty interval is describing an interval the true value may *not* lie in?”
I said, may or may not. Of course the true value may not lie in the interval, that’s why it’s defined in terms of a confidence interval.
“Is an impulse function a probability distribution?”
As I said before I don’t think so. It doesn’t have a PDF, but it does have a CDF. But I still don’t know what your point is. It’s irrelevant to a discussion of uncertainty because that is based on the uncertainty of measurements, not the probability of the true value. I also think it’s irrelevant because what you are describing doesn’t exist in the real world, and certainly isn’t applicable for describing the length of a piece of wood.
This is an important point, which probably needs to be raised again at a later date. QM aside, the uncertainty lies in the measurement, not in the dimension of the object of interest.
Of course you can have a standard uncertainty without a probability distribution.
3.3.5 The estimated variance u2 characterizing an uncertainty component obtained from a Type A
evaluation is calculated from series of repeated observations and is the familiar statistically estimated variance s2 (see 4.2). The estimated standard deviation (C.2.12, C.2.21, C.3.3) u, the positive square root of u2, is thus u = s and for convenience is sometimes called a Type A standard uncertainty. For an uncertainty
component obtained from a Type B evaluation, the estimated variance u2 is evaluated using available knowledge (see 4.3), and the estimated standard deviation u is sometimes called a Type B standard
uncertainty.
What do you think a Type B uncertainty is?
“What do you think a Type B uncertainty is?”
It’s all described in the GUM. Basically you either on the information provided, or you have to make assumptions about the distribution.
For example
and, applicable to Kip’s view of uncertainty
“ one can only assume that it is equally probable for X_i to lie anywhere within it”
What does the word “it” imply? There is one, and ONLY ONE, “it”.
And, yes, it can be anywhere in the interval. The GUM is correct based on their assumptions, but those assumptions do not hold in the real world. Not all values in the interval are equally likely to be the true value. Only ONE value in the interval can be the true value. Equal probability implies that you can *know* the probability distribution inside the uncertainty interval. But like I keep pointing out, SYSTEMATIC BIAS at the very least, keeps you from knowing what that probability distribution actually is. That means that *NOT* all values in the uncertainty interval are equally possible.
You remain stuck in the meme that all measurement uncertainty is random and Gaussian. That allows you to assume it all cancels.
You claim you don’t believe that but it just comes shining through EVERY SINGLE TIME you make a post on the subject.
Do *YOU* know the systematic bias in every temperature measuring station used to determine the global average temperature? Are you omnipotent? If you are then who is going to win the Super Bowl this year? If you aren’t then how can you possible know that every single temperature measuring station has an uncertainty interval with a uniform probability distribution?
“What does the word “it” imply? There is one, and ONLY ONE, “it”.”
Read the rest of the sentence. “it” is the interval.
“But like I keep pointing out, SYSTEMATIC BIAS at the very least, keeps you from knowing what that probability distribution actually is.”
So NIST is supplying a weight with a systematic bias are they? The problem is, even if that is true, you still don’t know what the bias is, and so it’s still reasonable to assume the true value can be anywhere with equal probability inside the interval.
“You remain stuck in the meme that all measurement uncertainty is random and Gaussian.”
Make your mind up. First you’re objecting to me quoting the GUM as saying you should assume the distribution is uniform, and in the next breath claim I think all distributions are Gaussian.
“You claim you don’t believe that but it just comes shining through EVERY SINGLE TIME you make a post on the subject.”
Yes, you got me. EVERY SINGLE TIME I say assume a uniform distribution, I really mean it’s Gaussian. Lucky we have Tim the mind reader to figure out what I really mean.
“If there is no specific knowledge about the possible values of X_i within the interval, one can only assume that it is equally probable for X_i to lie anywhere within it “
*IT* is Xi! *IT* can lie anywhere within the interval!
Next trying asking mr. experto here what the standard deviation of an impulse function is. The cloud of nonsense should be amusing.
He won’t answer. That’s his answer to all the hard questions – just don’t answer.
Or blow another smokescreen…
Not one of them has figured out the significance of the impulse function question yet.
“No, it is not a measurand.”
And you still fail to explain how it can have a measurement uncertainty in that case.
And you avoid answering the question about the sum of two things. Is equation 10 of the GUM the correct equation to use when adding two values together with independent uncertainties? You know why I’m asking you this, and what conclusions can be drawn from your avoidance of it.
Kip says of adding in quadrature “The statistical approach uses a definition that does not agree with the real physical world”. Is he correct or is the GUM (and every other authority on metrology) correct?
The hamster wheel squeaks and squeaks and come back around for another exciting dose of “stump the professor!”
In the case of an average, the uncertainty interval is not truly a *measurement uncertainty*. It is *still* a statement of uncertainty calculated from the measurement uncertainty.
When you can *truly* understand why the average uncertainty is not the uncertainty of the average perhaps this will make sense to you.
Too many responses on here to keep up.
Your question is ill posed and therefore unanswerable. Why? I have added an image of a portion of text from Dr. Taylor’s book, An Introduction to Measurement Error, Page 59.
This section describes why the GUM Eq. 10 may or may not be the correct equation to use. In any case the simple addition of the uncertainty is an upper bound. The true uncertainty may be less.
Even the GUM anticipates that uncertainties cancel in quadrature. However this is many times just a lower bound and the true uncertainty combines differently. This is the primary reason for performing experimental uncertainties. One can not just make generalities about what probabilities distributions of uncertainties truly are.
This whole conversation points out your lack of experience in physical measurements. I grew up with a micrometer in one hand and a dial gage in the other. Tim’s and my father was a master diesel mechanic and didn’t brook haphazard measurements.
Yes, Taylor’s talking about systematic error – i.e non independent uncertainties. But my question is what happens when you add two measurements with independent uncertainties.
Would you accept that if the uncertainties are independent, then equation 10 is the correct one? And would you also accept that when Taylor says the uncertainty can be no larger than u(x) + u(y) he is not agreeing with Kip who insists that it can be no smaller than that?
Did you not read the last sentence?
The simple sum ∂x + ∂y is an upper bound.
Quit trying to fit dice which are discreet values into an error/uncertainty frame that is continuous. What Kip is trying to illustrate is uncertainty that is absolute.
Read this again, and notice the word RANGE!
The RANGE is not an expectation of a statistical parameter of the routine mean/SD statistical description of a distribution.
The RANGE is a part of a 5 number statistical description.
You are trying to pound a round peg into a square hole.
What you should be recognizing is that a mean of 3.5 is a non-physical description of a series of discrete data. It has no meaning in the physical world. A mean is NOT a measurand, even if it duplicates a data point. It is a statistical parameter. That is all it is.
Operative part being “no larger”
“The simple sum ∂x + ∂y is an upper bound. ”
Exactly. Implying they may and probably are going to smaller. Something Kip says they can’t be.
“Quit trying to fit dice which are discreet values into an error/uncertainty frame that is continuous.”
The dice are Kip’s way of explaining uncertainty. Yes they are discreet rather than continuous, but that doesn’t effect the principle.
“The RANGE is not an expectation of a statistical parameter of the routine mean/SD statistical description of a distribution.”
And it’s also not what your sources say is usually the best value for uncertainty.
“You are trying to pound a round peg into a square hole.”
I’m trying to understand why you et.al have decided that all the works on metrology are wrong in their definition of uncertainty and are happy to let Kip define what uncertainty really is.
“What you should be recognizing is that a mean of 3.5 is a non-physical description of a series of discrete data.”
Stop bringing up that distraction. Again, I’m not talking at this point about averages. All I’m interested in
“From the GUM.”
I see nothing in those definitions that rule out an average as being a measured. If you can call the sum of N values a measurand then why isn’t the sum divided by N a measurand?
“The GUM deals with physical MEASUREMENTS.”
But also how to combine those physical measurements to get a combined measurand.
“The GUM only applies to the action of measuring a temperature, i.e., obtaining a single reading at a given time.”
Example 4.4 has twenty different temperature measurements ranging from (96.90°C – 102.72°C) being used to obtain a best estimate (mean) of the temperature.
“In other words, use the standard uncertainty developed by experts like the NWS or NOAA.”
You are talking about the uncertainty of an individual measurement here, not of the mean.
“I suggest that you start to learn what a random variable…”
Please stop with this patronizing nonsense. I’ve spent months trying to explain how random variables work, and it just keeps falling on deaf ears. Do you still think that the variance of the mean of a set of random variables is equal to the sum of the variances?
Even Pat Frank admitted that they effed up. I assumed that he would have set them straight, but – holidays I suppose..
Really? Where did he do that? He certainly hasn’t done it in this thread.
https://wattsupwiththat.com/2022/12/16/limitations-of-the-central-limit-theorem/#comment-3653232
Almost a day of Radio silence from Tim Gorman here. Of course he made many other comments ITMT. Awkward…..
Still skimming the Fluke PDF for loopholes, blob?
Some of us have a life. Guess you don’t.
Says the guy commenting in a pretty mid PM. I already recommended fresh air and exercise for you all. I repeat it – again.
You’re just another clown, blob.
HTH
“If you can call the sum of N values a measurand then why isn’t the sum divided by N a measurand?”
Because the average is a statistical descriptor. It is not a measurand.
It’s why you think the uncertainty in the length of an average length board is not .08 or .04 but is something different like .06.
The average is *NOT* telling you anything about the measurands! It is a statistical descriptor of the distribution. It will still have uncertainty because the elements used to calculate it have uncertainty. But that does not make it a measurand. It remains a statistical descriptor.
Addendum:
If Kip’s exact uncertainty values are taken as expanded uncertainties (i.e. U=±ku), what coverage factors k would correspond to the GUM standard uncertainty values?
k(1d) = 2.5 / 1.73 = 1.44
k(2d) = 5 / 2 = 2.5
I believe that because these are so different, this is another indication that it isn’t possible to calculate the 2-dice case from two instances of the 1-die case.
The k=2 figure is not the correct value to use for a triangular distribution. It should be k=1.93 for 95.45% confidence (the equivalent of k=2 for a Normal distribution).
I think you are right that one of the difficulties here is that the result has a triangular (not Normal) distribution.
If we used more dice, the result would conform more and more closely to a Normal distribution as the number of dice is increased.
k=2 is the standard coverage factor for measurements reported under ISO 17025.
This is *only* true if you have perfectly fair dice, no systematic bias.
Do *YOU* own any perfectly fair dice? I don’t. And I have a *lot* of dice of various sizes. I have some dice that I almost always use in my RPG games because they typically give me “better” outcomes!
In this particular brain experiment we are assuming fair dice.
“In this particular brain experiment we are assuming fair dice.”
And now we are back into the statistical dimension which doesn’t touch the reality we live in.
Just create assumptions, willy nilly, to make things come out the way you want them. Do you have even the faintest of clues why they change dice on a crap table? Do you even know what casinos do to new dice to make them roll more erratically?
We create assumption to analyze the ideal case. If you can’t correctly analyze the ideal case first then you certainly aren’t going to fair any better with real world cases.
Herr Professor Doktor Hubris speaks!
Y’all better listen up.
Of course the “k” factor differs depending on the degrees of freedom.
As I’ve been trying to tell others the roll of some number of dice can not be predicted by a probability distribution. If it could, no casino would have a dice game.
Each number on a die has a 1/6th chance of appearing on each roll. With 2 dice any number combination such as 1/6 is just as likely as a 1/1 or a 3/5.
They are independent and mutually exclusive events. What happens on one has no effect on the other each time you roll.
Statisticians like to point out that 7 is a more likely combination than any other number and that is true if you are examining a pattern over a large number of trials.
Let’s examine that by listing out combinations.
1 – N/A
2 – 1/1
3 – 1/2 & 2/1
4 – 3/1 & 1/3 & 2/2
5 – 4/1 & 1/4 & 3/2 & 2/3
6 – 5/1 & 1/5 & 4/2 & 2/4 & 3/3
7 – 6/1 & 1/6 & 5/2 & 2/5 & 4/3 & 3/4
8 – 6/2 & 2/6 & 5/3 & 3/5 & 4/4
9 – 5/4 & 4/5 & 6/3 & 3/6
Whoa, guess what we got, Pascal’s Triangle. Do you really think that on any single roll that 7 is much, much more likely than a 6 or 8? There is only one more combination for a 7 than for 6 or 8?
Well did I ever claim that a 7 is much more likely? No, I would say the probabilities are exactly how how show them above.
Also notice that the extremes of the distribution (2 and 12) are very unlikely to occur, even with only two dice being combined.
KM,
A better description is to use a 5 number description (box plot) for describing the distribution.
Kips unique description allow determining the range (min/max) and the median (same as the mean,). The quartiles you can calculate.
Yeah; I was trying reconcile the blind RSS of two 1-die into the 2-dice. It doesn’t work, as I found out.
Statistical analysis for dice rolls would apply the discrete random variable case
For one die (which is a uniform discrete distribution), the mean would be 3.5 and the standard deviation (stdev) would be 1.7 (sqrt(2) to be precise). The range of the values as being within +/- 2.5 from the mean has no bearing on the standard deviation, it is like apples and oranges. The +/- stdev happens to encompass the range [2..5], or 4 of 6 (67%) of the possible outcomes in this case.
For two dice (which is a triangular discrete distribution), the mean would be 7 and the stdev would be 2.4. The +/- stdev happens to encompass the range [5..9] which is 24 of 36 (67%) of the possible outcomes. A +/- 2*stdev spans the range [3..11] which is 34 of 36 (94%) of the possible outcomes.
Note the span of values for one die (N=1), +/- 2.5 is +/- 1.77*stdev but the span of values for two dice (N=2), which is +/- 6 from the mean of 7, is +/- 2.48*stdev. So comparing span of values to stdev for two different types of probability distributions also is not consistent.
The normal distribution you overlaid in your figure has stdev of about 1.9. If you overlaid one with 2.4 it would be more fairly representative of reality (though still not exact as the actual distribution converges to triangular not Gaussian/normal as number of trials=>infinity).
Compare the standard deviations with the GUM Type B standard uncertainty evaluations for the 1-die (rectangular) and 2-dice (triangular) cases from Figure 2:
u(1d) = 1.73, s(1d) = 1.73
u(2d) = 2.0, s(2d) = 2.4
If Kip’s exact uncertainty values are taken as expanded uncertainties, what coverage factors would correspond to the GUM standard uncertainty values?
k(1d) = 2.5 / 1.73 = 1.44
k(2d) = 5 / 2 = 2.5
That these are so different is another indication that it isn’t possible to calculate the 2-dice case from two instances of the 1-die case.
This ongoing discussion involves measurement in physics and how to represent the results of measurements in a quantitative way so that others can understand results for what they are as well as what they are not. Different people approach this subject with different backgrounds and different desires. It is helpful to appreciate the different ways in which measurements are approached in physics.
Many commenters here appear to come from a metrology perspective as embodied in the NIST and JCGM context. That community is generally geared towards measurements of fundamental physical constants or properties such that the measurements can be used for calibration purposes or for use in other calculations as a “given.” These measurands are generally isolated by very complex and sophisticated devices and great attention is paid to minute details of the measurement apparatus. A good example of the kind of things considered is a nice recent paper on measurement of surface temperature distributions from some NIST folks obtainable at
https://nvlpubs.nist.gov/nistpubs/jres/126/jres.126.013.pdf
Some may be disappointed that the paper does not treat climatological temperatures, but it illustrates the level of detail the metrology community considers necessary in evaluating the uncertainty of the measurements it considers. In this approach, many sources of uncertainty are identified and characterized and ultimately combined to represent the uncertainty in the final result. Much time and effort are spent on the functional dependence of the uncertainties on details of the measurement process, the presence or absence of randomness, etc. Whatever mathematical tools, approximations, or assumptions are used are explicitly stated and bounded. If some sources of uncertainty are still unresolved, that is also stated to the extent they are known.
Another community here seems to be working scientists and engineers from non-climatological disciplines. This community often is focused on determining if a particular controlled experiment demonstrates that a proposed new theory is “wrong” by showing that the value predicted by the theory is far enough away from the experimental value or values that the theory is in some sense unlikely to be correct or that the predicted value of another theory is closer to the experimental value. This community shares many of the same approaches as the metrology community but may be ready to accept slightly less rigor, particularly if the measurement involved is an observation of a remote object rather than a controlled experiment in a laboratory. But the focus is nevertheless on quantifying somehow the notion of “unlikely” or “close enough,” which typically involves probabilistic considerations.
Climatology seems to me peculiar in the sciences in that everybody seems to think they can do it, and everyone likes to play with the numbers involved in it. While I myself fall in the category of working scientist in another discipline (physics) as far as climatology goes, I am more or less in this latter group. Sometimes I can’t resist playing with the numbers, but I rarely invest the time required to delve into the climatological measurement details to the level I do in my own field or that the metrology folks do in theirs. (Or biologists in theirs or chemists in theirs, etc.) I do recall spending some time with the measurement details in a paper Karl wrote some time ago on sea surface temperature adjustments and another one someone wrote on remote temperature stations in Australia, but it was just for fun.
But the details matter a great deal. In our own disciplines, stating results to unsupported accuracy is tantamount to lying (at least that is what I tell my students) and should be avoided even if it means understating one’s results. In a measurement system, key details include what is the phenomenon that is sensitive to the thing being measured, and what physical device responds to it; is the response accumulated as charge or current in a circuit, are those values accumulated over specific times, is rounding done by truncation, simple buffer overflow or averaged over certain times; is the cumulant recorded or do transitions occur at intervals; are all results stored or are they processed and categorized in some way; are data transmitted immediately or stored; are any of these processes correlated with other processes or with environmental variables; are periodic calibrations required and is the drift between calibrations known or recorded; are some signals primary measurements and others derived numerically from others, etc. etc. These kinds of things should be taken into account when one wants to use the output quantitatively.
Now, let’s look at the topic under discussion, combination of two temperature measurements (or any number of them). The details of the specific measurement apparatus concerned have not been discussed in detail here, but at the risk of oversimplification, the assumption seems to be made that the measurement uncertainty is limited by the resolution of the output numbers, which in some cases has been stated to be 0.5 deg C. I am not sure of the particular instrument that refers to, but the Gavin Schmidt reference given in a previous post assigned that number, 0.5, to uncertainty in the average global temperature, although looking at the GHCN data for one station, Phoenix AZ, the temperature minima and maxima are given to resolution 0.1 deg C.
For the moment, let’s assume that a measurement device exists, whose details we don’t know, but we do know its results are stated only in 0.5 deg C increments. So, the standard expression of a measurement and its uncertainty would be, for example, [20.0 deg C ± 0.5 deg C]. The temperature being measured presumably varies continuously with time and in fact for this single measurement we assume any value between 20.5 deg C and 19.5 deg C is equally likely. (Ignoring here any quantum issues about the size of the system under measurement, fluctuation spectra etc. we assume much more continuity in the actual temperature values than the resolution.) Now what can we say about the expected uncertainty in the sum of two such measurements (or of course the difference, ie Tmax ±Tmin) say
[20.0 deg C ± 0.5 deg C] + [28.0 ± 0.5 deg C] = ?
First note that our convention of writing ± is entirely arbitrary and we could just as easily write the – sign on top if we wish and when we wish, so naively we might think that the first 0.5 deg C above could be “canceled out” by the second 0.5 deg C if that uncertainty happened to be on the negative side and the full amount. In fact, assuming the addition distributes over the top (or bottom) of the ± sign is just that, an assumption. Of course, the above is just a very specific instance and not at all general. To be more general (if we know nothing about the details of the instrument) we might say that the uncertainties are equally likely to be below the reported number as above the reported number. In fact, if we blithely add them with the same sign and magnitude, we are making an unstated assumption that the uncertainties are all strictly positively correlated and maximum. Nothing wrong with making that assumption, so long as it is clearly stated. In that case one could argue that adding the uncertainties with the same sign should produce an upper bound for the expected uncertainty. (This would naturally extend to any number of additions or subtractions.)
In fact, a particular device might operate this way, for example if it reported temperatures when whatever physical quantity it was measuring (charge, current, etc.) exceeded a particular threshold, in which case the reported number might always be at a bound of the uncertainty. But we don’t know that. Nor do we know if the device reports measurements according to a time schedule, every so many seconds or whatever, which conceivably could be correlated with some environmental variables. It is likely the instrument reports according to some time schedule, such as time of day, or every so many seconds, etc. Without any a priori knowledge, our minimal assumption is that there is no specific correlation between time of measurement and whether the actual value lies at some specific fraction of a degree away from the resolution points of the device.
So, a proper statement of our ignorance is that the actual value of temperature should lie either below (- sign) or above (+ sign) the reported value by as much as 0.5 deg C and further, it should be equally likely for the uncertainty to be below as above. What would this do to an addition of say, 3 measurements? One example would be (just to pick one specific possible outcome)
[28.5 deg C + 0.3 deg C] + [27.5 – 0.2 deg C] + [26.0 – 0.2 deg C] = 82 deg C – 0.1 deg C
We might be wanting to divide by 3 to get an average or some such but that is not important. The point is that if the stated uncertainty due to resolution is truly random in the range given, it should be both positive and negative and sums such as these (if one had a referee instrument calibrating things) should often give measurement results closer to the referee instrument than the stated 0.5 deg C “absolute uncertainty,” which we take here to mean simply the resolution of the device in its natural units. What this means is that insisting on adding all the uncertainties at the maximum value with the same sign is deliberately choosing the upper bound of what might happen, which would under our assumptions here certainly be an overestimate of the final difference from a referee measurement.
What could we do to get a better estimate of the expected uncertainty (taken here to mean an expected difference between a referee instrument)? Without delving into the details of the device (which in fact we should) we could make the weakest assumption that the distribution of the actual value of the temperature being measured is equally likely to be anywhere in the interval [-0.5 deg C, +0.5 deg C] about the reported value. This is a very easy thing to estimate, being a simple continuous uniform distribution and it turns out that (somewhat like sums of dice rolls or coin tosses) the distribution of the sum of two uniform distributions is a triangular distribution easily calculated (or simulated, whichever is easier) and it has a variance that is proportional to 1/3 and the interval width squared. For example, the standard deviation (sqrt variance) of the sum of two numbers with uncertainties uniformly distributed on [-0.5,+0.5] is about 0.3. In other words, the expected uncertainty of the sum of two such numbers under the assumptions here is something less than 0.5. Given that we can calculate the distribution exactly, we can explicitly read off the probability of the uncertainty of the sum being within any specific interval we want, up to the interval size, which would of course be 1.
In other words, simply adding this kind of uncertainty without accounting for variability overestimates the expected difference between the reported values and the referee values. Further, haters of statistics (and sometimes I am included in that set) may not like that if one adds larger numbers of terms with this kind of uncertainty, the “triangular” distribution becomes somewhat (the horror!) normal-like and some approximations become even easier to make to quite sufficient accuracy.
But without actually delving into the physics and electronics of the measurement device, all of this could be viewed as playing around with numbers without much connection to whatever physics and data accumulation is actually being done in the specific instrument. Based on my experience with problems like this, my recommendation to anyone who is concerned about a measurement device precision and accuracy is that they should get to know the device as well or better than the manufacturer and know even more about the computer geeks who wrote the code embedded in it. Assumptions are a bit like opinions, everyone has them and, well you know.
This is a perfect description of the armchair metrologists who month-after-month show up in WUWT trying to defend the tiny and nonphysical uncertainty numbers claimed by climate science (which is really pseudoscience).
You say that “For the moment, let’s assume that a measurement device exists, whose details we don’t know, but we do know its results are stated only in 0.5 deg C increments. So, the standard expression of a measurement and its uncertainty would be, for example, [20.0 deg C ± 0.5 deg C].”
But that only holds for measurements (visually) rounded to the nearest whole degree. The uncertainty of an individual observation is 1/2 the index distance. For a Celsius met-thermometer which has 1/2-degree indices, it is 0.25 degC; for a Fahrenheit thermometer with whole-degree indices, it is 0.5 degF. However, as I explained above, considering the number of indices (bars on the thermometer) between freezing and boiling-points, conversion from F to C makes no practical difference.
The issue for electronic instruments, such as resistance probes is different because (i), their 30-second or whatever estimate is based on an average or other smoothing function of n samples, and, (ii) most (or at least the ones I played with) rely on internal non-linear calibrations which at their extremes may result in spikes or out-of-range values (uncertainty is a function of the value being measured). Having said that, how the difference between instruments is resolved theoretically, I don’t know.
Cheers,
Bill Johnston
Bill and fah ==> My example is a simulation of some data rounded to whole number values (would apply even to those rounded to 0.x or 0.ox etc).
Somewhere, far above, I linked to the official specs for ASOSs (automatic surface observation system). The spec’d uncertainty in each individual temperature measurement is:
In the range of “-58°F to +122°F the RMSE is 0.9CF (or 0.5°C) With a Max Error of – 1.8°F (or 1°C).”
The document is “Automated Surface Observing System (ASOS) User’s Guide National Oceanic and Atmospheric Administration Department of Defense Federal Aviation Administration United States Navy March 1998” (despite the date, it is the latest version and currently in use). Available as a .pdf here.
“we do know its results are stated only in 0.5 deg C increments.” This rounding is found in the official NOAA temperature records — in columns for Tavg (daily average temp, really (Tmax + Tmin)/2 ).
Thanks Kip,
While I read your whole piece, remember that I am only addressing the challenges you raised in the last two paragraphs.
The trend baseline is not set by ASOS, ARGO floats or the BoM’s automatic weather stations, it is set by data measured by eye, by mostly untrained observers, at largely uncontrolled sites, using Fahrenheit meteorological thermometers of varying quality, supposedly held 1.2m above the ground in Stevenson screens of various types and in various states of repair.
I have found no evidence that suggests Australian data are useful for detecting trend that could be attributed to CO2 or anything else, which is the focus of my http://www.bomwatch.com.au work.
All the best,
Bill
Bill,
Thanks for the info and I admit to immense ignorance about all the data sets used by the climate folks. I have typically found climatology way more time consuming than I had stomach for to dig into it on any regular basis. Give me an instrument design and I can deal with it, but start talking about merging datasets, spatial homogenization, and teleconnections (whatever they are) and I am out of my depth.
Mostly what I thought about was Kip’s proposed approach to dealing with what he called absolute uncertainties, which it seems he assumed for some specific case to be a fixed uncertainty (as would be from rounding or truncating at some level) and I wasn’t too worried about the specific level, whether 1, 0.5, 0.3 or lower, just any fixed arbitrary upper/lower bound with no specific information on the distribution within the bounds. I took that as a hypothetical, general case, without any more detailed tech spec. It is an interesting abstract problem and I think in the end, it likely still can lead to the ability to sum errors in quadrature when summing up measurements, but I don’t want to gore anyone’s ox right now.
Even though I have a lot of other stuff I need to do, I did poke around a little bit following Kip’s pointers to ASOS and the like. I think it is fair to say my ignorance is still as large as it was (something about reductions of an infinite quantity). It looks to me like the docs suggest the temperature reporting from the ASOS system is automated and the user manuals state things in terms of RMSE on the order of 0.5 deg C and reported resolution at 0.1 deg C. I downloaded data that was alleged to be part of the GHCN data set, in particular data from Phoenix AZ. Some lines from that data look like this:
STATION NAME DATE TAVG TMAX TMIN TOBS
USW00023183 PHOENIX AIRPORT, AZ US 1/15/2022 15.8 21.7 9.4
USW00023183 PHOENIX AIRPORT, AZ US 1/16/2022 15.2 23.3 7.2
USW00023183 PHOENIX AIRPORT, AZ US 1/17/2022 15.1 21.7 8.3
USW00023183 PHOENIX AIRPORT, AZ US 1/18/2022 15.5 20.6 11.7
USW00023183 PHOENIX AIRPORT, AZ US 1/19/2022 15.1 21.1 10.6
USW00023183 PHOENIX AIRPORT, AZ US 1/20/2022 14.6 21.7 8.3
USW00023183 PHOENIX AIRPORT, AZ US 1/21/2022 14.4 21.1 8.3
I frankly have no clue if this is data that is actually used by the climate folks but it is reported to 0.1 deg C, like the ASOS manual claims. I also found some docs dealing with issues they have had with the thermometer instrumentation (apparently a basic Pt thermistor). One I liked that spoke my language is
https://www.researchgate.net/publication/249604712_Sensor_and_Electronic_BiasesErrors_in_Air_Temperature_Measurements_in_Common_Weather_Station_Networks
Another I was not as crazy about but it highlighted bias, which seems to be the major problem
https://journals.ametsoc.org/view/journals/bams/77/12/1520-0477_1996_077_2865_eaotdb_2_0_co_2.xml
There were several others that were not hard to find but the general gist seemed to be that the thermistor circuit needs calibrations done regularly but if done many of the “errors” involved are the random type that can be added in quadrature and the software cumulates a running 5 minute average of measurements, but the biggest of them all looks like a bias problem (order of degrees C) that did not look like it was resolved in any automated fashion.
All the docs read as if the data from ASOS is generated and reported automatically and not visually obtained by observers and then reported manually. It also looks like there is a lot of massaging the data by downstream computer codes and people afterwards.
I think I have used up my meagre time budget for climate stuff.
Regards and thanks again.
Resolution of 0.1C is not directly the measurement uncertainty.
A temp of 9.4 can’t be extended to the hundredths digit. But its uncertainty can tell you that the actual temperature could be between 8.9 to 9.9 (assuming a +/- .5 uncertainty).
The stated value of a measurement typically shouldn’t have more digits after the decimal point than the uncertainty has. Otherwise you are implying more resolution than you can be certain of.
fah ==> I investigated the issue of rounding in temp records long ago…given some time, I will resurrect the data for you. The issue is that when the Tavg gets reported and used in regainal and national averages, they use the rounded to a whole number figure.
Showing this requires downloading never-ending text files of data from the U.S. Historical Climate Network (USHCN) site. Just haven’t the time today.
Good post. The only thing you need to consider is that the variance of the distributions is totally ignored. Also, it is assumed that “errors and uncertainty” cancel if enough temperatures from different measuring devices are averaged together. This let’s averages of stated values be used with no concern about uncertainty.
The really big problem is that up until about 1980, temps were recorded as integers. Yet climate science routinely uses calculations that result in “anomalies” with 3 decimal places. This is adding resolution that was not originally measured. Uncertainties become important at these levels when NWS/NOAA specify accuracy of measurements of ±1°F. The boundaries of uncertainty generate error bars that subsume temperatures calculated to 0.001° or even 0.01° when the measured values are integer.
Three long posts and hundreds of comments later, it has yet to occur to Kip that (imperfectly correlated) errors cancel.
The variance of the sum of random variables was first derived by Ronald Fisher in a 1918 paper in the Transactions of the Royal Society of Edinburgh. The full implications of this work were acknowledged by the 1990 Nobel Prize for Harry Markowitz.
As Maryam Mirzakhani said, the beauty of mathematics shows itself to patient followers.
We all struggle to learn mathematics. It took clever people 5000 years to construct mathematics. Trying to get your head around even a small part of that is hard.
As an old friend said, mathematics is like sex, best done in private by consenting adults.
While Kip is clearly a consulting adult, it may be better if he would try to grasp 105-year-old papers in private.
Prefer Feynman’s observation: “Physics is like sex, sure it may give some practical results, but that’s not why we do it.”
Feynman’s always good. He did not suffer fools gladly. He would probably not be impressed by Kip confusing confidence interval and support.
Kip’s not confusing anything. You don’t know the distribution inside a closed box so you can’t form a confidence level. All you can do is determine the support – i.e. the range of possible values.
Perhaps you should come down from your perch and learn what Kip is trying t show. It is more akin to a 5 number description of a distribution than a mean/variance description.
As a person that has the ability to criticize other’s math, you should be very aware of the different descriptive statistics methods. I am surprised you are unable to think outside your little box and accept new ways of examining information.
Your statement that ” … (imperfectly correlated) errors cancel.” Reveals your inability to separate errors and uncertainty. Your statement is also only true for multiple, repeated measurements of a single measurand. You can not take an error measuring one measurand and use it to “offset” a measurement error on a different measurand. Would you take a measurement from one cylinder in an engine and use it to determine the value of another cylinder? If so, you would not be my mechanic for long.
You would not want to hire me as a mechanic in the first place. I trained as a statistician.
Kip is confusing support, standard deviation, and standard error.
This blog post and the previous one can be summarized as “the equation that people use for the standard error does not give the support.” Well duh.
As a statistician you probably simply don’t understand measurement uncertainty which is what Kip is getting at.
Uncertainty means YOU DON’T KNOW. You don’t know the probability distribution for the values contained within the CLOSED BOX that is the uncertainty interval. If you did you could make a best estimate of the true value and wouldn’t have to quote an uncertainty interval.
I would even extend Kip’s example further. Take a bag of dice and dump them into a plinko board with different closed boxes at the bottom. Take the closed boxes and try to figure out what is happening in the boxes. You might be able to identify some boxes with no dice but that’s about it. You won’t know if some of the dice are loaded (i.e. systematic bias) or not. You won’t know how many dice are in each box. You could even throw in a few 10-sided dice as outliers in the bag.
Now you have uncertainty intervals. A bunch of closed boxes and you don’t know what is going on inside them. NO PROBABILITY DISTRIBUTION CAN BE KNOWN.
Those are measurement uncertainty intervals. If you knew the distribution in each box there wouldn’t be any need for an uncertainty interval. You would just use the distributions to make a best estimate of the true value and put that down as the true value with nothing else attached.
That is what happens when you assume that all measurement uncertainty cancels. You just throw the boxes away and take the stated values as the best estimate of the true value.
Um, no. Bias errors do NOT cancel in general and continue to contribute to uncertainty.
as Taylor Swift says, goalposters gonna goalpost
Another clown who doesn’t understand that uncertainty is not error.
Next…
Only -3??
Shirley you clowns can do better than this, I demand it. Get busy.
Given your bitterness, I doubt you did purposefully spello’d this to be witty.
You are now the guy in line behind Woody Allen. Guess who, in this scene, is Richard S J Tol. Actually, it’s now the scene that got cut. The scene where the tie guy argued with Marshall McLuhan…
Another inane blob-post, ignored.
Don’t you have more down arrows that need clicking?
I usually would not want to be compared to Woody Allen but I’ll make an exception in this case.
Don’t blush, you were McLuhan.
If you are the guy I knew nothing about, until searching for you today, then your econometrics experience makes you a sickly good statistician. I have a bro with about the same CV as yours, and he is. He runs Banking and Finance education at an overpriced executive MBA school, and takes his students around the world to check out international banking systems.*
And if you read an earlier post of mine about how BS Geologists need more statistical training, well, I have a bro like that too…
No, all measurment uncertainty is random and Gaussian and cancels. /sarc
Sorry, my bad!
I think you forgot the /sarc.
Tol ==> Silly — but in a sense, you are right. Globally, temperatures themselves average out eventually to resolve to Earth’s expected Earth-like Global Average temperature of about 15°C.
However, I am not speaking of errors, but of intentional (rounding) original measurement uncertainty (clearly defined), in which the uncertainty is by definition evenly spread across the range of uncertainty, and since each measurement is uniquely of a different “thing”, the measurement uncertainty does not and cannot “cancel out”.
Many above have tried to explain that the measurement uncertainty should be statistically reduced to 3SD (95%) (with no physical reasoning to back up the claim).
This results in a false certainty:
EV Car Salesman: “Safety? dependability? Of course this new battery-powered EV is safe and dependable! It operates safely and dependably to 3SD! That’s 95%!”
Savvy buyer: “3SD, huh? 95%, right? So that means this car has a 1-in-20 chance of either bursting into flames and frying my family or stopping without heat in a snow storm leaving my family to freeze. I think I’ll pass…..”
~~~~
Bungee Jump Operator: “Yes, sir, statistics show that this activity is perfectly safe, tested to 3SD, 95% perfect every time.”
Prospective Bungee Jumper: “Hmmm, but what about the other 5%? What happens in those cases?”
Bungee Jump Operator: “Statistics, which is a SCIENCE, tells us we needn’t worry about those outliers, the cases where the bungee was too long and where the bungee was too short. They cancel out to leave ‘just the right length’. Remember, certified by statisticians (professionals!) to be safe to 3 Standard deviations!”
Prospective Bungee Jumper: “Gee. So 5% of the time, 1-in-20 jumps, your customers smash their head to pieces on the rocks below or end up hanging upside down far above the ground needing to be rescued by a fire truck with a 3 story ladder?. I think I’ll just to ride the roller coaster — much safer.”
Try building a bridge with such slop in the measurements, sir. A steel beam necessary to span 1000 feet would end up 50 feet too short or too long!
Errors introduced by rounding also cancel out, because you are just as likely to round up as round down.
In your second objection, you switched from confidence to inference. Different game, different rules.
Probability and statistics are seriously difficult subjects, typically requiring years of hard graft at university level to master. Exposing your ignorance in the way that you do serves nothing but your exhibitionism.
Tol ==> You use “cancel out” as if that actually reduces uncertainty, when you really mean, “just ignore the real measurements”.
You have a head full of “alternate reality definitions” — perfectly true to the World of Statisics, but whacky when applied on the real world. If your rules cannot and should not be used to “infer” things about the real world, to help us make decisions about the truth of measurements or make decisions that depend on them, what should we use them for?
Send me list of bridges you have built that use measurements that you are confident in — to 3SD/95% — and I’ll know which ones to avoid.
I am often amused by academics who disdain the world as lived in and experienced by 99.99% of humanity.
Inference is a synonym for hypothesis testing.
Do you have a recommended glossary of statistical terms? I have never bothered, and now that I’m hunting I can’t find the terms you’re using here. Yes, I should know them already….
I’m coping, somehow, with the d.t.’er that clicked as soon as they saw my nom de WUWT. But I wish that they would have at least provided a link to what I requested.
The Normal distribution is used a lot in engineering, particularly in production engineering.
Up to now I have focussed on the thermometer reading subtraction problem. But OK let’s consider other uncertainties.
Calibration certificates will give you an uncertainty based on the Normal distribution. Many other uncertainties also have close to Normal distributions.
So how exactly are you going to include Normal distributions in your “absolute” uncertainty calculation?
The Normal distribution goes on to infinity on both sides.
By your logic everything would have an infinite uncertainty range. Is that what you want?
You are missing the point entirely. This example started with discreet values from a die. An histogram is the appropriate way of illustrating the discreet values and their probability. Some folks are stuck in a rut and want to fit a continuous function of probability to those discreet values. I even see references to combining two dice probabilities to a triangular probability function.
Quit being a mathematician dealing with continuous probability functions and deal in the real, physical world. A die only gives discreet values – integers from 1 to 6. I don’t care how many you roll in order to generate enough data points to define the frequencies of combinations. You can not predict what the next roll will be because each die is independent and mutually exclusive. Each die, even from hundreds of dice, will still have equal probabilities for 1 to 6. The combinations will still be integer values even though a continuous probability curve can be “fit” to the data, you will never get combinations of 72.75 or 457.33. look up the equation for a normal curve. Does it have a way to define only discreet values?
Everyone is stuck in the mean/standard deviation box and can’t think of anything else. When I studied statistics one of the things we studied was combinations. That is the appropriate solution to THIS problem.
That’s because the dice experiment is put forward as an analogue for temperature uncertainty determination.
Temperature is a continuous function (OK let’s not get into quantum science here). I said previously, the dice are not an appropriate analogy because of that very fact.
You missed Kip’s point entirely, read section 3.1.2:
https://www.weather.gov/media/asos/aum-toc.pdf
You just don’t get it. The issue is not whether you have a continuous or discrete probability distribution. The issue is that you don’t know what the probability distribution *is*. The uncertainty interval is a closed box!
Temperature itself may be a continuous curve but its measurement is not! The measurements are discrete in time. Every second. Every minute. Every 5 minutes. Every ten minutes. Every hour. Twice a day.
Those generate a discrete histogram. Attached is a histogram of my measured temperatures for August, 2019 taken every five minutes.
Tell me that is a continuous probability distribution. Tell me that is a Gaussian probability distribution. Tell me that the average tells you what that distribution is like. Tell me that is not a skewed distribution. Tell me the mid-range value of all those temp measurements is a good index for global warming.
One of the things this graph points out is that daily mid-range temps versus time IS NOT a probability function that can be described as a PDF with all the connotations that implies.
This is a PDF. A monthly (or annual) temperature graph is a “time-series” and not a PDF.
I see that not a single CAGW advocate on here has addressed the data presented in this post!
Is it so damning of the assumptions behind the use of mid-range temp assumptions and averaging temperatures that no one dares to touch the issues the histogram brings up?
The real world of temperatures isn’t Gaussian. The assumption that it is just invalidates much of what climate science does today. It all starts with assuming that mid-range temperatures actually represent “averages” of anything.
Where are the usual suspects on this? bdgwx? bellman? KP? Stokes? bigoilbob?
Of course they haven’t, their only purpose is obfuscation and noise generation with meaningless smokescreens of illogic and sophistry. Starting with and especially the now-trite phrase: “cherry picking”. I predict the hamster wheel will circle back around at some point in the future to eject another instance of this smoke bomb.
Nobody advocates for catastrophic warming.
What data you think anyone has ignored. The only data presented is that for the sum two dice, that clearly shows you do not get a uniform distribution. Everything else is just Kip’s assertion that the only correct uncertainty interval is one that covers all possible values, regardless of how insignificantly implausible the are.
“Everything else is just Kip’s assertion that the only correct uncertainty interval is one that covers all possible values, regardless of how insignificantly implausible the are.”
If you know nothing about the probability distribution, i.e. a CLOSED BOX, then you *have* to try and determine the possible range of values. That basically covers all possible values.
There are no *implausible” values if you don’t know the distribution. It *could* be that the systematic bias drives the true value right out to the edge of the assumed interval of possible values.
As usual, you circle right back around to assuming that all uncertainty is random, normal, and cancels.
No matter how many times you deny you do that you just do it EVERY SINGLE TIME!
“If you know nothing about the probability distribution, i.e. a CLOSED BOX, then you *have* to try and determine the possible range of values.”
But you do know about the probability distribution. It’s what the despised statistics says. You know that it’s more likely to be a 7 than a 12, and just saying it’s in a closed box so we will never know isn’t an argument to pretend that actually 12 is as likely as 7.
Really this whole closed box thing is pretty nonsensical. You are saying the dice in the closed box are the “true vale” which you can never know. But if there is any point in estimating the true value and uncertainty, it’s because at some point you want to bring that value into reality, that is “open the box”. If this is a wooden plank, the reason for estimating the length and the uncertainty of that estimate is that at some point you are going to use the board for some purpose and you want to make sure it’s probably an acceptable length. And that’s the point you open the box. If it turns out your estimate was way of, you know the board is the wrong length. If you uncertainty was huge and it turns out each time the actual length was closed to your estimate, you’ve possibly wasted a lot of money.
“It *could* be that the systematic bias drives the true value right out to the edge of the assumed interval of possible values.”
But if you don’t know what systematic biases there are you don’t and can never know what the full range is. How do you know the dice don’t have some extra dots on one side, or someone hasn’t sneaked a 20 sided die in. And more realistically, if you are just measuring one of your 2m boards, how do you know what the total range is?
And, none of this has anything to do with Kip’s essay. He never mentions or suggests there is any bias in the dice. He explicitly demonstrates they are fair dice. He just wants to say that the correct “absolute measurement uncertainty” is one covering all possible values.
“EVERY SINGLE TIME!”
Every time you right that in all caps it’s obvious you have just lied. EVERY SINGLE TIME.
How about answering how you would deal with a Normal distribution in this “absolute uncertainty” calculation ?
It goes to infinity on each side, so your absolute uncertainty range will be infinite.
So what “in the real world” are you going to do with that please?
You are totally changing the problem. I have tried to tell you this example is built upon a RANGE of discreet values, 1 to 6. Look up the equation for a normal distribution and tell us if it has the ability to be evaluated for a finite range with limited discrete values.
A histogram is the appropriate graphical presentation. Can you fit a normal curve over the histogram? Sure you can but that doesn’t make the distribution a continuous normal distribution. It remains a finite range with discrete values.
The normal curve is a continuous function and can be integrated from -∞ to +∞.
God, this is so basic it shouldn’t be necessary to discuss it.
Bullshit. You are talking out of your hat, without knowledge.
No I’m not. I know what I am talking about and you do not.
I bow before the Great Man…
You’ve not posted *ANYTHING* on here to back your claim up at all!
Calibration certificates do *NOT* have to assume a normal distribution of uncertainty. Any measurement device that has an inherent hysteresis factor *can’t* generate a normal distribution and that should be noted in the calibration certificate. If it isn’t then you need to find a different calibration laboratory.
Think of a LIG thermometer that reads differently when the temperature is falling than when it is rising!
Many measurements can have hysteresis, assuming a normal distribution with errors canceling will be wrong, every time.
And again, spammed directly from the GUM:
That is instructive. You think 3SD corresponds to 95% confidence.
KB => As you know, despite the error (which I am happy to admit) most journal studies use 1 SD….
Did you notice he thinks 3SD corresponds to 95% confidence? !
Your last paragraph is very apposite.
However, I do not purport to be an expert in this field. I’m just someone who has to do uncertainty evaluations as part of my job. I would never presume to write articles in this field with my level of understanding.
I am at least at the level where I know what I don’t know.
But Kip is not even at that stage.
If you are experienced with uncertainty, then you should be familiar with a calibration chart that provides correction values. Something like for this reading add this much, for that reading subtract this much, i.e., a nonlinear curve. In other words, discrete correction values rather than a % or range. As an EE, we deal with that all the time due to nonlinear transfer functions.
If this guy really “has to do uncertainty evaluations as part of my job”, I pity anyone who has to deal him.
I pity any students who believe any of this stuff on this thread. They will fail their exams if they put any of this in their answers. Also any climate sceptic who uses this in arguments will embarrass themselves in public.
Ah yes, the truth reveals itself—KB is just another CO2 global warming propagandist.
I suspect the term “cultist” might be a better description.
Absolutely agree.
Ain’t that the truth! Following a flowchart to accomplish a goal is not the same as designing something and having to evaluate it.
After he stepped on me and elevated himself with his vast knowledge of uncertainty (a very religious act), I was tempted to let him know that, while I’m no expert, I have written multiple UAs that were required and critical for a laboratory accreditation under the ISO 17025 scheme. The calibrations were quite involved with multiple separate measurements and the UA documentation ran to scores of pages. Understanding every part of the measurement processes was critical.
ISO 17025 requires adherence to the GUM, so I was thrown into the deep end and had to learn. Fortunately I had a mathematician friend who had gone through the GUM I was able to consult. It was he who taught me that the probability distribution for a GUM combined uncertainty in general isn’t known. He rejected using the term U_95 for expanded uncertainty for this reason.
The UAs had to be evaluated by the accreditation agency, and were also made public so that the other labs doing similar calibrations could see and judge if they were bogus. In addition under ISO 17025, labs are required to do intercomparison testing with other labs, which can show problems right away.
We also did SPC charting of control measurements for quality control.
But I quickly realized that playing dueling credentials with this guy would be a fool’s errand.
“laboratory accreditation”
Which has absolutely nothing to do with field measurement instruments.
“It was he who taught me that the probability distribution for a GUM combined uncertainty in general isn’t known.”
Then why do you claim that it is?
KB: “Calibration certificates will give you an uncertainty based on the Normal distribution. Many other uncertainties also have close to Normal distributions.”
True, but ISO 17025 requires a laboratory to report calibration/measurement results with uncertainties using the GUM.
This would KB, not I; the GUM directly contradicts this claim.
Accreditation can indeed include field measurement instruments.
“Accreditation can indeed include field measurement instruments.”
That only applies immediately after installation and even that can be questioned based on the vagaries of handling during shipping and installation. Once the instrument leaves the calibration lab all bets are off.
Field thermometers are exposed to different environments than what was done in a calibration lab. Different pressure, temperatures, humidities, winds, enclosures, bugs, dirt, etc.
Those are all reasons why NWS specifies such large uncertainty intervals.
Temperature and time are merciless…
So if you are so clever why do you resort to ad hominens and trolling?
Because arguing with the noise generated by AGW cultists is pointless, so instead I just point and laugh.
blob pulls out more psych projection.
I don’t work in EE and I don’t use such charts.
However, using a correction factor is no problem in uncertainty evaluation if you have an estimate of the uncertainty on the correction factor.
Ideally you would also know the distribution shape of the correction factor uncertainty.
If you don’t, and the correction factor uncertainty is not one of the largest uncertainties in the calculation, it makes little difference what shape of distribution you assume.
What is the probability distribution of a combined standard uncertainty for a digital voltmeter?
I’d have to look that up, but I hesitate to waste my time with a troll such as you. I do have a recollection that U-shaped distributions can occur in EE.
A sufficient number of U-shaped distributions still ends up as close to Gaussian when combined, which is one of the more impressive demonstrations I watched.
The point you missed (again) is that a real DVM uncertainty calculation has to include use conditions which insert bias errors into the uncertainty which cannot be ignored or removed by averaging — all that can be done is to estimate their total ranges.
They are not Gaussian, they change with time in unknown magnitudes and directions.
The characteristics of the internal A-D converter can studied and complete distribution developed. But the A-D uncertainties are generally tiny.
Once the instrument leaves the manufacturing floor, use conditions dominate the measure results, and the manufacturer cannot tell you the answer.
+100.
If they could there would be no use for uncertainties and uncertainty analysis.
And what is under the hood of any modern temperature measurement?
A digital voltage measurement—far from being just “EE” esoterica, this is a critical subject for real-world uncertainty.
Yet climate science as a whole just ignores it and pretends it isn’t there.
Possibly but we were discussing the rounding error.
From what you say the voltage measurement uncertainty would need looking into.
But it is a completely separate point to what we are discussiing.
You are *still* assuming that uncertainty has a probability distribution. Uncertainty means *unknown*. How do you combine unknowns? I know you and bellman like to assume they will be normal but most of us in the real world don’t assume that.
Many instruments have a correction factor that covers only the internal components. When using it, the external connections to a DUT can cause considerable uncertainty.
Here is a document describing 4-wire measurements to eliminate some uncertainty but not all. Imagine what a simple two lead multimeter might have for uncertainty especially when you need 6 or 8 leads to reach measurement points.
KB ==> Did I make that error? — appears I did, too many charts, too many numbers. It is 2 SD that is 95% — 3 SD is 99.7%. Sorry for the confusion.
An aside detail. Something called an SD, which is taken to be a “standard deviation” can be calculated for any set of numbers, not just a normal probability distribution. (It is just the sqrt of the variance, which is just the sum of the differences between the mean and the values themselves. It can be any set of numbers, it does not have to be any kind of distribution at all). Formulas are available (or one can calculate them oneself) for all the generally used pdfs. Only the normal distribution has the property that +-1SD ~ 68%, +-2SD = ~95% etc. Other sets of numbers have different probabilities associated with increments of SD. For example the uniform (on 0,1) has +-1 SD ~ 58%, +-2SD = 100%. Other distributions or sets of numbers have a variety of relationships of cumulative fractions of the set to increments of SD. So just be aware that when one says +-2SD ~ 95%, +-3SD ~ 99% etc. one is specifically and explicitly and only talking about a normal distribution (or gaussian) and the statement is restricted to application to numbers that are so distributed.
What is the standard deviation of an impulse function?
You can calculate it yourself using simple arithmetic.
Gosh, I would never have known this until now.
Why don’t you work it out and tell us the answer?
fah ==> Thanks, was vaguely aware of that appreciate the clarifiction.
What does the average and standard deviation of a bi-modal or multi-modal set of numbers tell you about the set of numbers?
Statistical descriptors are only useful if they actually tell you something about the numbers you develop them from.
That is a great question, and an excuse to throw another Feynman quote into the conversation. A discussion of his lectures summarized his thinking that to just copy or imitate the method of the past is indeed to not be doing science. Feynman said we learn from science that you must doubt the experts: “Science is the belief in the ignorance of experts. When someone says ‘science teaches such and such’, he is using the word incorrectly. Science doesn’t teach it; experience teaches it”. Substitute “math” or “statistics’ for Science and it still applies. The point here is that no specific number “tells” you anything necessarily, you have to experience the data personally to learn anything.
This is a good example why doing math or science requires thought and ultimately direct experience. For example, for data that turns out to be bi-model or multi-modal one needs to look at it besides calculating one number and running with some conclusion. One can poke around a little calculating a mean and standard deviation and might notice (depending on the data) that the standard deviation seems large in the sense that it incorporates more of the range of the data than you would expect if it were a sharply peaked unimodal. But you don’t stop there, you LOOK at the data and experience it in a different way. If you look at the data you can often simply see bi- or multi-modality. You could calculate the median and look at the data to see if the median corresponded to a mode or modes, and that experience will suggest more things about the data. You could do what are called qqplots and look at the quantiles of the data compared to quantiles of various distributions you thought might apply. Or you could do an empirical cdf plot and see (experience) if the progression of the cumulants looks stepwise. The main point is to experience the data in as many ways as you can, to think about it before you blindly apply some numerical technique.
One of the exercises I give my students is a handful of simulated data sets. It is part of making probabilistic judgements about confidence intervals for data. If the data is or seems normal, then there are lots of tools to give you numbers, but if the data is not normal, then you have to use empirical estimates from the actual data distribution. One of the sets of data I give them is actually data merged from two normal distributions, fairly widely, but not grossly, separated. They are supposed to look at histograms of the data, qqplots (versus normal distributions), and do a simple Jarque-Bera test for normality on the data. All of this they do with matlab so it is easy. Depending on how you choose the bins, the histogram might look uniform or something else, but they are supposed to try a couple of binning methods, such as Sturges to see if structure is revealed. Sometimes they do, sometimes they don’t tease out the bimodality in that. The qqplots invariably show two distinct and characteristic squiggles around the intended straight line of the ideal qqplot. The normality test rejects the normal hypothesis rather roundly. Most of the students conclude that the data not normally distributed about the mean and any further estimates assuming that should not be trusted. Some of the students examine subsets of the data, looking at the data within each mode and figure out that the data actually looks more like it comes from two distinct sets that each are fairly closely normal (using the same visual techniques) and that something to think about would be whether the experiment in question may have involved several distinct phenomena. That is as far as they can go since it is only simulated data, but in an actual experiment, one would revisit the apparatus or system and look for things that might lead to that behavior.
The point is that no number one calculates from data “tells” you something, experiencing the data via multiple looks at it and considering the experiment it came from might tell you something.
Agreed, that is a point worth making.
It’s also often the case in uncertainty evaluation that the number of values is limited. In other words the degrees of freedom are too small to get a good estimate of the population standard deviation.
In those cases we have to use the t-distribution to find the multiplier for the desired confidence level.
I’m sure you know this already, but it has not been raised in this discussion previously.
Now I am remembering why I more or less stopped commenting here long ago. It seems much more effort gets spent figuring out things to call other people than clearly identifying a specific problem and then solving it. Lots of folks appear to bring unspoken assumed definitions of terms that I think vary a great deal in the particulars from discipline to discipline and person to person, like error, uncertainty, even things like standard deviation. That is a recipe for endless unresolved dispute in my experience. An example would be stating that +-2SD corresponds to 95%, where I think SD was meant as standard deviation. True enough if one is speaking strictly about a normal distribution, which is one reason a normal distribution is often useful, but in a conversation in which non-normality is a key feature, this can trigger someone who was thinking of non-normal interval distributions, such as a uniform distribution, in which +- 2SD encompasses 100% (I think). This is not a “statistical” thing per se, it reflects how the actual physical values occur in whatever problem is being considered. For a distribution like the normal, the possible values range to +- infinity and any number of SDs would be associated with probabilities distinct from each other and in the open interval (0,1). But all that stuff is beside the point, which seems to have become lost. I am much more at home with a clearly defined specific problem, down to the design of a specific instrument and a specific purpose or decision for which one wants to use the output.
Working to a purpose is one thing. Arguing for the sheer devilry of it is another. For example, if I was in an arguing mood, I might argue with calling statistics a “science.” It would be fun to argue that it is a part of math and math is invented, not discovered, whereas science is discovered. Or we could just argue about free will and entanglement, or whether a charge held at rest in a gravitational field radiates (and which observers would see the radiation) since acceleration and gravity are equivalent and accelerated charges radiate, and on and on. Personally, I think I have learned some things here and can move on.
It’s difficult to tell how much of it is the use of the same terms in different fields to mean slightly different things, and how much is argument for argument’s sake.
It’s good to see some new perspectives and insights this time.
I believe this belongs here:
https://diagrammonkey.wordpress.com/2023/01/04/on-the-twelfth-day-of-christmas-wuwt-gave-to-me-another-dreadful-analysis-of-uncertainty/
Happy New Years everyone!
That should be required reading for everyone on here !
It says what I have been saying, but with very helpful diagrams.
Clear and concise summary of what many have pointed out here. One very insightful statement tweaked something that had been in the back of my mind, but not yet risen to the surface, namely the visceral dislike for “statistics” or “complicated math” many voice here. As you point out, statistics is nothing more than math specifically developed to deal with this kind of problem. But the aversion to statistics and insistence via things like “the Challenge” to use nothing that is more complicated than one or two lines of arithmetic seems to be oblivious to the fact that virtually all of mathematics that relies on analysis of real numbers is derived with nothing more than arithmetic, sums and differences of real numbers. Computers calculate everything they produce from simple arithmetic. Sometimes deriving things can take more than a simple one or two lines of arithmetic, and sometimes people give shortcut names to things derived from some lines of arithmetic (like limits, integrals, derivatives, probability distributions, etc.) but ultimately everything involving real numbers is derived from the simple operations allowed between real numbers at the arithmetic level. The disdain for mathematical tools specifically designed for purpose, and ultimately based on nothing more than simple arithmetic, is puzzling in an otherwise technical community.
fah ==> If you read clearly, arithmetic comes up as the only maths needed to resolve the originally presented problem.
But resorting to definitions that are only found in statistics to deny simple facts proven by arithmetic is the problem.
First, I am learning a lot from much of the give and take on this and your other posts. Thanks for stimulating people to think.
That said, I think the reluctance to include things that “are only found in statistics” may be denying oneself the use of tools that have benefit. The point I was trying to make is that anything from what some here call, in somewhat of a derisive way, sophisticated math (for example statistics) are only things that themselves are derived from simple arithmetic. That means that simple arithmetic and statistics are not two things, they are one and the same, consistent with each other. If some math appropriately derived from arithmetic for specific purposes can be appropriately applied to a problem, and saves time and effort doing so, as well as avoids replicating rather long sequences of simple arithmetic, one should use it. Of course, it requires making sure that any underlying assumptions are also consistent.
Analogies can be misleading, but an example might be that one wants some carpentry work done, say an additional room to one’s house, and two people offer to do it. One has a hammer and a rip saw only and uses them to do anything they do, eschewing any other tools. Another has a truck full of tools, many specialized, and has done some very beautiful work for people you know, and if they have needed a better tool to do some specialized work, they have been known to acquire such tools. The second choice would likely be the one to take. (There could be others to consider, for example a specialized artisan who uses only authentic tools and materials from another era and takes their time doing things carefully and with art. One might choose such a workman and accept the time taken to do things in that fashion.)
One problem with “the problem” is stating it clearly and completely enough that the underlying definitions and conditions are absolutely limited and defined clearly enough that almost anyone could not fail to understand exactly what you mean as you mean it. I am sure you are exquisitely aware how many commenters here can read into statements conditions, possibilities, alternative definitions, etc. which you did not explicitly state or rule out, but which take the problem into domains you did not intend. In the classes I teach I myself write all of the problems, exercise instructions, etc that I use and I am continually learning how students can understand perfectly acceptable alternative interpretations of what I wrote and wind up taking paths that don’t work or sometimes go in more productive ways than I had intended. In a real sense, we are all students and should learn from each other. I had the good fortune to work with a fairly eminent scientist who once asked us if we knew why students came to university. His answer was “so the professors can learn from them.”
It is too bad that lots of folks on here (and in general) seem more interested in twisting content so as to denigrate or demean other people, rather than understanding and learning, but it seems to be a common human failing. A little humility can go a long way. I am sure that your posts will continue to take into account what you are learning from your students here.
fah ==> Just a quick note, I have to abandon this comment thread to its own devices, but I believe and have always believed and stated that statistics has its place in the tool box of science. But what it does not have is some over-riding, preeminent place in the tool box. It does not have exclusive rights to definition used in science, does not have the exclusive right to be
THE
ONLY CORRECT VIEWPOINT for looking at every problem.
I LIKE real proper statistics and real proper statistical approaches — I don’t like “hammer-sticians” — statisticians who claim the right to be the only tool in the box — a hammer — and the right demean, denigrate and scoff at all other approaches.
Third time was not the charm, as you can see from the responses. Even in this fora.
Perhaps read the WordPress critique, and ponder. Then, if you still hold this POV, have the convictional courage to write it up in superterranea.
Otherwise, you have the CV to tell sea stories, and others here do…
Can someone translate this word salad into English?
Anyone?
“The point I was trying to make is that anything from what some here call, in somewhat of a derisive way, sophisticated math (for example statistics) are only things that themselves are derived from simple arithmetic. “
Statistics are a tool. Just as a hammer doesn’t drill a hole very well, statistics that don’t describe reality don’t work very well.
You have to know when the statistical tool is applicable and when it isn’t.
Assuming all uncertainty is random and Gaussian so you can assume it all cancels is misusing a tool. The GUM only goes so far in describing the real world. For the most part it assumes that all systematic bias is corrected leaving only random error. That brings into question as to when the GUM is the proper tool to use in analyzing temperature data. It’s a hammer when you need a drill.
And people keep forgetting what is in the title of the GUM! It is a standard for expressing uncertainty, it can’t give each and every answer to every problem.
And once again I get a downvote from the $cientology zoo for stating the obvious.
Success.
Gosh, a whole down vote. No wonder you are so tetchy this morning.
Hypocrite. And again you mistake “techy” for laughter.
Where is YOUR air temperature uncertainty analysis? Must be in the place as blob’s.
Ha!
“I’m not tetchy” he stamped.
“You’re the one who’s tetchy” he whispered.
“I’m just laughing at you” he said through trembling lips.
Here, have an up vote from me, and try to take a brief nap. You’ll feel better for it.
It also misses the whole use of what variance provides. I will be doing an essay on that which I hope WUWT will post.
The basics. (Tmax+Tmin)/2 is a distribution. It is used as a random variable with a mean and a variance. Then some 30 days of random variables are added. When random variables are added, their variances add. Then an anomaly is calculated by subtracting a “random variable” made of ten years of the common month, each of which are made up of 30 random variables. When random variables are subtracted, the variances add.
When the comment is made that averages hide data and is pooh poohed, somehow pooh poohers never want to discuss what has happened to the variances that allow one to make coherent conclusions.
I look forward to this essay. But hopefully you’ll correct the error from this comment and avoid confusing adding with averaging. When you add random variables their variances add, but when you scale a variable by a constant, it variance is scaled by the square of that constant. I’m sur you can figure out what that means to the variance of an average of random variables.
I had missed Mr. Gorman’s offer to discuss this issue before, but it would be nice to see. I left a comment to him just now which might interest you, involving the ratio of two normally distributed variables. It is a somewhat obscure thing, but real. You may or may not be aware of it. A quick matlab calculation would illustrate the peculiarity if you were so inclined.
In general, I think it would be a nice thing if there were some kind of discussion forum either here at WUWT or somewhere focused on fundamental data analysis, particularly as relevant to climatology. Especially if it could be an honest effort to learn more about it rather than what seems to happen in the regular posts, trying to one-up others in various ways. I am immensely ignorant about the details of what is actually done in practice with climate data.
There tends to be some interesting discussion for a while, then Michael Palin and John Cleese take over as everybody becomes frustrated.
My take, fwiw, is that the various protagonists come from different fields which have subtly different understandings of the same technical terms.
It has been painfully obvious for quite some time that bellman & bgw have no real metrology experience, so to ascribe them to a relevant field is not accurate.
I’ve repeatedly told you have no metrology experience. My only “expertise” is that I understand a little statistics and and can read an equation.
So when someone claims the uncertainty of an average increases with sampling, I’m suspicious. When they justify this by pointing out the uncertainty of the sum, with no mention of an average, I can see where the problem is. When they insist that you never divide the uncertainty of the sum by the sample size, and justify that with equations that clearly show that the uncertainty will reduce with sample size, I assume they are confused. But when they continually refuse to accept the equations mean what they say, and just respond with ad hominems, distractions and insults I begin to wonder if they are the experts in their fields they claim to be.
“I’ve repeatedly told you have no metrology experience. My only “expertise” is that I understand a little statistics and and can read an equation.”
Which explains why you continually try to pound the square peg of metrology into the round hole of statistics. They just don’t fit!
“So when someone claims the uncertainty of an average increases with sampling, I’m suspicious.”
Uncertainty adds, even in GUM Eq 10. It never cancels, not even if you are evaluating an average. You can calculate an AVERAGE UNCERTAINTY but that is *NOT* the total uncertainty attributed to a measurand, not even through a functional relationship.
Look one more time at Possolo’s development of the volume of a cylinder. There’s no “averaging” done to get the uncertainty of the volume. The uncertainties of the contributing measurements ADD!
if q/n = x/n then u^2(q/n) = u^2(x/n) – then what is the uncertainty of x/n? What is u(x/n)?
u^2(x/n)/q = u^2)x)/x + u^2(n)/n — the relative uncertainties of each term. u(n) = 0 so you wind up with u^2(x/n)/q = u^2(x)/x
u^2(x)/x is the relative uncertainty of the average, not u(x)/n which is the average uncertainty.
If x = x1 + x2 + x3 + … then the uncertainty of x is the sum of the uncertainties of x1, x2, x3, ….
The more measurements of separate elements you add the larger the uncertainty is going to get.
You just can’t get out of the box where you assume that all measurement uncertainty cancels and the standard deviation of the sample means is the uncertainty of the average. It isn’t. It’s how close you are to the population mean. And the population mean can be *very* inaccurate, especially if you have systematic uncertainty!
“Which explains why you continually try to pound the square peg of metrology into the round hole of statistics. They just don’t fit!”
Yet all the metrology sources use the same statistics to derive the propagation of uncertainty / error.
“Uncertainty adds, even in GUM Eq 10. It never cancels, not even if you are evaluating an average. You can calculate an AVERAGE UNCERTAINTY but that is *NOT* the total uncertainty attributed to a measurand, not even through a functional relationship.”
Take it up with Kip Hansen. Even he agrees that the uncertainty of an average requires dividing the uncertainty of the sum by sthe sample size.
“Look one more time at Possolo’s development of the volume of a cylinder. There’s no “averaging” done to get the uncertainty of the volume.”
Of course there’s no averaging. It’s multiplication.
And I spent ages trying to explain how you were completely misunderstanding how the general rule of propagation (equation 10) was used to get that result, and you completely failed to understand because you don’t understand how to calculate a partial derivative, and then you tried to pretend that equation 10 was about relative uncertainty, just to get your misunderstanding make sense.
I have no intention of going through all that again at this late stage in these comments.
“if q/n = x/n then u^2(q/n) = u^2(x/n) – then what is the uncertainty of x/n? What is u(x/n)?”
Do you just have some random equation generator? What do you think you are saying there. If q/n = x/n, then q = x. Little point ploughing through the rest of your gobbledygook.
There is a lot of crossover between mathematics, statistics and metrology, but to paraphrase Evelyn Waugh, “three great fields separated by a common language”.
There are enough differences that proficiency in one doesn’t guarantee proficiency in either of the others.
Quite true, well stated.
From left to right, wouldn’t proficiency in one area be required for proficiency in the next?
That’s a good question. It has more to do with breadth than depth.
There are intersections between the fields, but they aren’t concentric circles.
Mathematics is a very broad field.
Statistics is a specialised field of applied mathematics, so there are many areas of pure mathematics (for example) which aren’t relevant.
Metrology appears to be more of a field of experimental physics / engineering with a strong statistical component.
I don’t know that I’d want a metrologist doing questionnaire design, for example.
Literally, the Study of Metrics or Measurements.
I agree with your post. But it didn’t answer my question. My point is that you can’t do the stuff on the right without mastering the stuff on the left.
This runs counter to those “realists” who claim that there are situations where their “real world” intuition is superior to established statistical laws. Yes, those laws can be incorrectly applied, but the “real worlders” here have yet to provide an example of a measurement problem in which their “common sense” prevails against those statistical laws.
Good examples of nonintuitive truth are the multiple applications of Bayes theorem to drug and medical outcomes. One is linked here, and there are others*. What these all have in common is the fact that they are commonly posed to experienced medical professionals and the right answers are offered infrequently. The fact that the right answers are nonintuitive don’t make them any less correct.
https://towardsdatascience.com/bayes-rule-with-a-simple-and-practical-example-2bce3d0f4ad0#:~:text=For%20example%2C%20if%20a%20disease,knowledge%20of%20the%20person's%20age.
*By properly chaining Bayes Rule, this problem can be solved either from use of the theorem on a piece of paper, or with a decision tree. Software is not required.
I think we’re looking at this from different perspectives.
Yes, you have to be able to “do” (some) mathematics for the field of statistics, and you have to be able to “do” (some) statistics for the field of metrology.
That’s quite a different proposition to “being” a mathematician (pure or applied), statistician or metrologist.
There is a lot in common between the fields, and also a lot of differences.
My proposition is that the differences in understanding and thought processes between the different fields introduce enough “uncertainties” that people end up talking past each other.
and become frustrated, and can’t understand why the people from the other fields are so dumb, and move the goal posts and lie and …
What’s frustrating to me is that they aren’t dumb, lying, etc, but the fundamentals are so deeply embedded (and almost the same) that it’s impossible to reconcile the differences.
It’s like the analogy of fuzzy images in a microscope.
Lots to agree on there.
But the exchanges here are on the bogus proposition that statistical analysis is “just a hammer” and not the whole toolbox. It is indeed the latter. When you question statistical laws developed, step by step, for over a century, those laws are either (1) not being properly applied by the evaluator, and/or (2) not being considered by the metrologist with flawed intuition.
Since brilliant, hard working medical pro’s get these Bayes problems wrong so often, these “realists” are in good company. But the difference is that – I’m guessing here – most of those medical pro’s don’t pull Dan Kahan Type 2 hysterical blindness when called out on algebra errors so obvious that a journeyman like me can catch them (when guided, step by step). Rather, they learn the lesson.
Yep.
“In general, I think it would be a nice thing if there were some kind of discussion forum either here at WUWT or somewhere focused on fundamental data analysis, particularly as relevant to climatology.”
https://wattsupwiththat.com/2023/01/03/unknown-uncertain-or-both/#comment-3662955
You can’t really work without tools. Or the willingness to use them. Bellman’s grandiose claim (sarc) that he can follow someone else’s derivation of equations is about the minimum level of expertise required to contribute. Him and bdgwx exceed that, I do so just barely. The pity is that the rest – at least some – seem to have the horsepower, but are doomed by Dan Kahan Type 2 hysterical blindness. Their very acumen is put to use to enforce prejudgments rather than to advance their understandings.
…says the clown who sees DJ Trump under his bed at night.
I’ll give you a hint.
https://intellipaat.com/blog/tutorial/statistics-and-probability-tutorial/sampling-and-combination-of-variables/
Think about what “n” is when averaging two temperatures. Think about what a random variable is when it comes to a collection of temperatures and their averaging.
I wasn’t talking about averaging two temperatures, but about you adding 30 random variables (the mean daily temperatures).
These “hints” in the form of some random internet page of unspecified authority are not helpful, unless you say exactly what meaning you want me to draw from them. And they usually just demonstrate you are wrong. This particular page makes my point.
If W = aX + bY
then
σ²_W = a²σ²_X + b²σ²_Y
What’s your point?
It isn’t [a²σ²_X + b²σ²_Y] / 2 like you seem to want to do all the time!
“It isn’t [a²σ²_X + b²σ²_Y] / 2”
Correct. It’s a²σ²_X + b²σ²_Y, as I clearly wrote. Why you have to keep corrupting all these simple equations I don’t know.
“…like you seem to want to do all the time!”
Stop lying. You actually copied what I wrote, added a meaningless division by zero and then claimed that’s what to do all the time. Can’t you see how absurd this vendetta is? You have just claimed I want to do something whilst quoting me as doing something else.
“When you add random variables their variances add, but when you scale a variable by a constant, it variance is scaled by the square of that constant. I’m sur you can figure out what that means to the variance of an average of random variables.”
Finding the average variance is just like finding the average uncertainty. You are *NOT* finding out anything about the population. All you are doing is taking a set of random variables, which may have widely differing variances, and finding an average value you can apply instead of using the differing values themselves.
It is the variance of the population that you are interested in, not some “equal” value of variance you can apply to each member of the population.
Just like the average uncertainty is not the uncertainty of the average, the average variance is not the variance of the population and may not be the variance of any individual member of the population.
It’s like the example of the fifty 6′ +/- 0.08′ boards and the fifty 8′ +/- 0.04′ boards.
The average length is 7′ – WHICH DOESN’T EXIST IN YOUR PILE OF BOARDS, and the average uncertainty is +/- 0.06′ – WHICH DOESN’T EXIST IN YOUR PILE OF BOARDS.
Finding the average length and the average uncertainty tells you 1. nothing about the population, 2. tells you nothing about any of the individual boards, and 3. doesn’t describe reality. Statistics are only a useful tool if the tool describes reality.
If the statistical tools don’t work every time then they aren’t functional relationships. If the average length and the average uncertainty doesn’t physically exist then the equation used to find them isn’t a functional relationship. It may be a function but it isn’t a functional RELATIONSHIP.
Saying it works sometimes doesn’t help. πR^2H works EVERY TIME. Width x Length works EVERY TIME. Velocity of a falling object = gravity x time works EVERY TIME.
You somehow can’t differentiate between an equation and a functional relationship. An equation may be a function but it does *not* always describe a functional relationship.
Why can you never stick to a point? One of you talked about adding variances and was using that for an average. I explain how to actually calculate the variance for an average, and you immediately jump on it and go into another stream of well worn nonsense.
“Finding the average variance is just like finding the average uncertainty. ”
I’m not finding the average variance, but the variance of an average. Why do you always get these two concepts mixed up?
“You are *NOT* finding out anything about the population.”
What population. I was talking about measurement uncertainties. It’s just as applicable to sampling from a population, but throughout the rest of your comments you seem very unsure about which you are talking about.
“All you are doing is taking a set of random variables, which may have widely differing variances”
You see, if I’m taking an iid sample from a population, all the variances would be the same.
“ and finding an average value you can apply instead of using the differing values themselves.”
Garbled nonsense. I’m finding the variance of the average, and I don’t know what you mean by using the values themselves. Why are you taking the average? How would you achieve the same goals by looking at the different values themselves?
“It is the variance of the population that you are interested in”
You may or may not be interested in the variance of the population. But if I’m taking an average I am interested in how good an estimate it is of the population mean, which is the purpose of finding the variance of the average of the sample.
“not some “equal” value of variance you can apply to each member of the population.”
No idea what you think you are saying here. I’m not applying the variance of the mean to “each member of the population”.
“Just like the average uncertainty is not the uncertainty of the average”
Good to see you agree with me on that.
“the average variance is not the variance of the population”
But it is. If I take N random variables from the same population distribution, all will have the same variance and the average variance will be the same as the population variance, as they are all the same value.
“and may not be the variance of any individual member of the population.”
What do you think the variance of an individual member of the population is? The variance of an individual member will be 0.
I’ll take a breather here, before you start obsessing about 6′ boards again.
“It’s like the example of the fifty 6′ +/- 0.08′ boards and the fifty 8′ +/- 0.04′ boards.
The average length is 7′ – WHICH DOESN’T EXIST IN YOUR PILE OF BOARDS, and the average uncertainty is +/- 0.06′ – WHICH DOESN’T EXIST IN YOUR PILE OF BOARDS.”
Could you for once take a breath and explain what you are trying to do with these boards. If you have as many 6 and 8′ boards the average is 7′, self evidently, but I don’t know what you are getting at with the uncertainty. If you mean the measurement uncertainty, then the uncertainty of that average is ±0.006′, not 0.06. If on the other hand you are talking about a sample of 100 boards (I assume not as you have exactly 50 of each) the standard uncertainty is ±0.1′, with an extended uncertainty of lets say ±0.2′, not including measurement errors.
The fact that you don;t actually have a 7′ board is irrelevant, we are not talking about an individual board of average length but the average length of your two sets of board.
“Finding the average length and the average uncertainty tells you 1. nothing about the population”
It gives me the best estimate of mean of the population and the SEM tells me how much confidence to put in that estimate. What ever you may think, that is not nothing.
“2. tells you nothing about any of the individual boards”
It tells me what the average length of an individual board is, but it isn’t really the point of the mean to tell me the distribution of the boards. If I wanted to know that I’d look at – guess what =- the distribution of the boards.
“3. doesn’t describe reality.”
In your small box you may not consider it reality. In the wider world it’s part of a highly useful way of describing and investigating parts of reality.
As always you try to come up with a pointless example of averaging and then claim if you personally cannot figure out why it might not be useful in this case, all averaging must be entirely useless. I feel sorry for you. You want to shut yourself up in a closed box, and call it the whole world.
“If the statistical tools don’t work every time then they aren’t functional relationships.”
Meaningless non sequitur .
“If the average length and the average uncertainty doesn’t physically exist then the equation used to find them isn’t a functional relationship.”
Again, show me a reference for your definition of functional relationship. I’m beginning to suspect you just guessed at what it meant and now refuse to accept you might be wrong.
“It may be a function but it isn’t a functional RELATIONSHIP”
What do you think the word functional means in the term functional relationship? What do you think a relationship is?
“Saying it works sometimes doesn’t help.”
(x + y) / 2 works every time.
And you still won’t say if you accept that x + y is a functional relationship. It’ should be obvious by now you are just trying to make a special point about the mean to avoid answering the question I’m actually asking, which is whether you agree with Kip or the GUM when it comes to adding (not averaging) random independent measurements.
“You somehow can’t differentiate between an equation and a functional relationship.”
Sure I can. If an equation produces a single specific value for any specific set of variables, it describes a functional relationship. If it doesn’t it doesn’t.
“An equation may be a function but it does *not* always describe a functional relationship.”
It does in every definition of functional relationship I’ve seen (apart from it’s use in psychology).
Just noticed your comment. Sounds like it would be a great thing to discuss in a reasonably rigorous fashion. I know I would like to see the discussion.
It might be useful to do a google scholar search on “geary hinkley transformation,” if you are so inclined. A decade or two ago I was involved in an experiment looking at the decay of certain particles in which the quantity of interest was the fraction of surviving particles after a particular fluence of radiation of a certain type. The fraction was expressed as the ratio of number of remaining particles divided by the number of particles before the fluence. (The details don’t really matter) The measurement process generated numbers of surviving particles and numbers of initial particles such that each of those quantities were distributed fairly closely following a normal distribution. I got interested in the statistics of the resulting surviving fraction, which was calculated as the ratio of the two (normally distributed) variables. It turns out there is a fairly obscure nuance that the ratio of two normally distributed random variables is not normally distributed, and is a little complicated depending on independence, correlations, etc. However, there is a transformation called the Geary Hinkley (no relation to John Hinckley) transformation that, when applied to the variables, renders the result to be a normally distributed variable. It is a bit messy but it works. As I recall, I concluded that for the data I was looking at, the difference was small enough to ignore, given the effort required. It is of interest now (which you will see if you do the search) mostly to biologists studying cancers, survival of microorganisms, and things like that. But it is nice to be aware of if you are dealing with ratios of random variables. I have a number of refs on it, but you can find them just as easily via the search.
Looking forward to seeing your discussion.
Any mathematical procedure applied to reasoning about physical processes (which I will call for brevity physics) is a tool. The major leap in science, in my view largely due to that wonder of a thinker, Galileo, has been the realization that physics is most clearly and non arbitrarily advanced when the reasoning is quantified. For which the tool currently is mathematics. And of course, the tool is a representation of the physical reality, not the physical entity itself. For example, a wave equation is sometimes a good tool for conceptualizing physical properties of things like electrons or mesons or Higgs bosons, but it is just a tool not the thing itself. Sometimes a particulate description is better.
It is a misconception to ascribe to statistics a notion that it necessarily involves distributions (such as a particular one, the normal distribution). Statistics includes basic concepts aimed at the collection, organization, analysis, interpretation, and presentation of data, consistent with the discipline in which the data was obtained. If the data is obtained from physics then the use of any mathematical tool must be matched to the proper representation of the underlying physics. If it is biology, or psychology, or economics, it is the same, the math is only useful insofar as its use reflects the underlying discipline.
Statistics includes more than fitting a normal distribution to some data, or using an equation or set of values that depend on a normal distribution. In fact, that is a relatively minute part of statistics. The equation that Kip uses is in fact a statistic, which means nothing more than an estimator of a property of any set of numbers, irrespective of any knowledge of how the numbers may be arranged (as in distributed). Things like arithmetic means, medians, geometric means, variances, covariances, etc. are all simple “statistics” which do not depend on any underlying probability distribution, they are simply descriptors of the set of numbers. They tell us about the properties the set has in general. So to casually dismiss using “statistics” to analyze data is summarily dismissing the use of mathematics in toto to represent physics.
Sometimes people split statistics into exploratory statistics and inferential statistics. This is largely due to the influence of John Tukey, who I had the good fortune and pleasure to work around for a time. Exploratory statistics is just what it says, looking around at data and seeing what you can see, versus trying to calculate some number that tests some hypothesis. It has a heavy graphical component, which is now sometimes called data visualization. In the physics classes I teach, I include a heavy component of simply looking at one’s data to see if there appears to be any structure. If one wants to use some tool or make an inference that assumes a particular distribution (such as, but not only, a normal distribution) does the data LOOK normal? Does it pass any tests for normality? Is it even unimodal? If not there may be multiple phenomena going on and math tools to analyze it need to take that into account. The main point is not to use what you call a drill or a hammer if the underlying data (or they physics) don’t call for it.
So the notion that statistics “blindly” applies tools to data is simply not the general case. True, some scientists use computer tools they have not written to estimate numbers for which they have not carefully considered for applicability. But thoughtful ones do not. Using computer programs of any sort, whether statistical packages, or simulations, or CFD, etc. always requires careful consideration of whether the math is representing the physics.
I think we found the core of Kip’s problem. Statistics and probability go well beyond arithmetics. You need series, limits, derivatives, integrals, etc to scratch the surface.
(This explains why the development of statistics really started only in the early 20th century. Earlier mathematicians did not have the tools. It also explains why you need more than rusty high-school maths to know what is going on.)
The more subtle points of statistics and probability require a thorough understanding of filtration and Borel algebra.
Who are “we”?
That sums it up pretty well.
Let’s consider what happens if we increase the number of dice in each throw.
Fairly arbitrarily I will do this with just FIVE dice.
The full range of outcomes is 5 x 1 =5 to 5 x 6 =30
Every possibility from 5 to 30 inclusive can occur, so the range is 26. In the rather strange terminology of Kip, this is the “absolute uncertainty” range.
But let’s look at the extremes of this “absolute” uncertainty range.
The probability of all five dice in a throw showing “1” is (1/6)^5 = 0.00013, or 0.013%
It is exactly the same for all five dice showing “6” of course, so 0.013%
It’s about 1 in 7,800 chance that either 5 or 30 will be thrown.
If we repeat the exercise with 10 dice, the range is 10 to 60. The probability of either 10 or 60 being thrown is about 1 in 60 million each.
This shows that using this so-called “absolute uncertainty” could quickly become useless when we have several distributions to combine. The probability of the extremes of the range being the “true” answer becomes very small as the number of individual uncertainties increases.
It also becomes impossible to conclude anything. The ten dice example has a mean of 35 which would have +/- 25 in “absolute” uncertainty. Yet the extremes of that range have very little chance (0.0000017%) of being the “true” result.
Statistics über alles.
Epilogue:
One final synopsis and then I’m done following this comment thread. Note that in the fog of battle, I incorrectly confused 3SD and 2SD — it is 2SD that represents ~95% of values.
The detractors, mostly self-declared statisticians, scoff and mock. Given enough rope, they in the end admit that what my very simple examples show is that the real full uncertainty of two added value and their original absolute measurement uncertainty is exactly as I give in the essay.
BUT, they don’t like it — they don’t want “full uncertainty” they want (and when I say want, it is just a preference, not based on science). They want, in the end, what one detractor finally admits is nothing other than “what’s reasonable to expect”.
Reasonable is NOT a scientific term.
The statistician’s fight here is to prevent us from coming to the conclusion that when we add two values with absolute measurement uncertainty by adding the values and adding the uncertainties, then it follows that when we find the mean of the added values using the standard arithmetic process is:(using an example)
ex. (4.05 +/- 0.3) / 3
answer: (4.05 +/- 0.3) / 3 = (4.08 to 4.02) / 3 = 1.36 to 1.34 = 1.35 +/- 0.1
[Reference]
or in other words:
Arithmetic Mean = ((x1 + x2 + …. + xn) / n) +/- (summed uncertainty / n)
The significance of this is the average of many added rounded values (say Temperatures Values, rounded to whole degrees, to be X° +/- 0.5°C) will carry the same original measurement uncertainty value.
The full uncertainty will not be reduced by averaging.
This is TRUE and can be easily demonstrated arithmetically and physically.
But it runs contrary to current statistical beliefs and practices. And, as we can see in all the above, is fought claw-tooth-and-nail.
Thank you for reading.
Thanks for the summary. My own, not necessarily final, thoughts:
“Given enough rope, they in the end admit that what my very simple examples show is that the real full uncertainty of two added value and their original absolute measurement uncertainty is exactly as I give in the essay.”
Never said it wasn’t. It’s just a question of how useful a figure it is.
“Reasonable is NOT a scientific term. ”
It is the term used by the GUM though.
As I’ve noted before, it’s strange that those who keep insisting that the GUM is the authority on the subject now seem unwilling to defend it.
“The statistician’s fight here is to prevent us from coming to the conclusion that when we add two values with absolute measurement uncertainty by adding the values and adding the uncertainties, then it follows that when we find the mean of the added values using the standard arithmetic process is”
I’ve not argued that you cannot say that. It’s what a lot of introductions to measurements do. Taylor, again regarded as an authority by some until these essays came out, uses this as his provisional rule. Before explaining that under some circumstances this will give a result that is too wide, and introduces the standard rules for adding independent uncertainties – i.e. adding in quadrature. Again it would be helpful if those who spent so much time promoting Taylor, would now just say if they agree with Kip that Taylor was wrong to advocate adding in quadrature.
“Arithmetic Mean = ((x1 + x2 + …. + xn) / n) +/- (summed uncertainty / n)”
It would also be useful if anyone who insisted that you never divide the uncertainty by n, when looking for the uncertainty of the mean, would say if they accept they were wrong, or if they disagree with Kip on this point.
“This is TRUE and can be easily demonstrated arithmetically and physically.”
It’s true only if everyone accepts your definition of “absolute measurement uncertainty”. Most sources on metrology do no accept that, and I’d say with good reason. Claiming something is TRUE isn’t interesting if your truth leads to meaningless results. It is true that 2 + 2 < 100000000. That doesn’t mean it’s useful.
“But it runs contrary to current statistical beliefs and practices.”
If I was writing multiple essays on a subject I might wonder why current statistical practices disagreed with what I said.
I am surprised (and perhaps a little disappointed) by how many commenters here get into what they think is in other people’s minds or other people’s intentions, or even moral turpitude. Someone or some folks here quoted John Taylor’s charming discussion of uncertainty analysis, now quite old but still delightful to read. I think many who have participated here could be refreshed by reading (or re-reading) his Chapter 3 Propagation of Uncertainties. In it he gives a very simple and direct discussion of how to treat uncertainties in measurements, always basing the discussion on the physical nature of measurements, and only using simple arithmetic expressions. He describes when (in terms of what the physics of the measuring process imply) one should add uncertainty numbers directly and when (again in terms of the physics) one should add them in quadrature and what the significance might be in specific cases of either approach. He uses simple arithmetic everywhere and relies on no “sophisticated” math. I think if one steps back, takes a breath, and reads his treatment just for the pleasure of his writing, one can return to discussions of this topic feeling refreshed.
Best wishes.
The irony is that it was Tim Gorman who introduced me to Taylor’s book. He used it to justify his claim that uncertainties increase with sample size, and I’ve spent years trying to explain why Taylor doesn’t say that. Now he seems keen to ignore Taylor in order to agree with Kip.
But his argument is always that he has some profoundly deeper understanding of Taylor, because he’s done all the exercises, and quoting Taylor’s actual words is just cherry-picking.
I’ve given you the EXACT QUOTE from Taylor giving the restrictions that must be met for using RSS.
As usual, you just ignore the truth in favor of an undying belief that all uncertainty is random and Gaussian.
You never quote Taylor in context, only as cherry picked pieces you think you can use to prove someone wrong.
You’ve never once done the exercises and tried to figure out why you can’t get the answer in the back of the book.
Uncertainty *does* increase with sample size UNLESS the uncertainty is totally random and Gaussian. As Taylor points out specifically.
For some reason you simply can’t get it that the CLT only tells you how close you are to the population mean and not how uncertain the population mean is. You can’t even understand that the mean tells you nothing about the population by itself. That’s why climate scientists never bother with variance, standard deviation, kurtosis, skewness, etc. You and they just cling to the religious dogma that all uncertainty is random and Gaussian and therefore cancels – even for single measurements of a single thing!
Bellman said: “because he’s done all the exercises”
Which is baffling since we’re continuously spammed with the “equations only work when you’re measuring the same thing” arguments. Note that Taylor’s work has plenty of examples (including the obvious ones in section 3.9 and 3.10) that are decisively and unequivocally operating on measurements of different things. It couldn’t be more obviously that Taylor intends for the rules to be used on measurements of different things. He’s literally telling people to do so in section 3 via the examples.
Jeez, why don’t you actually study this tome instead of cherry-picking! Did you miss the sentence that says:
Why do you think this is a necessary assumption when considering the rule of “the same thing”!
Then of course the text with Eq. 3.10 discusses the uncertainty of a power. Specifically, the calculation of kinetic energy of ((1/2 • mv^2). Where v^2 mathematically is “v•v”. Thus, the total uncertainty would be “2 • uncertainty of v”.
Why do you think v•v are two different things?
Your ineptness with the basics of uncertainty is disappointing considering how you post as an expert.
Which is just a form of the Appeal to Authority logical fallacy, with himself as the authority. bellcurvewhineman uses this one also.
“Note that Taylor’s work has plenty of examples (including the obvious ones in section 3.9 and 3.10) that are decisively and unequivocally operating on measurements of different things.”
And what does he say to do when you are measuring different things?
See his final section on Chapter 3 where he gives the general formula for using RSS for error propagation.
“(provided all errors are independent and random)”
How many measurements provided by different measuring stations are “Random”?
It is sad that so many so-called experts on uncertainty that post here can’t understand how uncertainty relates to the real world. The apparently don’t even know the basic rule for combining random variables – VARIANCES ADD.
That includes *YOU*. You don’t even understand when Eq 10 of the GUM is used and when it isn’t.
TG said: “And what does he say to do when you are measuring different things?”
Use the rules he outlines in section.
TG said: ““(provided all errors are independent and random)””
That has nothing to do with the fact that Taylor tell us to use his rules on measurements of different things.
TG said: “How many measurements provided by different measuring stations are “Random”?”
That has nothing to do with the fact that Taylor tell us to use his rules on measurements of different things.
TG said: “The apparently don’t even know the basic rule for combining random variables.”
That is the pot calling the kettle black. You insist on using Taylor 3.16, which is the rule for sums (+), to assess the result of quotients (/) and make numerous algebra mistakes when applying any of Taylor’s rules in general. And yet you still have the hubris to gaslight us by saying the rules are basic.
Where does Taylor do *any* thing past Chapter 3 where he doesn’t assume that all measurement uncertainty is random and Gaussian and cancels with no systematic uncertainty.
In Chapter 4 onward he assumes you can do a statistical analysis of the stated values while ignoring measurement uncertainty.
How many measurement uncertainty values does he state in Tables 9.1 and 9.2? Don’t give us a bunch of word salad. Just answer “all of them” or “none of them”.
In chapter 3 Taylor *does* use “stated value +/- measurement uncertainty. He propagates the *measurement uncertainties”. Not the standard deviations of multiple measurements of different things.
You and your compatriots simply can’t seem to get the difference in these approaches straight in your head.
If I measure TempA and TempB and get a +/- u(a) and b +/- u(b) then I propagate the uncertainties of the two measurements, either directly, u(a) + u(b) or by RSS. I am including both the random error factor in u(a) and u(b) as well as the systematic bias in both u(a) and u(b).
I do *NOT* assume u(a) and u(b) are random, Gaussian, and cancel with no systematic bias in either.
But that is what *YOU*, bellman, KB, etc want to do EVERY SINGLE TIME.
You’ve been brainwashed into ignoring measurement uncertainty.
“Where does Taylor do *any* thing past Chapter 3 where he doesn’t assume that all measurement uncertainty is random and Gaussian and cancels with no systematic uncertainty.”
I’ve given you quotes and screenshots already. The sentence immediately under equation 9.9.
Taylor specifically states that adding in quadrature is only fully applicable when the uncertainty is random and Gaussian. That is hard to run away from but it seems to be the final cave in which statisticians try to hide.
He does not. And if he did he would be wrong. Unlike you I don’t regard Taylor as holy scripture that can only be understood by the devout who can ignore the words and just understand the meaning.
I’ve pointed you to chapter 9 where he explains what to do with non normal uncertainties, it ‘s the same as equation 11 in the GUM. If the uncertainties are completely independent the equation is the same as for equation 10. The uncertainties add in quadrature.
You can look at this from numerous different directions and the algebra is always the same. Add random variables, the variances add, or use the CLT, the sum tends to a normal distribution with standard deviation sqrt(N) * SD. In all cases the distribution of the random variables does not change the result, the SD of the sum is derived from adding the individual SDs in quadrature.
The truth of this doesn’t require a belief in a holy book, it can be understood from the maths or it can be demonstrated by rolling dice or on a computer. The rules assume independence they do not assume a normal distribution.
More projection.
I apologize for not seeing your comment until after I posted the little jpg from Taylor’s Chapter 9. It would be nice to be able to post extensively from Taylor because first, his writing is just superbly concise and clear, and secondly he truly does derive everything he shows from very basic math, essentially arithmetic (except for taking derivatives, which also can be derived by simple arithmetic, just a little more of it).
No need. It’s useful to have independent confirmation.
It won’t matter of course, we’ll just be accused of cherry-picking and not reading for the deeper meaning.
I am just glad I now have a pdf second edition of Taylor, which I give to my students, from which I can extract pages. My paper copy is a first edition, about 40 years old now, and was given to me as a student. I perhaps should not admit that now I often go to Kendall’s advanced theory of statistics volumes for things when I need the gory details. While much more encyclopedic, Kendall’s could never be accused of being clear and concise and some here might insist it is written in the language of Mordor, which should not be uttered here.
“Unlike you I don’t regard Taylor as holy scripture that can only be understood by the devout who can ignore the words and just understand the meaning.”
I gave you the words. Their meaning is obvious.
Taylor: “Suppose that to find a value for the function q(x,y), we measure the quantities x and y several times, obtaining N pairs of data, (x1,y1), …, (xn,yn). From the N measurements x1,…,xn we can compute the mean x_bar and standard deviation σx in the usual way; …”
“I’ve pointed you to chapter 9 where he explains what to do with non normal uncertainties, it ‘s the same as equation 11 in the GUM.”
I explained Chapter 9 to you using Taylor’s own words. you choose to ignore them. Chapter 9 is *NOT* about uncertainty interval probabilities. It’s about using the stated values only. That implies that the uncertainty intervals associated with the measurements are totally random and normal so that complete cancellation occurs – AND NO SYSTEMATIC UNCERTAINTY EXISTS.
Chapter 11 is about the Poisson distribution – COUNTING SITUATIONS.
Temperature measurements are not *counting*. They are *measuring*.
You just keep falling back on cherry picking stuff you have no basic understanding of and try to paint yourself as some kind of expert. Yet everything you post shows that you are even a novice in the field!
You just can’t get out of that statistical dimension you live in where everything is random, normal, and subject to statistical analysis – no matter what anyone says – even the experts!
Nor can he read and comprehend what is written, this is obvious. He just sees what he wants to see.
“I gave you the words. Their meaning is obvious.”
“Chapter 9 is *NOT* about uncertainty interval probabilities. It’s about using the stated values only. That implies that the uncertainty intervals associated with the measurements are totally random and normal so that complete cancellation occurs”
You keep accusing me of “not reading for meaning”. Maybe you should try it sometimes.
The introduction to Chapter 9 starts
Meaning he’s introducing statistical concepts in order to use them for error propagation. Error propagation means uncertainty intervals.
Section 9.1 starts by quoting the two equations for error propagation using his symbol δ to represent uncertainty. Then he points out that in Chapter 5 he showed how a good measure of the uncertainty was the standard deviation σ. Equation 9.3 is the same as equation 9.2, but using σ in place of δ. In that case he used the assumption of normality and independence to derive the equation for standard deviations in order to prove the equation involving uncertainty intervals was correct.
He then goes on to explain that in section 9.2 he will show how to use the concept of covariance to derive a formula that does not require the distributions to be independent or Gaussian.
Explicitly stating the formula will not depend on the assumptions you claim he requires.
The formula is 9.9 and immediately after it he repeats the point
Back to section 9.1, he points out how he assumed the errors were normally distributed in Chapter 5 and then says
I’m not sure how many more quotes you need before you realise I am not the one cherry-picking words whilst ignoring the meaning. The meaning is clear. The standard deviation is a measure of the random uncertainties in a measurement and the formula does not assume that the distribution of errors is normal.
You do realize that you can calculate a standard deviation for a non-normal distribution, right? Does the mean, and standard deviation allow you to determine the distribution? No, they don’t. Why do you think that an “expanded uncertainty” is calculated when you use an experimental standard deviation from a few “samples?
“You do realize that you can calculate a standard deviation for a non-normal distribution, right?”
That’s what I’ve been trying to tell you lot, every time you say the distributions must be normal. Taylor’s Chapter 9 is all about the standard deviations, not about the shape of the distribution.
“Does the mean, and standard deviation allow you to determine the distribution?”
No they do not.
“No, they don’t.”
As I said.
“Why do you think that an “expanded uncertainty” is calculated when you use an experimental standard deviation from a few “samples?”
We were not talking about expanded uncertainties, but thanks for offering up yet another distraction to cover up the mistakes in the previous distraction. Do you, or do you not accept that Chapter 9 of Taylor does not assume that all distributions are Gaussian, as Tim claims?
“Chapter 11 is about the Poisson distribution – COUNTING SITUATIONS.
Temperature measurements are not *counting*. They are *measuring*.”
Oh look, a squirrel.
Nobody mentioned chapter 11 or Poisson distributions.
“If the uncertainties are completely independent the equation is the same as for equation 10. The uncertainties add in quadrature.”
Equation 10 requires NO SYSTEMATIC UNCERTIANTY exist. How many times does that have to be explained to you?
“use the CLT, the sum tends to a normal distribution with standard deviation sqrt(N) * SD. “
And all that tells you is how close you are to the population mean. It does *NOT* tell you the accuracy (i.e. uncertainty) of the population mean!
You can’t even get that simple distinction straight in your mind!
“In all cases the distribution of the random variables”
And now we circle back to all uncertainty is random, normal, and cancels. No matter how many times you deny you assume that it just shows up EVERY SINGLE TIME!
“it can be understood from the maths or it can be demonstrated by rolling dice or on a computer. The rules assume independence they do not assume a normal distribution.”
And your math REQUIRES assuming that all uncertainty is random, normal, and cancels. You can then use the stated values for your statistical analysis.
“Equation 10 requires NO SYSTEMATIC UNCERTIANTY exist. How many times does that have to be explained to you?”
Could you point me to the section that says that?
The only reference in the GUM to SYSTEMATIC UNCERTAINTY is to say that it can be misleading and is a term that should be avoided.
Systematic errors should be corrected as much as possible, but uncertainty about that corrections is included in the overall description of the uncertainty.
“And all that tells you is how close you are to the population mean. ”
It’s the sum not the mean.
“You can’t even get that simple distinction straight in your mind!”
I wish you could get the argument straight in your mind. You keep trying to distract from the point I was making. I’m just trying to get you to say if or if not you agree with the concept that if uncertainties are independent, all sources explain how the correct thing to do is add them in quadrature. I am not saying that means you can juts do that and say “that’s the definitive uncertainty”. There may be other factors including systematic errors or uncertainty in the definition. But the general case says to add in quadrature and your constant attempt to distract from this by talking about means just makes it clear you cannot or will not answer the question.
“And now we circle back to all uncertainty is random, normal, and cancels. No matter how many times you deny you assume that it just shows up EVERY SINGLE TIME!”
And yet more deflection.
“And your math REQUIRES assuming that all uncertainty is random, normal, and cancels.”
It’s not my maths. It’s the maths detailed in Taylor et al. Like all maths it might not work in all cases and will always be an approximation of the real world. But there’s a reason why all the books describe it. The fact you seem to think it should be discarded in real world cases, and replaced with Taylor’s provisional rule, suggests to me you don’t have much confidence in the wisdom of the authors. If it don;t work in the real world, why mention it?
Taylor does not say that. Nor does Taylor say that you are free to make algebra mistakes when executing the formulas he provides. Taylor tells you exactly what to do when you see a sum (+) via 3.16, a quotient (/) via 3.18, and even a general rule via 3.47 and he expects the readers to be able to do so without making algebra mistakes.
I’ve given you the EXACT quotes from Taylor.
Here it is ONE MORE TIME:
Taylor: Section 3.5, Page 58:
“If the measurements of x and y are independent and subject only to random uncertainties, then the uncertainty ẟq in the calculated value q = x + y is the sum in quadrature or quadratic sum of the uncertainties ẟx and ẟy.” (bolding mine, tpg)
You stubbornly refuse to actually read what Taylor, Bevington, Possolo, and the GUM say.
They *all* say that measurements with systematic uncertainty are not amenable to statistical analysis.
It can’t be simpler. Only those living in statistical world instead of reality can’t seem to grasp that simple truth.
TG said: “I’ve given you the EXACT quotes from Taylor.”
You said “Taylor specifically states that adding in quadrature is only fully applicable when the uncertainty is random and Gaussian.”
I don’t see the word Gaussian in the quote you provided from Taylor.
TG said: “They *all* say that measurements with systematic uncertainty are not amenable to statistical analysis.”
That has nothing to do with the distribution.
Here is a short excerpt from Taylor’s chapter 9. He shows that the quadrature expression for the standard deviation of measurements (calculated from the basic algebraic definition, not as defined by any distribution) whether or not the measurements of x and y are independent or normally distributed. Then he demonstrates that if the measurements are independent and random (but not necessarily Gaussian) the result reduces to the simple quadrature expression. This result is for any function of the measurements, but reduces to the kind of expressions considered here for simple sums and differences.
I am unfamiliar with posting jpgs here and it looks like it will post but it may not.
Remember, this is an “Introductory” textbook. It has many applications but is not comprehensive.
Many assumptions are made.
Always provide an upper bound means there are times when this is appropriate. Taylor mentions at this at least twice.
No systematic errors, is ok for teaching but is not applicable in the real world.
Not all “random” errors result in orthogonal errors that can be reduced by quadrature.
The multiple measurements is the critical point.
First, multiple measurements in textbooks and the GUM and in the real world always require the same measurand. You simply can not take measurements of different measurands, add them in quadrature and then say they each have the same uncertainty. Would you buy a half-million dollar racing engine if the mechanic told you that he measured the crankshaft journals one time each, found the combined uncertainty of them, and ordered bearings using the average tolerance?
You can not have a distribution of random errors with single measurements such as atmospheric temperatures of Tmax and Tmin because you don’t have multiple readings. You certaintly can’t go ten miles away, take a reading and claim you now have a distribution so that you can find the uncertainty of both. You can’t even use the excuse of using the same device because you aren’t measuring the same thing.
Much of the conversation on this and some other threads has devolved into esoteric references about quantum mechanics, etc.
The real issue boils down to quoting temperature anomalies to the one-thousandths of a degree when the recorded temperatures only have units resolution. Systematic uncertainties dominant as do instrument resolution.
One must recognize these are field instruments subject to loose calibration intervals and large environmental differences. NWS/NOAA have specified rather large uncertainty values that far exceed the quoted anomaly values.
Several years ago when this subject was started the same issue was discussed and metrology texts and the GUM were brought into the discussions primarily to show how error reductions required multiple measurements of the same thing.
And I’m accused of cherry-picking.
It’s a simple point. Tim insists that quadrature only works if all distributions are Gaussian. I and fah both point out that Taylor, along with everyone else, shows that this is not the case. And you do what you keep doing and search for loopholes. None of which have anything to do with the question of adding non-Gaussian distributions.
“Always provide an upper bound means there are times when this is appropriate.”
Yes, and shows when this upper bound will be reached. Given Kip insists this is the only permissible equation for combining uncertainty, I’m not sure why you are so insistent on pointing out that it is just the upper bound.
“P 211, we measure the two quantities x and y several times.”
Context. Taylor starts that sentence with the word “suppose”.
He’s not saying you have to measure the two quantities several times. He’s showing how the equation is derived.
“P 212, “after many measurements, the covariance σ(x,y) should approach zero:””
Context. “If the measurements x and y are independent … covariance should approach zero”.
“You simply can not take measurements of different measurands, add them in quadrature and then say they each have the same uncertainty.”
They don’t need to have the same uncertainty.
“You can not have a distribution of random errors with single measurements such as atmospheric temperatures of Tmax and Tmin because you don’t have multiple readings.”
You keep confusing things. The distributions being added in quadrature are the measurement uncertainties. The assumption is you know that from previous experiments with the instruments. You don’t need to have multiple measurements to know that a single measurement comes from a distribution.
“The real issue boils down to quoting temperature anomalies to the one-thousandths of a degree when the recorded temperatures only have units resolution.”
You’re engaging in a fallacy here. You don’t like some imagined conclusion so you reject any step that might lead to it. Forget about averaging for the moment, the only question is whether you agree or not that in some circumstances adding quadrature is correct and and if those circumstance apply even when the uncertainty distributions are not Gaussian.
“Several years ago when this subject was started the same issue was discussed and metrology texts and the GUM were brought into the discussions primarily to show how error reductions required multiple measurements of the same thing.”
No. The texts and GUM were brought in because people were using them to justify a claim you never divided the uncertainty of the sum when calculating the uncertainty of the average – and hence the uncertainty of the average will increase the greater the sample size.
But again, there is nothing in any of these texts that has ever shown you can only reduce errors when multiply the same thing. That’s just something that has been asserted from the start.
“They don’t need to have the same uncertainty.”
Your reading comprehension is showing again!
“The distributions being added in quadrature are the measurement uncertainties. The assumption is you know that from previous experiments with the instruments.”
Really? You think all these temperature measuring devices around the globe get calibration experiments done on them on a continuous basis? You are still living in statistical world!
“You don’t need to have multiple measurements to know that a single measurement comes from a distribution.”
What distribution? Calibration drift is a time function. How do you get a distribution from a time function that can be used to calibrate something in the present? You are still living in statistical world!
” Forget about averaging for the moment, “
I’m not surprised you want to forget about it! It’s the main issue!
” the only question is whether you agree or not that in some circumstances adding quadrature is correct and and if those circumstance apply even when the uncertainty distributions are not Gaussian.”
I’ll be interested in seeing what Jim says. In some circumstances adding in quadrature is correct. You’ve been given Taylor’s statement on that at least THREE TIMES! The problem is that you just won’t accept what it says.
As Bevington points out, and Taylor too, repeated observations with no systematic bias are typically assumed to be a Gaussian distribution. Using quadrature is correct in those cases. When you have systematic bias all bets are off. And you can probably count on the fingers of one hand how many field temperature measurement devices have no systematic bias – and I mean globally!
Here is what I think. The discussion of uncertainty has gotten really esoteric and is really useless. I have grown tired of debating the minutia and am close to deciding that each temperature reading is a “random variable” with a standard deviation as quoted by NWS/NOAA. If it wasn’t a random variable, it would have no standard deviation. This will solve the debate about what the uncertainty should be and will basically say we will use the Type B uncertainty from NWS/NOAA.
Random variables have very defined ways to be averaged and their variances added as “fah” has outlined in a post. You can not simply take ±1°F, divide it by the square root of the number of stations to get an SEM with a one-thousandths number. “n” should be the size of the samples, i.e., 12 months, and not the number of stations. That is where the real argument begins.
“It’s a simple point. Tim insists that quadrature only works if all distributions are Gaussian”
You are going to fight it right down to the last man in the foxhole, right?
I just posted you a quote from Bevington on this subject!
“Context. Taylor starts that sentence with the word “suppose”.”
OMG! Why do you think he is using that example?
“Context. “If the measurements x and y are independent … covariance should approach zero”.”
Are measurements made with the same instrument independent?
Concur completely. The discussions here have a hard time staying on point, any point. Personally I am most at home dealing with controlled physics experiments or sometimes general notions about sets of numbers. Two words that I think should be the first ones of any comment on here would be “it depends.” And then folks go on about dice rolls or carpentry and hammers and drills sometimes. I am comfortable dealing with laboratory or observatory instruments that measure essentially the same physical quantity repeatedly, but I have long had things I am not sure of about the manipulations done with the set of global temperature measurements. It has always been tough for me to reconcile those manipulations with a notion of what the actual physical quantity is that they are ostensibly measuring. For example, what is the “true” physical value of the global average that one is attempting to calculate, and what is the theoretical prediction we are trying to compare it to? And of course the issue of the intensive nature of temperature and what physics does it reflect in the first place. To me, it does not meet the requirements for what I am used to in a physical quantity subject to measurement. I have been much more comfortable thinking of the so-called “global average temperature” more as an “average of global temperatures” or even a global temperature index. To me the quantity has more in common with things like a stock market index, consumer price index, or fertility index in which issues of “accuracy” are somewhat different and have more to do with consistency in calculation rules and comparison to previous values than considerations of physics. Small increments in indices, when the indices are consistently calculated, can allow inferences about the behavior of the populations involved, but shoe horning those things into techniques designed for assessing physical measurements seems a bit off to me. It could well be meaningful to consider very small changes in indices that are quite distinct from any notion of physical measurement accuracy. It seems to me they are two distinct conversations, both worth having.
And prior to these guys being introduced to GUM-style UA, they maintained up down and sideways that sigma/root(N) was THE uncertainty of any and all averages of air temperatures.
Then they proceeded to find loopholes around uncertainty always increasing, and they found the perfect loophole in GUM Eq. 10.
Success! Who cares about reality, mission accomplished.
And now they pawn themselves off as UA experts.
You are making the same mistake Bellman does. In Taylor, everything after Chapter 3 assumes the individual measurements have small, random-only, uncertainties. And that these cancel and the stated values of multiple measurements can be used to determine an uncertainty value and mean. Random-only implies no systematic bias in any measurement.
That is simply not a valid assumption for single temperatures made by different measuring devices. But it is the assumption made by bellmen in every assertion.
Read Chptrs 3 and 9 for meaning. I can provide you quotes backing this up if you need them.
If you read Chapter 9.2 you will find Taylor discusses standard deviation as the uncertainty. The standard deviation is of the stated values of the measurements. The unwritten assumption is that there is no systematic bias in any measurement.
Taylor, Chapter 9.1, page 210.
“As in Chapter 5, I will suppose that all systematic errors have been identified and reduced to a negligible level, so that all remaining errors are random.”
Taylor, Bevington, Possolo, and the GUM state that the presence of systematic bias is *NOT* amenable to statistical analysis.
Yet all the statisticians on here want to argue that temperature measurements have no systematic biases and can therefore be studied using statistical analysis of the stated values – after first assuming that all errors are random and cancel!
Statistical world is not congruent with real world.
“Taylor, Bevington, Possolo, and the GUM state that the presence of systematic bias is *NOT* amenable to statistical analysis.”
Which is why you don’t include it in your analysis. If you know there is systematic bias you remove it, but you still have to include the uncertainty about the correction in the uncertainty analysis.
“Yet all the statisticians on here want to argue that temperature measurements have no systematic biases and can therefore be studied using statistical analysis of the stated values”
Nobody, certainly not I, have argued such a thing. There are lot’s of systematic biases that are corrected for, and may well be others unknown. All I’ve argued is that it makes no sense just to assume there is a 0.5°C systematic error that is present in all stations with no explanation as to what that error is. (And to ignore the fact that if it did exist in all readings it would cancel out when looking at the trend or taking anomalies.)
I’ve argued several times that if there is a systematic bias that maters it’s something that is changing over time, and you only have to compare UAH and other trends to suspect there must be some systematic bias somewhere.
But as always you are trying to distract with specific issues, whilst refusing to accept the general case, which is the simple question of how in general to propagate uncertainties. Whether you can only do so by direct addition as Kip claims, or in appropriate case using quadrature as statisticians and metrologists claim.
“Statistical world is not congruent with real world.”
The more I hear about your particular real world the less I like it.
Fanning from the hip yet again.
“Which is why you don’t include it in your analysis. If you know there is systematic bias you remove it, but you still have to include the uncertainty about the correction in the uncertainty analysis.”
How in Pete’s name do you remove something that is unknown!!
I asked you before what the systematic bias is for the temperature measuring device at Topeka Forbes Airfield. You didn’t know. How then can you expect for it to be removed?
total uncertainty = random uncertainty + systematic uncertainty
You typically can’t define either one in an untended, uncalibrated field measurement device. At least no one I know can. Are you God that you can?
“Nobody, certainly not I, have argued such a thing.”
Bullshite! That’s your entire argument!
“There are lot’s of systematic biases that are corrected for, and may well be others unknown.”
Again, what systematic bias is known and corrected for in field temperature measuring devices?
“All I’ve argued is that it makes no sense just to assume there is a 0.5°C systematic error that is present in all stations with no explanation as to what that error is”
How much random error do you expect from an automated, electronic modern temperature measuring device? It’s not a human – tall/short, near/far sighted, old/young, etc – reading that device every few seconds or minute. What do you think the major source of uncertainty is in such a device?
Why don’t you get out of statistical world and join the rest of us in the real world someday?
“I’ve argued several times that if there is a systematic bias that maters it’s something that is changing over time, and you only have to compare UAH and other trends to suspect there must be some systematic bias somewhere.”
Why then do you argue that all uncertainty is random and normal? Why do you argue that addition in quadrature is perfectly fine in the face of systematic bias? Why do you argue that the GUM does *not* base most of its work on repeated measurement of the same thing?
Just like saying you don’t argue that all measurement error is random and normal but put the lie to it with what you post, the same thing applies here. You say out of one side of your mouth that you can remove systematic bias in measurement and also claim you can identify systematic bias by comparing trends when you *should* know from all the quotes you’ve been given that identifying systematic bias is difficult at best and impossible at worst. Who says: “Errors of this type are not easy to detect and not easily studied by statistical analysis”? You’ve been given the quote several times? Did it stick with you at all?
UHI is a systematic bias. How do you identify it and correct for it?
“How in Pete’s name do you remove something that is unknown!!”
You need to decide what systematic errors you are considering. It’s a complicated issue, and as the GUM says the distinction between systematic and random errors isn’t always clear cut.
Trying to think thought he problem it seems there are three possibilities, some or all of which might apply to a given situation.
Think about what you are saying in terms of the temperature database.
Similar devices probably drift in the same direction. If they read higher than they should that is a systematic bias.
Where it really comes into play is when you might have a systematic error of say 0.1 degree yet quote anomalies to a 0.001 resolution?
Absolutely correct (except perhaps for the part about “making the same mistake as Bellman”). Systematic errors in physical measurements are an always present nettlesome problem. I think Taylor has a very straightforward and practical discussion in his Section 4.6 with particular emphasis on what it means for physics students. He does generally explicitly state when he is dealing with random variables and he typically just says such quantities are “random” and if the additional information that they are distributed in some distribution, he generally explicitly states that is the case. And his conclusion, like most experimentalists is that every experiment (i.e. repeated measurement of the same quantity) will inevitably have systematic and random contributions to the results. These are often called the accuracy and precision, respectively, as I am sure you are aware.
The major saving grace with controlled experiments with repeated measurements of the same quantity is one can repeat it enough that you can just look at the data and see if some pattern is emerging. Sometimes experiments are designed expecting to get a zero result, sometimes expecting to get a non-zero value, and in physics, these values are generally predicted by competing theories. I have written a few matlab apps for my students in which they repeatedly measure things like sizes of spores, or distances between objects, etc. by moving crosshairs across the images in different orientations a number of times. If they do it enough times, and carefully, their values invariably do miraculously become very gaussian looking the more they measure.
When they do experiments, it is often the case that they get results that are very normally distributed about some value, but the value they get is sometimes a little far from what is predicted by theory. The analysis they next do is just what Taylor discusses in his Section 4.6. First, does the random (often normal) spread of the results about the mean (it might be from fitting or some other process, but its the same thing) accord with a propagation of errors of the guess for the random errors from what is known about the components of the apparatus. Some statistics tools are useful here to indicate whether there is reason to think the experiment performed as well as you thought it would, or if there is some reason to doubt that it did. Most of the time careful students will get random errors quite close to the estimates.
The next thing students do (and I am describing this because it is exactly what working physicists do in “real” experiments) is compare the result to 1) what a theory predicts and/or 2) a “true” value obtained either from a separate referee instrument or, often, the known true value from big lab measurements (e.g charge of the electron, acceleration of gravity etc.). Usually there is some difference between the students result and the comparison “true” or predicted values. Their first task is to see if the random component is such that they would be very unlikely to be as far away from the “true” value as they are. This is fairly simple if the random components are nicely normal, but even if they are not one can do a little empirical estimation. If the difference is unlikely to have happened by chance, then the students turn to looking for possible sources of systematic error. The usual approach is to examine the effects of small changes in the different parts of the apparatus that one thought was “controlled” to see what magnitudes and directions those changes would have in the measured values and then see if those changes could have escaped notice or involved some quirky part of the experiment that was problematic anyway. Sometimes there is time to repeat measurements with additional instruments or techniques to tie down the suspected sources and see if it makes a difference, more often not. Sometimes detailed examination of the measurement device can tease out the source. The Lin and Hubbard analysis I linked to somewhere here on the details of the ASOS temperature measuring device is a good example of that kind of thing. They teased out the source and behavior of the bias many workers have noted that apparently has to do with a small variation in background instrumental temperatures and its effect on the current through a specific resistor due to time of day and insolation. The NIST paper I also linked to somewhere here on temperature measurements of surfaces is another good example of that kind of thing.
But the bottom line is that dealing with systematic errors is often not easy and does not lend itself to simple prescriptive approaches. There is the interesting famous case of Millikan’s oil drop experiment in which he measured the charge of the electron, reported the value and got a Nobel prize. It turns out his experiment had a systematic error (he did not accurately measure the viscosity of the air in the chamber) and his result was in fact wrong by a fair amount. The funny part is that people repeated his measurements in various ways and kept coming up with slightly different results, but Millian’s results were so authoritative that the other groups fiddled around looking for systematic errors in their experiments, trying to show how they themselves got it a little wrong (since they did not agree with Millikan) and moving their results closer to his. It took a number of years for people to finally sheepishly admit that, well yes Millikan’s results were wrong due to a systematic error he had. There are lots of other cases where this kind of thing happened such as with gravitational light deflection or detection of gravitational waves and other things. One of the best ways of dealing with systematic errors is to be able to repeat the measurement in question with a completely different experiment, based on completely different technology. An example here would be measuring the deflection of light by the sun with microwave telescopes rather than optical telescopes, but there are lots of cases where this has been applied.
So systematic errors are a different thing from random errors, agree wholeheartedly, but both have ways to deal with them. The main thing I think is what Feynman kept saying, that the first rule is not to fool yourself, and you are the easiest person to fool. An open mind, respect for thoughts of others, and a readiness to find errors in one’s own way of thinking will in the end make progress.
All that said, an interesting thing to think about is how much any analysis suitable for repeated measurements could or should apply to climatological measurements, which I think in many ways, are one time observations that intrinsically can not be repeated. But that is a longer conversation which I hope to see people make because I have a lot of ignorance about it. As I said somewhere here before, I have had some thoughts on the utility of averages of temperatures from different times and places and generating anomalies and the like. I am not at all sure such numbers should be approached and used in the same ways as physical measurements, but perhaps should be simply considered indexes, in the manner of stock exchange or fertility indexes and different things should be done with them than trying to link to global thermodynamic quantities. In an index the useful utility of the number (and its precision of expression) is not directly related to the variations of the individual contributing numbers. But that is a longer conversation I think.
Those indices tend to be derived from large samples of discrete values. Does that have an effect on the treatment?
Yes and no. Some, like the fertility index are constructed from discrete numbers, namely the number of children had by women during specific time intervals, but others like the stock indexes are obtained from sets of numbers which often have several decimal places or more. It is more that indices of populations like that (distinct from other things called indices such as index of refraction or some such, which are repeatable physical measurements) and are meant to indicates something about the population at large and usually are not oriented to measurement of physical quantities as we typical treat in physic measurements. A change at a several decimal place level in a fertility (or stock) index for a population of 1 million women would reflect a large and noticeable effect in the dynamic of the population and might be examined for issues relating to nutrition, availability of health care, presence of conflict, etc. In a certain sense the accuracty of the individual members of the set of numbers is largely irrelevant. Couple that with the notion that often what is of interest is the change in the index and it is just different animal from repeated physical measurements. That is part of why I have been somewhat hesitant to get into the precision and accuracy of the average of global temperatures in the same way as the PA of individual measurements.
BTW, I apologize if I am tardy in replying to comments here, I am still not very woke with this new commenting system. With the old one I used to get notifications, but I had not been aware how to do that here. I think however I just noticed a little notification widget at the bottom of the comments that I can click such that I might not have to just look around for replies to my comments.
Thank you.
Excellent post.
The problem with climatological measurements is their resolution. Climate science takes records from 1910 which were recorded as integers and through averaging and subtraction end up with anomaly values to at least one-hundredth of a degree and sometimes to the one-thousandth of a degree.
Take a look at charts from some of the NOAA web sites where they show anomalies and you can see this. To many of us engineers and physical scientists this is fantasy to the ultimate.
I don’t know about your students, but when I went to school, the lab instructors would fail your experiments if you did not deal with resolution and significant digits in a proper fashion. Subtracting a “baseline” temperature with 3 decimal places from an integer temperature recorded in 1910 and ending up with an anomaly containing 3 decimal places is not only a joke, it should be considered fantasy fiction.
This is where and why all this began. The original NWS uncertainty interval of ±1°F which should tell folks not to quote temperatures beyond units digits. Even with the latest CRN stations the uncertainty interval quoted by NOAA is ±0.3°F which should tell one to only quote tenths of a degree. Quoting a temperature anomaly like 0.025 ±0.3°F is beyond ridiculous.
Chapter 9 in Taylor assumes all measurement uncertainty is random and Gaussian, that it cancels, that there is no systematic uncertainty, and the uncertainty of the stated values can be analyzed statistically.
That is *NOT* the case for temperatures used to create a global temperature average.
Do NONE of the statisticians on here wonder why in Chapter 9 Taylor gives *NO* measurements as “stated value +/- measurement uncertainty”?
For instance, see Table 9.1! Not a single measurement is given as anything except a stated value! Table 9.2 is the same!
This is a MAJOR miss in all statistics textbooks I have purchased. Not a single one ever offers up analysis of anything except stated values. The implicit assumption, never actually stated, is that measurement uncertainty is always totally random, Gaussian, and cancels with no systematic uncertainty.
It’s no wonder that the statisticians on here can’t conceive of how to handle single measurements of multiple different things using different devices.
A single measurement of a single thing provides no set of measurements which can be analyzed to get a standard deviation or mean. All you can state is an uncertainty interval for that single measurement. Combining that single measurement with a measurement uncertainty with a single measurement of something different is like trying to combine single measurements of the heights of a Shetland Pony and an Arabian stallion and saying the average value of the height of a horse is the average of the two measurements. Not only does that ignore the uncertainties associated with each measurement, you are trying to jam two different populations together and then analyze them statistically!
It’s the same for temperature. Jamming temperatures from Canada in with temperatures from Brazil, all from the same month is like jamming the heights of Shetland ponies and Arabian stallions together in order to get the average height of a horse.
I’m not amazed the statisticians on here can’t make this distinction. Most apparently have very little real world experience with metrology, especially from scattered measuring instruments with unknown accuracy and environments.
That is one of the blind spots of most statistics courses, in that they tend to concentrate on discrete values. It may well be that most general statistics work does indeed concentrate on areas where discrete values dominate. That’s certainly the case with Econometrics.
Metrology is a specialised field with a large overlap with general statistics, and the added complication of working with continuous values instead of discrete. That tends to be more of a factor in Engineering and the physical sciences.
I beg to differ on that. Measurement uncertainty doesn’t come into play with discrete values, but something which was banged into our heads was the rather critical effects of distributions, and to not assumes Gaussian.
Survey design, sampling techniques and determining if a sample belongs to the population of interest are more of factors.
Great comment!
“Chapter 9 in Taylor assumes all measurement uncertainty is random and Gaussian, that it cancels, that there is no systematic uncertainty, and the uncertainty of the stated values can be analyzed statistically.”
How many times does Taylor have to explain that the equation in chapter 9 does not assume a normal distribution before you accept it? A rhetorical question as we know it’s impossible for you to accept anything that runs counter to your presumptions.
“For instance, see Table 9.1! Not a single measurement is given as anything except a stated value! Table 9.2 is the same!”
The point of the tables is to estimate the standard deviation and covariance. When you apply these figures to 9.9 you are not using just the stated values, you are assuming there is uncertainty int he measurements.
Well stated, Kip, thanks.
“-1” — word salad blob is in da house! Heh
The “reference” is about dividing the total by a constant, not by n. Again Kip shows fundamental misconceptions.
If you have a list of temperatures to average, you do not even need to look at the uncertainty on each value (provided they are all similar; if not you would have to look into doing a “weighted mean”).
In this case we are considering only the rounding error, which is the same for each measurement.
That being the case, the average is the arithmetic mean, and the uncertainty is a multiple of the observed standard deviation. The individual uncertainties are not even included in this calculation.
The act of rounding the individual temperatures simply increases the observed standard deviation. The extra variance caused by rounding is already included in the observed standard deviation. It would be incorrect to add extra uncertainty caused by rounding, because it has already been included. It has already caused the observed standard deviation to be larger than it would have been without the rounding.
And yes the observed standard deviation (i.e the measure of uncertainty) can in principle be smaller than the uncertainty on an individual result. It does not need to be, but it can be. The more numbers that are averaged the more likely this is to occur.
Here again the CO2 global warming zealots toss the instrumental temperature uncertainty into the trash: its all random, it all cancels.
“Nothing to see here, folks! Move along!”
Again you misrepresent what I am trying to explain. I am isolating the effect of rounding on a list of temperatures. It is but one component of the overall uncertainty of the final result.
If I were to include the instrumental uncertainty, I would do it at a later stage. If all the measurements were done on the same instrument and under the same calibration, it would be an individual uncertainty component that I would add in quadrature to the to the observed relative standard deviation.
No problem with doing that at all.
Uncertainty needs to be considered at the very start. That’s why measurements are written “stated value +/- uncertainty.
“If all the measurements were done on the same instrument and under the same calibration,”
This whole blog post is how to handle uncertainty in relation to the global average temperature. *NOT* the same instrument and *NOT* the same calibration.
“And yes the observed standard deviation”
How do you observe the standard deviation of something you can’t see?
All of the sources on how to handle uncertainty state that measurements with systematic bias cannot be analyzed using statistical tools.
You are the same as all the other statisticians trying to justify the global average temperature having uncertainty in the hundredths digit. You assume all uncertainty is random and Gaussian and, therefore, cancels. You can then use the stated values, i.e. the stated values of the temperature to determine an uncertainty value by substituting the standard deviation of the stated values.
You simply cannot do that with field instruments that have systematic biases due to calibration drift. It’s why Hubbard and Lin found in 2002 that you can’t adjust for calibration drift by using surrounding measuring stations. There is too much variation in microclimate environments to identify systematic bias in that manner. You have to apply corrections for systematic bias on a station-by-station basis and applying those corrections over past time is questionable because you don’t know the drift curve over time, only the current drift value.
Bottom line: Statistical analysis of stated temperature values simply doesn’t work the way most assume it does!
Kip Hansen said: “The significance of this is the average of many added rounded values (say Temperatures Values, rounded to whole degrees, to be X° +/- 0.5°C) will carry the same original measurement uncertainty value. “
Patently False. This is a direct and blatant contradiction to what 100 years of mathematics, NIST, UKAS, ISO, JCGM, Bevington, Taylor, and others.
You can call all of those groups/individuals detractors all you want. It does not change the fact from the law of propagation of uncertainty (wiki or JCGM 100:2008 equation E.3 and the like) that when you average a sample of uncorrelated measurements the uncertainty of the average will be less than the uncertainty of the individual measurements themselves. That is an indisputable fact.
And the only ones supporting you here are those that either don’t bother doing the mathematics (or even just using the NIST uncertainty machine) or those that continually make algebra mistakes some so trivial that even middle schoolers would be able to quickly identify. And some of the arguments being made are so absurd that they confuse sums (+) with quotients (/) or challenge the Σ[x_i, 1, N] / N definition of an arithmetic mean.
Kip Hansen said: “But it runs contrary to current statistical beliefs and practices. And, as we can see in all the above, is fought claw-tooth-and-nail.”
Your hubris almost defies credulity.
He claims to have left the set – twice. “I can’t work like this. I’ll be in my trailer”*.
Stolen from Charlie Brennan joking around on channel 9, Donnybrook.
https://www.ninepbs.org/donnybrook/
“It does not change the fact from the law of propagation of uncertainty (wiki or JCGM 100:2008 equation E.3 and the like) that when you average a sample of uncorrelated measurements the uncertainty of the average will be less than the uncertainty of the individual measurements themselves. That is an indisputable fact.”
All you are doing is showing off your ignorance.
All of these require multiple measurements of the same thing creating a set of stated values as a distribution by assuming all measurement uncertainty is random, normal, and cancels and that no systematic uncertainty exists.
That is hardly a set of real world measurements from a set of widely separated individual measurement devices of unknown calibration.
All you are stating here is that the sample means get closer to the population mean as you increase the number of samples you are averaging.
That has ABSOLUTELY NOTHING to do with the accuracy of that population mean. You can calculate the population mean very, very closely with a set of large sample sizes but the accuracy of that population mean may be very, very bad if a lot of systematic bias exists in the data and you ignore it.
For some reason you simply cannot seem to make that connection in your mind. You are stuck in a statistics dimension where systematic uncertainty doesn’t exist and where all measurement uncertainty cancels.
Thank the heavens you don’t design things for the public that could kill or injure them!
He’d find himself in court PDQ.
TG said: “All of these require multiple measurements of the same thing”
And as you get told repeatedly the GUM makes no such mandate and even provides numerous examples where E.3 and derivations 10 and 13 are applied to measurements of different things.
Burn this into your brain…the GUM has numerous examples where equation 10 is used on measurement models where the inputs are measurements of different things.
The next time you feel the desire to claim that the GUM method only works on measurements of the same thing remember the examples that contradict your claim.
TG said: “For some reason you simply cannot seem to make that connection in your mind.”
The reason isn’t mysterious. The reason is because the GUM applies its own rules to measurement models incorporating inputs of measurements of different things.
Section E deals with A MEASURAND. That is a single value calculated by using multple measurements of of various components that make up the measurand.
“””””E.2.1 When the value of A MEASURAND is reported, the best estimate of its value and the best evaluation of the uncertainty of that estimate must be given, for if the uncertainty is to err, it is not normally possible to decide in which direction it should err “safely”.””””” (CAPS by me)
Read this and wrap your head around the fact that a random variable is a single measurand with INDEPENDENT REPEATED OBSERVATIONS. IT IS NOT,REPEATED OBSERVATIONS OF DIFFERENT THINGS.
“””””E.3.4 Consider the following example: z depends on only one input quantity w, z = f(w), where w is estimated by averaging n values wk of w; these n values are obtained from n independent repeated observations qk of a random variable q; and wk and qk are related by “”””””
I already have burned that content in my brain. None of that says E.3 and derivations 10 and 13 cannot be used on a measurement model incorporating multiple measurands. Nor does it invalidate the GUM’s use of these equations and examples of doing just that.
Let me ask you this…do you think the GUM’s examples of measurement models incorporating multiple measurands have been incorrectly added to the document?
“”””” … when you average a sample of uncorrelated measurements the uncertainty of the average will be less than the uncertainty of the individual measurements themselves. That is an indisputable fact.””””
You betray your lack of interest in learning. You have obviously scanned a number books to cherry pick stuff about uncertainty but failed to decipher that what you end up with is a random variable that includes a variance.
Let’s look at two temps, Tmax and Tmin. These are not unique numbers that are exact so you can just take an average of the two. They are random variables with a mean, variance, and standard deviation. You need to learn how to average two or more random variables means and variances. Then you move into monthly averages of random variables from the days. That gets even more complicated as covariance suddenly rear its head.
I’m so glad you brought up your beloved tide gauge measurements, as they are a perfect example of why this article is misleading. In your first article, you get your errors confused (a simple mistake), the absolute measurement uncertainty of the tide gauge would be the resolution of the instrument itself (a aquatrak transducer) which is 1mm, so actually if you took a single measurement the error is +/- 0.001m and then if you averaged over say a 1000 measurements its error is still +/- 0.001m… Now if you’ve ever been to the sea, you’ll quickly realise that basing your uncertainty on how well you think you know the height of the water rather than the random variation in the height of the water is really dumb.
The NOAA know this too and thats why they use standard deviation, because it reflects the range of your values better. Absolute uncertainty is the uncertainty of your instrument, if your data fluctuates randomly, you wouldn’t give a toss about how accurate your thermometer is if your random deviation is going to be bigger.
Also on the same note you definitely wouldn’t use absolute uncertainty on the height of the water either (as in the range of lowest tide to highest) as a single boat near the tide gauge is gonna up your whole data set and that would be incredibly dumb thing to do where as standard deviation is designed to account for events like that.
KB EPILOGUE
(1) If there are any students reading this, for heavens sake DO NOT think Kip Hansen or any of his acolytes are teaching you anything here. If you use these ideas in your exams, you will likely be failed.
(2) To fellow climate sceptics, for heavens sake DO NOT use the Kip Hansen ideas on here to back up any position you might have, in discussions with climate alarmists. You will be scorned and laughed at. It will certainly not do the cause any good. Quite the reverse.
(3) I am not saying that data processing and statistics should not be questioned. Not at all. In fact it is vital that this area is reviewed and critiqued. But that requires someone who is speaking from a position of deep knowledge of the subject. To criticise something, you need to know what you are criticising. It’s patently obvious by now that Kip and his acolytes do not.
I’ll echo this. If you want to know how to work with uncertainty then
1) Read JCGM 100:2008 and JCGM 6:2020.
2) Read An Introduction to Error Analysis by Taylor.
3) Read Data Reduction and Error Analysis by Bevington.
4) Read The Expression of Uncertainty and Confidence in Measurement by UKAS.
5) Read NIST TN 1297.
6) Use a computer algebra system (like MATLAB, Mathematica, symbolab.com, etc) to assist with the calculations.
7) Or use the NIST Uncertainty Machine which will do the calculations for you.
Nice set of things, but you forgot the most important step: THINK about what you want to do.
Long ago I had the occasion to visit some folks in the UK who were doing some fairly large scale calculations of a specific kind of problem which we were also doing in the US. At the time, we in the US had a tremendous advantage over the UK in raw computer power and we were frankly a bit cocky about our capabilities versus the UK. One Brit was presenting some results of a particularly difficult calculation and demonstrating remarkably impressive results. We Americans were surprised and taken aback by this and someone asked how they could get such results with the relatively weak computational power they had. The Englishman replied, “Well, before we do a calculation we think about it a bit first.” It was a fairly humbling, but educational, experience.
Understatement is an art form
I can echo that with another example. It used to be the case that American numerical weather prediction models were the best in the world. But because of our not-invented-here attitude we shunned techniques proven to improve forecasting skill and reduce forecasting uncertainty. As a result the GFS now ranks 3rd behind UKMET and ECMWF. If it weren’t for the Europeans we would have had 2 days less lead time on Sandy’s capture by the trough and dramatic left turn into the CONUS. And 10 years later we’re still in the same situation.
Thanks for the links. I’m not commenting in this thread again, because I think it would be more productive to bang my head against a brick wall repeatedly.
Anyhow thanks again.
Yeah! Read Taylor!
“If the measurements x and y are independent and SUBJECT ONLY TO RANDOM UNCERTAINTIES, then the uncertainty ẟq in the calculated value q = x + y is the sum in quadrature or quadratic sum of the uncertainties ẟx and ẟy.” (bolding and caps mine, tpg)
You and your compatriots just simply can’t accept the fact that not all uncertainties are subject only to random uncertainties – especially when using field measurement devices of unknown calibration and environment.
This restriction by Taylor is echoed in every single reference you provide.
I’ve given you the quote from Bevington as well.
“The accuracy of an experiment, as we have defined it, is generally dependent on how well we can control or compensate for systematic errors, errors that will make our results different from the “true” values with reproducible discrepancies. Errors of this type are not easy to detect and not easily studied by statistical analysis”
The GUM gives exactly the same restriction.
For some reason you and your compatriots want to live in statistics world where all measurement uncertainty is random, Gaussian, and cancels and there is no systematic uncertainty in any physical measrement.
I’m glad you don’t design any bridge or auto that I use in the real world!
I did read Taylor. I’m not challenging anything he said.
I think we all agree that systematic error would change the error assessments of both averages and trends. What Mr. Gorman spaces on is the fact that those changes wouldn’t materially affect the evaluations of either the GAT or sea level averages or (especially) trends under constant discussion here.
I’ll paraphrase myself on the functionally impossible conditions under which they would make a difference. For trending, those errors (1) would have to be much larger than those even the biggest feces throwers here fling, (2) they would also require an even, constant change in those errors for the life of the evaluation. So, since 30 years seems to be a fairly good, basic, ROT, for the minimum physical/statistical evaluation of either GAT or sea level trends, this Big Foot “systematic error” whining is utterly inconsequential.
Many sources of this “systematic error” have been noted and attempts made to quantify them here. But no one seems willing to provide a the statistically/physically significant data – with both systematic and random error estimates – that would withstand scrutiny. And since this is their claim, they should back it up, as is customary above ground.
This is pure BS.I failed my first EE lab along with 7 other students because we followed the tactic you, bdgwx, and bellman champion.
We each had to build an amplifier circuit and measure its characteristics. We all though we would be smart and average all the results together to get the *right* answer. We all turned in our lab sheets with the same values for all characteristics.
WE FAILED THE LAB.
That is what you and your buddies are advocating. Just average all the stated values and you’ll get the right answer because all measurement uncertainty is random, Gaussian, and cancels!
The correct answer would have been to propagate the uncertainty for each amplifier and the instruments at each work station based on the tolerances of all the components used and instruments used.
If you don’t teach your students this then when they get in the real world, especially as an engineer, they could easily be endangering people’s lives.
It’s truly sad that there are people on here that live exclusively in statistics world where measurement uncertainty all cancels.
Hey, don’t blame me for your educational failings. I’ve never advocated cheating in exams.
I thought it might be helpful to post a simple algebraic path to the standard deviation of a sum of random variables. There has been a lot of discussion of this, particularly with the “sum in quadrature” thing. This can be derived lots of fancy ways with what Feynman would call fancy-shmancy math (characteristic functions, convolutions, etc), but simple algebra will do.
For two random variables X and Y, where random variable here just means that one can define the domain of all the possible values, and assign intervals of those values (or values themselves if they are discrete) to the interval [0,1], which means it might be some named distribution or it could just be any set of numbers that met these requirements (in particular it does not need to be gaussian or anything like that), then the variance is derived in the first set of equations, and the standard deviation, defined as the square root of the variance is given in the second set of equations. If the covariance is zero (typically described as independent) the simple quadrature formula holds. The variables do not have to obey any particular probability distribution they simply need to be random variables.
It looks like algebra but it is not. You treat the expectation operator E as if it were a variable. This is correct (in this case) but you need a lot more than algebra to understand that you can do this.
Agree, but the folks here seem to like to see algebra, plus I think they are mostly interested in sets of values that are relatively small and finite. I think as long as we restrict to small finite, at worst countable, sets of values the expectation can just be defined as a weighted sum of the values, weighted by the empirically obtained frequency (probability) of occurrence of that value in the set. I was thinking the unweighted mean and variance would just be finite sets of values for the cases the folks are interested in here. I wasn’t thinking to include any pathological or larger sets of values, so everything is happy in algebra-land. No need going where people see dragons. No?
Fair enough. If E is the sample mean (rather than the population expectation) and Var is the sample variance (rather than the population variance), then it is just a matter of rearranging a quadratic equation.
Just a comment lest people misunderstand.
σ^2(X+Y) = σ^2(X) + σ^2(Y). That is, [Var(X+Y) = Var(X) + Var(Y)]
σ(X+Y) = √[σ^2(X+Y)]
And, the sum of values must be done using the probability of each element in the random variable. Such that for daily temps:
(0.5)Tmax + (0.5)Tmin = Tavg. where 0.5 is the probability of each.
If ±1°F is taken as the standard deviation, we have:
σ^2 = 1^2 + 1^2 = 2. and
σ = √2 = 1.41
Why is it so difficult to follow a simple equation?
“σ(X+Y) = √[σ^2(X+Y)]”
Correct.
“And, the sum of values must be done using the probability of each element in the random variable.”
Not sure what you mean by that. The equation already embodies the idea of adding all possible values of each variable in the correct probability.
“(0.5)Tmax + (0.5)Tmin = Tavg. where 0.5 is the probability of each.”
And now your argument goes completely off the rails. What do you mean “the probability of Tmax and Tmin? You are not randomly selecting one or the other, you are treating each as a random variable and taking the average of the two.
But the equation is fine, just stating what the average is.
“If ±1°F is taken as the standard deviation,”
To be clear that’s looking at the measurement uncertainty? So 1°F is the standard measurement uncertainty.
“σ^2 = 1^2 + 1^2 = 2”
That’s given you the variance of Tmax + Tmin, not (0.5)Tmax + (0.5)Tmin. As I keep trying to tell you, when you scale a random variable by a constant c you have to scale the variable by the square of the c. So what you want is
σ^2(Tavg) = σ^2(Tmax) / 4 + σ^2(Tmin) / 4 = (1 + 1) / 4 = 1/2
So
σ(Tavg) = 1 / √2 = 0.71
which is
σ(Tavg) = [√(σ^2(Tmax) + σ^2(Tmin))] / 2
Nobody is challenging that. What we are challenging is your assertion that:
σ^2(X/2+Y/2) = σ^2(X) + σ^2(Y).
It’s actually:
σ^2(X/2+Y/2) = σ^2(X/2) + σ^2(Y/2)
Where:
σ^2(X/2) = σ^2(X) / 2 and σ^2(Y/2) = σ^2(Y) / 2.
Thus:
σ^2(X/2+Y/2) = (σ^2(X) + σ^2(Y)) / 2.
Resulting in:
σ(X/2+Y/2) = (σ(X) + σ(Y)) / sqrt(2)
Doh. I butchered that.
It should have read…
Where:
σ^2(X/2) = σ^2(X) / 4 and σ^2(Y/2) = σ^2(Y) / 4.
Thus:
σ^2(X/2+Y/2) = (σ^2(X) + σ^2(Y)) / 4.
Resulting in:
σ(X/2+Y/2) = sqrt[σ(X)^2 + σ(Y)^2] / sqrt(4)
Did you not read “fah’s” post? He even showed a proof that is all over the internet in university classes and notes. Not a one scales the variances.
Var(X+Y) = Var(X) + Var(Y) + 2cov(x,y)
=> Var(X+Y) = Var(X) + Var(Y) (when cov(X,Y) = 0)
I don’t see a scaling factor here anywhere. Dig out your statistics book fellows. Not one example I have examined shows scaling variances when the random variables are added.
I suspect you have seen multiplying/dividing by a weighting factor in order properly add random variables with different sizes. We don’t have that here.
“Not one example I have examined shows scaling variances when the random variables are added.”
It’s true most examples on the web only give the simple case, here’s a references you wanted me to read.
https://apcentral.collegeboard.org/courses/ap-statistics/classroom-resources/why-variances-add-and-why-it-matters
Look at the section on the Central Limit Theorem. It starts by treating the mean as the sum of values divided by N, so the variance of the mean is the variance of the sum divided by N^2.
And here’s another one
https://chris-said.io/2019/05/18/variance_after_scaling_and_summing/
See “Part 2: Weighted sums of uncorrelated random variables: Applications to machine learning and scientific meta-analysis”
But really it should be obvious that any scaling of a random variable by a constant must result in the variance being squared by the square of the constant. It just follows from the definition of variance. Variance is just the average of the squares of the differences between all points and the variables expected value. What happens to those differences if you scale all points by the same constant?
You have done nothing to explain WHY you need to scale. If you read closely, scaling is done to assign PROBABILITIES to different variables. Scaling is not done just to change numbers. You need to explain WHY your use of scaling is appropriate for combining temperature random variables.
Here is a section from your second reference.
“””””To gain some intuition for this rule, it’s helpful to think about outliers. We know that outliers have a huge effect on variance. That’s because the formula used to compute variance, , squares all the deviations, and so we get really big variances when we square large deviations. With that as background, let’s think about what happens if we scale our data by 2. The outliers will spread out twice as far, which means they will have even more than twice as much impact on the variance. Similarly, if we multiply our data by 0.5, we will squash the most “damaging” part of the outliers, and so we will reduce our variance by more than a factor of two.”””””
There are no outliers in climate data, so why scale?
I have your 1st reference in my list of statistical references. Notice where it says:
,””””””Because rolls of the dice are independent, we can apply the Pythagorean theorem to find the variance of the total, and that gives us the standard deviation.”””””
There is no scaling/weighting done here either.
You have failed to show ANY reason for scaling! The only reason I can see is that it can cause a reduction in the total variance. You are cherry picking again without understanding.
Read this site for a good explanation of probability weighting and scale factor. Notice that the probabilities must add to 1.
http://www.stat.yale.edu/Courses/1997-98/101/rvmnvar.htm
The reason we are talking about scaling is because you are talking about an average. An average is a sum scaled by the number of measurements.
This has nothing to do with probability. You are confusing this with a mixing function.
As usual none of your quotes mean what you want them to mean.
Look at your last reference under the section properties of variance.
Var(a + bX) = b^2 Var(X)
You didn’t answer the question I asked.
“””””You need to explain WHY your use of scaling is appropriate for combining temperature random variables.”””””
If you can’t give a reason, then you have no basis for claiming that scaling is appropriate.
Your reference to Var(a+bx) = b^2Var(X) lacks context.
The reference says:
“””””If a random variable X is adjusted by multiplying by the value b … .”
Exactly what are you adjusting and why?
If you would not just cherry pick you would see that the adjustment in the example is changing the values of the components of the random variable, i.e., changing the temperatures to give a new mean value.
In other words, “if the random variable X is adjusted” doesn’t mean just adjusting the variance, you must also adjust the values that make up the mean μ.
The mean of the values of a random variable is:
μ(x) = x1p1 + x2p2 + … + x(k)p(k)
A simple mean is:
μ(x) = [x1 + x2 + … + x(n)]/ n
From the reference:
The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
σ(x)^2 = Σ[x(i) – μ(x)]^2p(i)
You can’t change the value of μ(x) without changing the individual x(i) values that make up the mean. The “b” is applied to the x(i).
I forgot to add that the equation you quoted gives this away.
Var(a + bx) = b^2Var(X)
“bx” IS MULTIPLYING each individual term, x(i), by “b”. Var(X) is the variance in the mean of the random variable.
Trying to teach you in this blog is growing tiring because you won’t take the time to study and learn.
“Trying to teach you in this blog is growing tiring because you won’t take the time to study and learn.”
Or maybe it’s because you are trying to teach something you don’t understand.
““bx” IS MULTIPLYING each individual term, x(i), by “b”. Var(X) is the variance in the mean of the random variable.”
Not sure what point you think you are making here. But saying Var(X) is the variance of the mean is just wrong. It’s the variance of the random variable X.
Each individual term in a variance calculation is (x(i) – μ(X))^2
You end up multiplying both the value of each term but the mean also.
What purpose does this accomplish with temperatures. Doubling a temperature or halving it makes no sense.
“You end up multiplying both the value of each term but the mean also.”
Weary sigh, Yes, that’s why you end up with Var(bX) = b^2Var(X).
“Doubling a temperature or halving it makes no sense.”
Unless, (this might be a wild idea, but lets run with it) you actual want to get an average of two temperatures.
“You need to explain WHY your use of scaling is appropriate for combining temperature random variables.”
I’m not obliged to answer any of your increasingly frantic questions.
But I thought is was patently obvious why you need to scale variables to get a mean. It’s in the definition of a mean.
(x + y) / 2.
What does the divide by 2 mean to you, other than you are scaling the result by 1/2? It doesn’t matter if you think of this as multiplying the sum by 1/2, or multiplying each element by 1/2 and adding them together the result is the same.
“Exactly what are you adjusting and why?”
What bit of multiply the variable by b don’t understand. You are adjusting the random variable X by multiplying it by a constant b. Why you are doing that is irrelevant, it’s just maths. It works why ever you do it. You might be converting some ancient measurement into the proper metric units, you might require a weighted sum of values, you might be taking an average. It’s all the same.
“If you would not just cherry pick you would see that the adjustment in the example is changing the values of the components of the random variable”
Yes, that’s what I mean by scaling the variable. You are saying every value you get from that variable will be adjusted by being multiplied by a constant.
“i.e., changing the temperatures to give a new mean value.”
You are not changing the temperature. You are simply saying the average requires adding all the temperatures and dividing by N.
“In other words, “if the random variable X is adjusted” doesn’t mean just adjusting the variance, you must also adjust the values that make up the mean μ.”
Which is why I, and everyone else, is saying the effect of scaling a random variable is to scale the variance of that variable by the square of the constant.
It’s possible you are getting confused as to what means are being used here. Adding a number of random variables to get a mean, should not be confused with the mean of each random variable (more correctly that’s called the expected value).
“But I thought is was patently obvious why you need to scale variables to get a mean. It’s in the definition of a mean.”
Variance isn’t a mean. A mean is part of calculating a variance. [x(i) – μ(X)]^2.
Why do you think the calculation of a mean applies to the calculation of a variance?
“The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. ”
Do you not understand what this says? The mean of a random variable is a weighted average THAT IS DETERMINED BY THE PROBABILITY OF EACH TERM. You DO NOT divide by the number of terms to get a random variable mean.
“Variance isn’t a mean.”
Well, it’s the mean of the squares of the deviations. But no it’;s not the same as the mean of a random variable. Not sure what point you are making.
“Why do you think the calculation of a mean applies to the calculation of a variance?”
The variance is the mean of the squares of the deviations, the deviations are the distance any value is from the mean of the variable. Hence it’s quite important to know the mean of the variable.
“Do you not understand what this says?”
Yes, but you obviously don’t.
“You DO NOT divide by the number of terms to get a random variable mean.”
Of course not. Still no idea why you think this has any relevance to the variance of an average of random variables.
For the last time (I can but hope) when you take an average of random variables you are not working out the mean of an individual variable. You are just combining multiple variables into a new variable which is the mean of all those variables. It’s no different in principle than adding multiple random variables to get a new random variable that is the sum of those variables. The only is that in the case of averaging you also have to divide by N.
Ask him for a copy of his GAT uncertainty analysis.
The only reason I bring it up at all is to give them a head start on finding the variance with the GAT. Somehow dividing a hosed standard deviation by the number of samples and not the size of the samples is just the start.
Yes. I read his post. There’s no challenge from me regarding it.
What I’m challenging is your assertion that Var(X/2+Y/2) = Var(X) + Var(Y). The correct answer is Var(X/2+Y/2) = Var(X/2) + Var(Y/2) where Var(X/2) = Var(X) / 4 and Var(Y/2) = Var(Y) / 4.
I have never said that:
“Var(X/2+Y/2) = Var(X) + Var(Y)”
I did say that:
σ^2(X+Y) = σ^2(X) + σ^2(Y). That is, [Var(X+Y) = Var(X) + Var(Y)]
You have the same problem as Bellman, cherry picking with no understanding.
The short version –
The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
σ(x)^2 = Σ[x(i) – μ(x)]^2p(i)
You can’t change the value of μ(x) without changing the individual x(i) values that make up the mean. The “b” is applied to the x(i).
From the equation
Var(a + bx) = b^2Var(X)
“bx” IS MULTIPLYING each individual term, x(i), by “b”. Var(X) is the variance in the mean of the random variable.
YOUR JOB IS TO EXPLAIN WHY YOU ARE ADJUSTING THE TEMPERATURES BEFORE DOING THE CALCULATIONS. Study the example at this link closely.
http://www.stat.yale.edu/Courses/1997-98/101/rvmnvar.htm
JG said: “I have never said that:
“Var(X/2+Y/2) = Var(X) + Var(Y)””
You’ve made repeated statements over the last two years that you believe u^2(X/2 + Y/2) = u^2(X) + u^2(Y). In other words you keep telling us that uncertainties add in quadrature without scaling even when the original terms are scaled by 1/n not unlike Var(X/2+Y/2) = Var(X) + Var(Y).
Your link is proof that you have been wrong the last two years because it is easily shown using the Var(a+bX) = b^2*Var(X) rule that the answer is u^2(X/2 + Y/2) = [ u^2(X) + u^2(Y) ] / 4.
BTW…you can use the Var(a+bX) = b^2*Var(X) rule in your link to solve this.
(1) Var(X/2 + Y/2) = Var(X/2) + Var(Y/2)
(2) Var(X/2 + Y/2) = Var(0 + 1/2*X) + Var(0 + 1/2*Y)
(3) Var(X/2 + Y/2) = 1/4*Var(X) + 1/4*Var(Y)
(4) Var(X/2 + Y/2) = [ Var(X) + Var(Y) ] / 4
(5) σ = sqrt[ Var(X) + Var(Y) ] / 2
And when V = Var(X) = Var(Y) we have:
(4) Var(X/2 + Y/2) = [ Var(X) + Var(Y) ] / 4
(6) Var(X/2 + Y/2) = [ V + V ] / 4
(7) Var(X/2 + Y/2) = [ 2 * V ] / 4
(8) Var(X/2 + Y/2) = V / 2
(9) σ = sqrt(V) / sqrt(2)
Let’s walk thru the process.
______________________________________
From http://www.stat.yale.edu/Courses/1997-98/101/rvmnvar.htm
μ(x) = x1p1 + x2p2 + … + x(k)p(k)
σ(x)^2 = Σ{[x(i) – μ(x)]^2 • p(i)}
where x(i) = temp values and p(k) = the probability assigned to each value.
______________________________________
I’ll take my Tmax and Tmin for today.
Tmax = 49 ±1°F => x(1)
Tmin = 23 ±1°F => x(2)
______________________________________
Since we have two temps, we will use a 50% probability for each (the sum = 1). Please note that this carries through all the calculations for Tavg.
______________________________________
μ(x) = (49 • 0.50) + (23 • 0.50) = 24.5 + 11.5 = 36
Tavg = 36
σ(x)^2 = [(49 – 36)^2 • 0.50 + (23 – 36^2) • 0.50] =
[(13^2) • 0.50 + (-13)^2 • 0.50] =
(169 • 0.50) + (169 • 0.50) =
84.5 + 84.5 = 169
σ(x) = √169 = 13
______________________________________
As I told Bellman, if you want to scale or do something else, you must multiply the constant “b” against the “xi” values and then start over. Your insistence on using 1/2 will result in the following.
μ(x) = (49 / 2) • 0.50 + (23 / 2) • 0.50 =
24.5 • 0.50 + 11.5 • 0.50 =
12.25 + 5.75 = 18
Tavg = 18 (This is a ridiculous value for a mean of 49 and 23.
σ(x)^2 = [(24.5 – 18)^2 • 0.5 + (11.5 – 18)^2 • 0.5] =
42.25 • 0.50 + 42.25 • 0.50 =
21.125 + 21.125 =
42.25
σ(x) = √42.25 = 6.5
______________________________________
Now you tell me which is correct.
μ(x) = 36 and a σ(x) = 13
or a
μ(x) = 18 and a σ(x) = 6.5
______________________________________
If you wish to disagree, please show your derivation and a reference.
The reference for this work is
http://www.stat.yale.edu/Courses/1997-98/101/rvmnvar.htm
and includes examples of changing x(i), i.e., multiplying by a constant “b” and then recalculating the variance with new x(i)’s.
You’re still completely confused about this. You are mixing up the way the mean of a random value is defined, with adding two random variables.
“Since we have two temps, we will use a 50% probability for each (the sum = 1).”
That only makes sense if you are saying the daily temperature is a discrete random variable consisting of just two values a max or min. Or giving you still want uncertainty for each, a mix of two specific random variables.
“As I told Bellman”
Whatever you told me was just as garbled as this. Everything you say here is nonsense because you don;t understand the distinction between the mean of a random variable and a random variable produced by taking the mean of several random variables. Did you not read the first couple of sentences?
As always you are prepared to jump through all manor of misunderstandings, rather than accept the simple truth that averaging random variables from the same distribution leads to the familiar σ / √N.
“””””As always you are prepared to jump through all manor of misunderstandings, rather than accept the simple truth that averaging random variables from the same distribution leads to the familiar
σ / √N.”””””
σ/√n. is a formula to obtain the SEM, i.e., the standard deviation of the SAMPLE MEAN.
As you and bdgwx recommended let’s look at what Dr. Possolo did in the NIST TN1900.
Tmax = 49
Tmin = 23
μ = (49 + 23) / 2 = 36
σ^2 = [(49 – 36)^2 + (23 – 36)^2] / 2 = 169
σ = √169 = 13
σ(sem) = σ / √2 = 9.2
k (from t-table) @2 degrees of freedom = 2.92 @ 95% confidence
Expanded σ = 9.2 • 2.92 = 26.9
The end result is 36 ± 26.9
You should note that this is the variance in the data. What most of this thread has been about is the variance in the measurement uncertainty. As you can see, the measurement uncertainty is pretty small compared to the variance in the data.
Why do you think I have been asking for the standard deviation of the GAT? This has been totally ignored in climate science.
“σ/√n. is a formula to obtain the SEM, i.e., the standard deviation of the SAMPLE MEAN.”
And the correct use of averaging random variables illustrates why that is true.
“As you and bdgwx recommended let’s look at what Dr. Possolo did in the NIST TN1900.”
I’ve never made any such recommendation. Until you started going about it I’d never heard of TN1900 or Dr Possolo. It all looks fine though.
But you don’t say where in the document your formula for daily mean temperature is explained. The only example involving surface temperature is Example 2, which is only using maximum temperature. As usual you seem to be more interested in finding some example you think proves your point than actually understanding how random variables work.
“Tmax = 49
Tmin = 23
μ = (49 + 23) / 2 = 36
σ^2 = [(49 – 36)^2 + (23 – 36)^2] / 2 = 169
σ = √169 = 13
σ(sem) = σ / √2 = 9.2”
It doesn’t make sense to use SEM with maximum and minimum temperatures as they are not random values taken from the daily temperature range. They are specifically the maximum and minimum values.
“As you can see, the measurement uncertainty is pretty small compared to the variance in the data.”
Again, you are not looking at the variance of the daily data, just at the entire range. Bui all you are saying here is what I’ve been trying to point out for years – the uncertainty caused by sampling is usually going to be much bigger than that from the measurement uncertainty.
“Why do you think I have been asking for the standard deviation of the GAT?”
Because you don;t understand the difference between the uncertainty of the mean, and the uncertainty of an individual measurement. Because you don’t understand the distinction between anomalies and temperature. Because you think that looking at the huge range in temperatures across the globe, you can convince yourself that the mean anomaly can’t be used to determine if temperatures are rising or not. Because you think that if you cannot see a problem, then the problem can’t exist. Is that enough to be getting on with? Or do you want to tell me why you are asking about the SD, rather than keep asking me?
“””””And the correct use of averaging random variables illustrates why that is true.”””””
You don’t AVERAGE random variables to gat “σ” or “σ^2”. You use the summation of [(values – mean)^2]. In the general case you can assign each “value” a probability weight also. For temps, the probability of each can be 1/n.
“””””But you don’t say where in the document your formula for daily mean temperature is explained. The only example involving surface temperature is Example 2, which is only using maximum temperature. As usual you seem to be more interested in finding some example you think proves your point than actually understanding how random variables work.”””””
Word salad! What does Tmax vs Tavg data points have to do with calculating the mean or variance of the distribution regardless of what it is made up of.?
I could have made up numbers from my head and how does that affect the mathematical process?
“””””It doesn’t make sense to use SEM with maximum and minimum temperatures as they are not random values taken from the daily temperature range.”””””
They are what we have. If SEM is not usable then neither is σ or σ^2 or even the mean. Your argument simply says that the data is not fit for purpose. Good luck.
Ask yourself where the measure uncertainty comes from that allows anomalies of one-thousandths of a degree when the measurements were in units.
“””””Because you don’t understand the distinction between anomalies and temperature.”””””
I do understand, and you’ve never heard me say anomalies are valid.
“””””because you think that looking at the huge range in temperatures across the globe, you can convince yourself that the mean anomaly can’t be used to determine if temperatures are rising or not.”””””
Let me repeat your words back to you. “””””It doesn’t make sense to use SEM with maximum and minimum temperatures as they are not random values taken from the daily temperature range.”””””
Yet somehow anomalies whose uncertainty is justified by using an SEM that is calculated wrongly is ok? Fyi, the σ of the sample means distribution IS THE SEM. You don’t divide it again by the square root of the number of samples, when it should be the size of the samples?
Lastly, your denigration of Dr. Possolo,’s use of the SEM puts you in the position of criticizing a NIST scientist! CONGRATULATIONS!!!!
“You don’t AVERAGE random variables to gat “σ” or “σ^2”.“#
You still won’t take a step back and consider you might not understand what is being said.
It’s really simple. You combine random variables to produce a new random variable. You can use simple rules to determine bit the mean and the mean and variance of this new random variable. You understand how this works when you combine by adding two or more random variables, but for some reason have to throw all sorts of nonsense into the argument as soon as you start talking about combining in a mean. But the only difference is that the mean is a scaled version of the sum, and the only rule you need to understand is that when you scale a random variable by a constant the variance is scaled by the square of that constant.
Taking just two random variables X and Y, when we sum them we get
Mean(X + Y) = Mean(X) + Mean(Y)
Var(X + Y) = Var(X) + Var(Y)
when taking the average of the two we get
Mean((X + Y) / 2) = (Mean(X) + Mean(Y)) / 2
Var((X + Y) / 2) = (Var(X) + Var(Y)) / 4
It does not matter what the random variables are or why you are averaging them. The rules are the same regardless.
The random variables can be representing different things in different context. When you are talking about averaging temperatures the variables can either represent the measurement uncertainty of a specific measurement, or they can represent a random sample from a population.
“You use the summation of [(values – mean)^2].”
And you keep confusing how you calculate the variance of a random variable withe the rules for calculating the variance of a random variable that is the result of combining multiple random variables. You can do it from first principles, but the advantage of having these simple rules is you don’t have to.
If you do want to calculate the average or sum of two variables like this you can’t just average each value. You have to look at all possible pairs of values. Something like
ΣΣ[(x + y)^2(p(x,y)]
where the double summation is over all xs and ys. And for the mean
ΣΣ[{(x + y)/2}^2(p(x,y)] = (1/4)ΣΣ[(x + y)^2(p(x,y)]
“””””Mean(X + Y) = Mean(X) + Mean(Y)
Var(X + Y) = Var(X) + Var(Y)”””””
We are not adding two random variables. You are just copying formulas.
You are saying Tmax=49 and Tmin = 23 are two separate variables.
X = {49} and Y = {23}.
What value does Mean(X) =. [49 + 23 = 72]
What value does Mean(Y) =. [0 + 0 = 0]
Use the formulas I have attached in an image.
Do you ever try to understand these things. I gave an illustration of the difference between adding and averaging. I don’t care what you are trying to do. I’m showing how to do it, whether adding of averaging. All you do is keep objecting to me “copying” the correct formulas, whilst you present me with things irrelevant to the point. I really don’t know what you think you are trying to do with these min and max temperatures.
“You are saying Tmax=49 and Tmin = 23 are two separate variables.”
It depends on what you are trying to say. If you are interested in an exact average with measurement uncertainty, yes they are two separate variables, if you want them to be a sample of two taken from the population of all daily values, then they are two random variables with the same distribution. But as I am trying to explain, they are not random variables in that case because you are explicitly choosing the highest and lowest values.
“What value does Mean(X) =. [49 + 23 = 72]
What value does Mean(Y) =. [0 + 0 = 0]”
I can’t help you because what you are writing is meaningless.
“Use the formulas I have attached in an image.”
Why? I’ve explained why you don’t need to use that in the previous comment, which you obviously decided to ignore. You are averaging to random variables. You use the formulas I gave you to determine the mean and variance of the new variable.
All you are doing is trying to evade the simple explanation for why the variance of an average of random variables decreases the more variables you include. You won’t accept it becasue it goes against some strange Gorman ideology, and so like your brother you just try to search for as many ways to misunderstand the simple explanation as you can, in a hope that it will go away.
“””””What value does Mean(X) =. [49 + 23 = 72]
What value does Mean(Y) =. [0 + 0 = 0]”””””
That will teach me to hurry while trying to leave.
“””””Why? I’ve explained why you don’t need to use that in the previous comment, which you obviously decided to ignore. You are averaging to random variables. You use the formulas I gave you to determine the mean and variance of the new variable.”””””
Variance = Σ(x(i) – μ)
X = {49} μ = 49/1 = 49
Y = {23} μ = 23/1 = 23
Var(X) = (49 – 49) = 0
Var(Y) = (23 – 23) = 0
What is the Variance of the sum?
As I told bdgwx, you need a distribution (a mean) to have a variance, single values have no variance!
Read the following document.
https://www.colorado.edu/amath/sites/default/files/attached-files/ch4.pdf
It says:
“””””In principle variables such as height, weight, and temperature are continuous, in practice the limitations of our measuring instruments restrict us to a discrete (though sometimes very finely subdivided) world. “””””
Notice the word discreet.
Could you for one stick to a point.
All I want you to do is say what you think the variance of an average of random variables is. All you do is try to bring up badly thought out examples and keep insisting I learn the basics of how variance is calculated.
First you wanted me to consider max and minimum temperatures with a known measurement uncertainty.
Then the max and min suddenly became random samples from a daily cycle.
Now they aren’t random at all and have zero variance.
Just what point do you want to make? It’s obvious you are just trying to avoid answering the question about the variance of the average of random variables.
“What is the Variance of the sum?”
0
And your point is?
“As I told bdgwx, you need a distribution (a mean) to have a variance, single values have no variance!”
Again suggesting you don’t know what a random variable is.
“Read the following document.”
Not another on.
“Notice the word discreet.”
Again, what is your point?
JG said: “We are not adding two random variables.”
Yes we are. And you seemed to accept it earlier. What changed?
JG said: “Use the formulas I have attached in an image.”
That is for discrete random variables. Neither Tmax nor Tmin are discrete random variables. Nevermind the fact that they are 2 distinct and separate random variables and measurands. That’s not say that you cannot combine them and treat them as one via the measurement model Y = Tavg = (Tmin + Tmax) / 2, but to declare they are individually the same is absurd.
“””””JG said: “We are not adding two random variables.”
Yes we are. And you seemed to accept it earlier. What changed?”””””
I indicated what was needed to add two random variables, I don’t recall having said the individual daily temps were random variables.
“””””JG said: “Use the formulas I have attached in an image.”
That is for discrete random variables. Neither Tmax nor Tmin are discrete random variables. Nevermind the fact that they are 2 distinct and separate random variables and measurands. That’s not say that you cannot combine them and treat them as one via the measurement model Y = Tavg = (Tmin + Tmax) / 2, but to declare they are individually the same is absurd.”””””
Download the PDF document at:
https://www.colorado.edu/amath/sites/default/files/attached-files/ch4.pdf
It says:
“””””In principle variables such as height, weight, and temperature are continuous, in practice the limitations of our measuring instruments restrict us to a discrete (though sometimes very finely subdivided) world.
However, continuous models often approximate real-world situations very well, and continuous mathematics (calculus) is frequently easier to work with than mathematics of discrete variables and distributions. “””””
Notice as you read the document, that measuring devices limit us to a DISCRETE set of numbers. You will notice that using a continuous variable requires defining a continuous probability function. As I have said before, temps throughout a day have two distinct functions, day is a sine from 0 => xπ and an exponential at night. Good luck on finding a functional probability relationship for temperature for days, months, or years.
“””””That’s not say that you cannot combine them and treat them as one via the measurement model Y = Tavg = (Tmin + Tmax) / 2, but to declare they are individually the same is absurd.”””””
The general equation I posted for μ (mean) using probabilities of 50% turns into the equation you have shown. That is what general equations are for.
However, your variance calculation is a problem. You can not simply add variances together. If you try to use with single values
Var(X + Y) = Var(X) + Var(Y)
you have to find the variance of each variable. As I tried, unsuccessfully apparently, to show you and Bellman, the variance of a single value is zero (0).
Var = (x(i) – μ)^2
What is the μ of a single value? The value! This gives
(x(i) – (x(i))^2 = 0
(y(i) – (y(i))^2 = 0
So,
Var(X) + Var(Y) = 0 + 0 = 0
The average of 0 is:
0 / 2 = 0.
Anyway you cut it single values do not have a variance around a mean.! You must have at least two values, however inaccurate that might be in establishing an actual probability function! Is 50/50 correct? I don’t know, use the probabilities you wish for each, they just have to add to 1.
I thought you understood math but now I don’t know. You CAN NOT treat variances as simple numbers, they are determined by calculations. When you do something externally that affects all the items in the calculation. Which brings us to the question – why do you want to change the initial values and the mean in finding an addition of random variables?
“As I tried, unsuccessfully apparently, to show you and Bellman, the variance of a single value is zero (0).”
We are not talking about adding single values, we are adding random variables.
When I tell you the temperature here was 58F you consider that to be a random variable rather than a single value?
What’s the probability of the current temperature being 40F?
How many times have you complained about books “only using the stated values” and ignoring measurement uncertainty?
58F is a random variable if you are claiming there is measurement uncertainty.
In case it gets lost amid all these evasions, this is what I was originally responding to
https://wattsupwiththat.com/2023/01/03/unknown-uncertain-or-both/#comment-3663777
Jim is claiming that he is taking an average of Tmax and Tmin, but also claims that multiplying each by 0.5 is to do with their probabilities. Which makes no sense.
He then implies that the act of taking the average increases the ±1°F to ±1.41°F, because he calculates the sum and not the average.
If you ignore the measurement uncertainty and just use the stated values in your data set then they are *NOT* random variables. You can’t have your cake and eat it too!
Not true. What do you think a die roll is?
Neither a single temperature or a single die roll is a random variable by itself. A group of die rolls using the same die might form a probability function, but combining the rolls from a group of dies, say a d20, d12, d8, d6, and a d4 die, doesn’t provide a very well defined probability distribution for just one of the dies. A group of temperatures, say from Anchorage and Buenos Aries, does not provide a useful probability distribution either. The temp in Anchorage carries no predictive value for determining the temp in BA, just like a die roll from a d20 doesn’t carry any predictive value for the next roll of a d6 die.
“They are what we have. If SEM is not usable then neither is σ or σ^2 or even the mean.”
You still haven’t said what part of TN1900 you are getting this from.
“Your argument simply says that the data is not fit for purpose. Good luck.”
Your missing the point. Having a maximum and minimum value is better than just having two random samples from throughout the day. With random samples you have a fair chance of picking two values from the middle of the night, or from the warmest part of the afternoon. That’s why you get such a large uncertainty as to how much the mean is representative of the daily mean. When you have the maximum and minimum you know they are spaced out around the mean. There’s no guarantee you will get the exact mean, but it’s more likely to be closer than using two completely random values.
“Ask yourself where the measure uncertainty comes from that allows anomalies of one-thousandths of a degree when the measurements were in units.”
You’ve jumped from talking about the sampling uncertainty of two temperature values, to the uncertainty of thousands of values taken around the world across a month. And nobody, certainly not I, suggests the uncertainty of the monthly anomaly is anywhere near a thousandth of a degree.
“I do understand, and you’ve never heard me say anomalies are valid.”
Which is your problem. The point, or one of the points, of anomalies is that the variance across the globe is much smaller at any time than that for temperatures. Hence less uncertainty in the monthly average.
“Yet somehow anomalies whose uncertainty is justified by using an SEM that is calculated wrongly is ok? Fyi, the σ of the sample means distribution IS THE SEM. You don’t divide it again by the square root of the number of samples, when it should be the size of the samples?”
Could you rephrase that? I’m sure you are trying to make a point that is wrong, but I honestly can’t figure out what that point is.
“Lastly, your denigration of Dr. Possolo,’s use of the SEM puts you in the position of criticizing a NIST scientist! CONGRATULATIONS!!!!”
When have I done that. I’m still waiting for you to quote the part where you claim TN1900 says what you claim. And, I know this may be a shock, but sometimes even authority figures get things wrong.
“””””You still haven’t said what part of TN1900 you are getting this from.”””””
From TN 1900.
1) “””””The average t = 25.59 ◦C of these readings is a commonly used estimate of the daily maximum temperature τ during that month. The adequacy of this choice is contingent on the definition of τ and on a model that explains the relationship between the thermometer read ings and τ.
2) “”””””If Ei denotes the combined result of such effects, then ti = τ + Ei where Ei denotes a random variable with mean 0, for i = 1, … ,m, where m = 22 denotes the number of days in which the thermometer was read. This so-called measurement error model (Freedman et al., 2007) may be specialized further by assuming that E1, …, Em are modeled independent random m variables with the same Gaussian distribution with mean 0 and standard deviation (. In these circumstances, the {ti} will be like a sample from a Gaussian distribution with mean τ and standard deviation σ ( (both unknown).”””””
3) “””””For example, proceeding as in the GUM (4.2.3, 4.4.3, G.3.2), the average of the m = 22 daily readings is t̄ = 25.6 ◦C, and the standard deviation is s = 4.1 ◦C. Therefore, the √ standard uncertainty associated with the average is u(τ) = s∕ m = 0.872 ◦C”””””
4)”””””The coverage factor for 95% coverage probability is k = 2.08, which is the 97.5th percentile of Student’s t distribution with 21 degrees of freedom. In this conformity, the shortest 95% coverage interval is t̄ ± ks∕√ n = (23.8 ◦C, 27.4 ◦C). “””””
Paragraph 1 -> t = 25.59
Paragraph 2 -> ” E1, …, Em are modeled independent random m variables ”
Paragraph 3 -> “the average of the m = 22 daily readings is t̄ = 25.6 ◦C, and the standard deviation is s = 4.1 ◦C. Therefore, the √ standard uncertainty associated with the average is u(τ) = s∕ √m = 0.872 ◦C”
Paragraph 4 -> “t̄ ± ks∕√ n = (23.8 ◦C, 27.4 ◦C).”
So,
t̄ 25.6 (t̄ = μ)
σ = 4.1 (s = standard deviation)
SEM = 0.872 (σ / √m)
k = 2.08 (m-1 degrees of freedom)
“””””That’s why you get such a large uncertainty as to how much the mean is representative of the daily mean.”””””
There is nothing new about max and min having a large effect on the variance.
Which is not what you were saying. You were talking about a daily mean from a max and min. This example is looking at a the monthly average of maximum temperatures from 22 daily maximum readings. Completely different.
Do you understand what random variables are?
Do you understand the difference between a measurement uncertainty distribution and a distribution of measurements?
No wonder KM has had it with you!
Better than you it would seem.
JG said: “As you and bdgwx recommended let’s look at what Dr. Possolo did in the NIST TN1900.”
The example in TN 1900 uses a type A evaluation to estimate the uncertainty of measurements and then divides the type A evaluation estimate by sqrt(N). If you use the exact same procedure for the UAH TLT Jan baseline grid you get 263.13 ± 0.13 K. And for the UAH TLT 2022/01 grid you get 0.03 ± 0.005 K.
Don’t hear what I never said. I do NOT think the monthly UAH TLT anomaly uncertainties are ± 0.005 K. I think they are much higher because I think a type B evaluation (like from Christy et al. 2003 showing 0.20 K) provides a better estimate.
JG said: “Why do you think I have been asking for the standard deviation of the GAT?”
I’ve given this to you multiple times. For the UAH baseline grid it is about 0.13 K depending on month. For the individual anomaly grids it is about 0.5 K depending on month.
“””””The example in TN 1900 uses a type A evaluation to estimate the uncertainty of measurements and then divides the type A evaluation estimate by sqrt(N). “””””
This example is not about measurement uncertainty. The measurand is the AVERAGE TEMPERATURE at this location. The document pretty much eliminate all uncertainty except the variation in the data.
From TN1900
“””””Assuming that the calibration uncertainty is negligible by comparison with the other uncertainty components, and that no other significant sources of uncertainty are in play”””””
Example 2 is to show an example of calculating an experimental uncertainty of a measurand.
“””””I’ve given this to you multiple times. For the UAH baseline grid it is about 0.13 K depending on month. For the individual anomaly grids it is about 0.5 K depending on month.”””””
Those figures are not the variance of the individual temperatures around the globe. One reason for starting this is to begin an examination of the data and how it is being used. Other than satellite temps, NWS/NOAA have LIG temps with a measurement uncertainty of ±1°F. I am agreeable to accepting that as the expanded experimental uncertainty in measurement. ASOS, MMTS, & CRN all have these specified.
Tn1900 certainly got me to considering the actual variance in temperatures. The variances in winter and summer daily temps are different. Those weightings should be applied as percentages when both computing a mean and variance. That brings us to using random variables and not simple means.
Dr. Possolo ‘s notes apply to average temps and is one process that needs to be weighed. Another is straight random variable means and variances.
JG said: “This example is not about measurement uncertainty. The measurand is the AVERAGE TEMPERATURE at this location.”
So you agree that an average temperature can be a measurand?
JG said: “Those figures are not the variance of the individual temperatures around the globe. One reason for starting this is to begin an examination of the data and how it is being used.”
First, typo…the baseline grid SD is 13 K; not 0.13 K. The 0.13 K comes from 13 / sqrt(9508) and is the type A uncertainty of the grid.
Second, you can get the variance by squaring so (13 K)^2 = 169 K^2.
And this comes directly from the grid files here.
JG said: “Tn1900 certainly got me to considering the actual variance in temperatures.”
And like I said…when you follow the example from TN 1900 and apply it to UAH you get 263.13 ± 0.13 K for the Jan baseline and 0.03 ± 0.005 K for the 2022/01 anomaly.
And like I said also…those are type A evaluations which I believe significantly underestimates the uncertainty. A better approach is to use a type B evaluation like what Christy et al. 2003 did.
“””””So you agree that an average temperature can be a measurand?”””””
If that is what you define it as. BUT, then you must use the measurements FROM WHICH YOU CALCULATED THE MEAN to calculate the variance. You can’t declare the mean is a measurement and use an average measurement uncertainty of the individual members as the variance of the mean!
“””””First, typo…the baseline grid SD is 13 K; not 0.13 K. The 0.13 K comes from 13 / sqrt(9508) and is the type A uncertainty of the grid.”””””
You do realize the “9508” you are using to calculate the SEM must be the size of the samples and NOT THE NUMBER OF SAMPLES, right?
The SEM is determined by the number of elements in each sample. If you are declaring 9508 samples with a size of 1, then you should divide by 1. This is the one area where I question Dr. Possolo calculation of standard uncertainty. However, I assume he is using a small number of samples, i.e., 1 and a sample size of 22. The low numbers of samples and sample size (<30) is why he 'expanded' the SEM. Plus, his claimed 2.08=k is a 97.5% confidence.
Now, if you know the population SD, then you already know both the population mean and the population SD. Why are you wanting to determine the interval (SEM) that may include the sample estimated means?
In fact if you have only one sample of 9508, the sample means of that one sample is the estimated population mean and the SEM is the standard deviation of that one sample. The population SD is then (SEM • √9508) What you are in essence doing with one sample is finding the SEM of the sample SEM. That makes no sense.
So you don’t agree with the rules Var(a+bX) = b^2*Var(X) and Var(X+Y) = Var(X) + Var(Y)?
And why are you treating Tmin and Tmax, which are two different random variables with two completely different values and two completely different and non-overlapping uncertainty envelopes, as if they are a single variable?
I don’t disagree with the formula. However you need to know what you are adjusting when you do Var(bx). You are adjusting both “x” and “μ” in the variance formula.
If Tmax = 49 and Tmin = 23
1. Uncertainty in measurement IS NOT the same as the variance in a distribution of data.
2. For a mean or variance to occur you need a minimum of two data points.
3. We are defining random variable “X = {49, 23}”. The mean of X and the variance of X define Tavg.
4. We are not dealing with X and Y.
When you do [(Var(X)] / 2) you are not just calculating a “mean” of two variances. You are doing
1/2 Var(X) =
Σ1/2 [x(i) – μ(i)]^2 =
Σ[1/4x(i) – 1/4μ(i)]^2
You are modifying both the data point values and the mean of the data points.
The big question is why?
You have not answered this at all. You are just spitting out formulas as a justification!
JG said: “I don’t disagree with the formula. However you need to know what you are adjusting when you do Var(bx).”
No you don’t. That’s point of identity rules. Var(a+bX) = b^2*Var(X). You don’t have know anything about X. That’s the whole point.
JG said: “3. We are defining random variable “X = {49, 23}”. The mean of X and the variance of X define Tavg. 4. We are not dealing with X and Y.”
Ok, so you’re moving the goalpost from X being Tmax and Y being Tmin to X being Tavg.
JG said: “When you do [(Var(X)] / 2) you are not just calculating a “mean” of two variances. You are doing”
Nobody is doing Var(X) / 2 where X:{49, 23} is Tavg. I don’t even know what that is. And dividing a single value by 2 does not turn it into a mean. It just divides it into 2. The formula for a mean is Σ[x_i, 1, N] / N.
JG said: “The big question is why? You have not answered this at all.”
That’s what I’d like to know. I’ve not answered it because it’s the first time you’ve presented the scenario Var({49, 23}) / 2. Like, I said. I have no idea what you are doing here.
“”””No you don’t. That’s point of identity rules. Var(a+bX) = b^2*Var(X). You don’t have know anything about X. That’s the whole point.”””””
Carry that on through the equation.
b^2Var(X) = b^2 • Σ(x(i) – μ)^2 • p(i) = Σb^2 • (x(i) – μ)^2 • p(i) =
Σ (b(x(i) – bμ)^2 • p(i)
The question remains, why are you altering the values that make up the variance?
.
Tmax and Tmin are not random variables. They are discreet values albeit with a measurement uncertainty.
To do a proper evaluation of measurement uncertainty effects one needs to run calculations with 0, +1, and -1 to see the affects.
JG said: “Tmax and Tmin are not random variables.”
Yes they are.
And you even said they were random variables here. Did you change your mind.
JG said: “To do a proper evaluation of measurement uncertainty effects one needs to run calculations with 0, +1, and -1 to see the affects.”
This is a new twist. Where do you see that in Taylor or the GUM?
Do you understand the difference between a measurement uncertainty and a distribution of measurements? You are getting more and more out into never- never land.
“For two random variables X and Y”
Systematic bias does not create a “random” variable. Systematic bias in a measurement creates an uncertainty that is not amenable to usual statistical analysis.
How many statistical analyses have you done where the data is given as “stated value +/- uncertainty” as opposed to just stated value?
A data set of
24 +/- 0.5, 25 +/- 0.3, 23 +/- 0.6, 28 +/- 0.1
is far different than one that is
24, 25, 23, 28
If you *know* that systematic bias exists in the uncertainty intervals how do you handle that when determining the uncertainty of the data set?
To my mind, a risk has a probability of occurrence of less than one, a risk may or may not happen. Whereas an uncertainty will happen, ie has a probability of occurrence of one, but the impact or outcome will vary. A risk’s impact or outcome, should it happen, can also have uncertainty