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.)