Unknown, Uncertain or Both?

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 unskewednormal 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”.

# # # # #

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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Tom Halla
January 3, 2023 6:10 am

Statistical reasoning somehow never feels right, which is why people play lotteries.

strativarius
Reply to  Kip Hansen
January 3, 2023 7:18 am

the gamblers still play.”

Hope springs eternal.

Thomas
Reply to  strativarius
January 3, 2023 7:34 am

Life is a gamble, they say.

Gunga Din
Reply to  Thomas
January 3, 2023 8:50 am

I thought it was “but a game”?
(Maybe a Vegas game?)

Drake
Reply to  Kip Hansen
January 3, 2023 10:31 am

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.

Clyde Spencer
Reply to  Kip Hansen
January 3, 2023 8:05 pm

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.

Richard Greene
Reply to  Kip Hansen
January 4, 2023 6:22 am

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

Michael S. Kelly
Reply to  Kip Hansen
January 4, 2023 4:20 pm

In Vegas, the casino owners like to tell the gamblers “This place wasn’t built on your winnings.”

Chasmsteed
Reply to  Tom Halla
January 3, 2023 6:58 am

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.

Ben Vorlich
Reply to  Chasmsteed
January 3, 2023 7:24 am

My dad used to say, when warning about the folly of gambling, “You never see a bookmaker on a bike”

karlomonte
Reply to  Kip Hansen
January 3, 2023 8:03 am

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.

Fran
Reply to  Kip Hansen
January 3, 2023 11:04 am

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.

Clyde Spencer
Reply to  Chasmsteed
January 3, 2023 8:11 pm

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.

Giving_Cat
Reply to  Tom Halla
January 3, 2023 11:53 am

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.

Richard S J Tol
Reply to  Tom Halla
January 5, 2023 1:22 am

I’d amend that: Statistical reasoning somehow never feels right, which is why people play lotteries and the house always wins.

DMacKenzie
January 3, 2023 6:22 am

When we roll two dice, we get what looks like an unskewed “normal distribution”.

No, a pair of dice get a triangular distribution, not a bell-shaped curve….

strativarius
January 3, 2023 6:57 am

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.

hiskorr
January 3, 2023 6:58 am

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!

Mr.
Reply to  hiskorr
January 3, 2023 9:35 am

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 –

birds eye chillis.JPG
Chasmsteed
January 3, 2023 7:03 am

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

Last edited 25 days ago by Chasmsteed
vuk
Reply to  Kip Hansen
January 3, 2023 9:50 am

When Einstein was discussing dice with Born, Schrodinger’s cat was already dead,

Last edited 24 days ago by vuk
cilo
Reply to  Kip Hansen
January 4, 2023 1:28 am

We cannot conclude that 100% of the respondents were slightly female.

And

dice are an example that readers can easily understand…

…as opposed to the abstract concept of male/ female gender?
Sigh…(insert smiling yellow circle)

Tim Gorman
Reply to  Chasmsteed
January 3, 2023 8:56 am

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.

cilo
Reply to  Tim Gorman
January 4, 2023 1:37 am

If I, and most engineers I know,…

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…

Bellman
January 3, 2023 7:17 am

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?

Thomas
Reply to  Bellman
January 3, 2023 7:38 am

Because, knowing that, a person wouldn’t gamble with dice?

karlomonte
Reply to  Bellman
January 3, 2023 7:40 am

In metrology, the preferred usage

/snort/ — the expert is on his soapbox.

Bellman
Reply to  karlomonte
January 3, 2023 8:12 am

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.

karlomonte
Reply to  Bellman
January 3, 2023 8:33 am

“Stop whining”—CMoB.

Bellman
Reply to  karlomonte
January 3, 2023 8:58 am

I’ll take that as a no.

karlomonte
Reply to  Bellman
January 3, 2023 9:12 am

No 10 mK T uncertainties today?

Bellman
Reply to  Kip Hansen
January 3, 2023 8:44 am

You keep showing that definition, and I don’t think it means what you seem to think it does.

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.

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
Reply to  Kip Hansen
January 3, 2023 1:06 pm

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.

Tim Gorman
Reply to  Bellman
January 4, 2023 2:46 pm

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.

Bellman
Reply to  Tim Gorman
January 4, 2023 3:46 pm

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.

Tim Gorman
Reply to  Bellman
January 5, 2023 8:39 am

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?

karlomonte
Reply to  Tim Gorman
January 5, 2023 8:55 am

“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.

And he still can’t grasp the concept of the true value!

Bellman
Reply to  Kip Hansen
January 3, 2023 8:48 am

Here’s a slightly clearer definition

Absolute uncertainty: This is the simple uncertainty in the value itself as we have discussed it up to now. It is the term used when we need to distinguish this uncertainty from relative or percent uncertainties. If there is no chance of confusion we may still simply say “uncertainty” when referring to the absolute uncertainty. Absolute uncertainty has the same units as the value. Thus it is:3.8 cm ± 0.1 cm.

https://www.bellevuecollege.edu/physics/resources/measure-sigfigsintro/f-uncert-percent/

Tim Gorman
Reply to  Bellman
January 3, 2023 9:08 am

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?

Bellman
January 3, 2023 7:27 am

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.

Tim Gorman
Reply to  Bellman
January 3, 2023 9:12 am

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
Reply to  Kip Hansen
January 3, 2023 12:44 pm

If I’m misunderstanding your point, maybe you need to be clearer.

You said

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.

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. .

Bellman
January 3, 2023 7:29 am

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.

karlomonte
Reply to  Kip Hansen
January 3, 2023 8:35 am

bellman uses the same tactics as Nickpick Nick Stokes…

Bellman
Reply to  Kip Hansen
January 3, 2023 9:08 am

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.

Tim Gorman
Reply to  Bellman
January 3, 2023 9:29 am

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”.

Bellman
Reply to  Tim Gorman
January 3, 2023 9:42 am

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:

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.

Bellman
Reply to  Kip Hansen
January 3, 2023 12:46 pm

Giving the results with 1 SD, is not the same thing as claiming you have eliminated all values outside that range.

Tim Gorman
Reply to  Bellman
January 4, 2023 8:28 am

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?

Bellman
Reply to  Tim Gorman
January 4, 2023 2:17 pm

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?

Tim Gorman
Reply to  Bellman
January 5, 2023 6:54 am

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.

Tim Gorman
Reply to  Bellman
January 3, 2023 9:13 am

Now you are back to using the argumentative fallacy of Equivocation. Trying to change the subject to something else. You are so transparent!

Bellman
January 3, 2023 7:32 am

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.

Tim Gorman
Reply to  Kip Hansen
January 3, 2023 9:31 am

Bellman has no definitions other that what fits at the time so he can argue someone is wrong.

Nick Stokes
Reply to  Kip Hansen
January 3, 2023 9:54 am

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.

karlomonte
Reply to  Nick Stokes
January 3, 2023 10:31 am

Got any 20 mK T uncertainties to quote today?

Nick Stokes
Reply to  Kip Hansen
January 3, 2023 1:33 pm

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 Stokes
Reply to  Kip Hansen
January 3, 2023 5:59 pm

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).

Tim Gorman
Reply to  Nick Stokes
January 4, 2023 9:34 am

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?

michael hart
January 3, 2023 7:37 am

“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.”

Bellman
January 3, 2023 7:39 am

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.

bigoilbob
Reply to  Bellman
January 3, 2023 7:50 am

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.”

Frank from NoVA
Reply to  bigoilbob
January 3, 2023 8:32 am

bob, are you suggesting that every ruling handed down by a SCOTUS majority is ‘constitutional’?

bigoilbob
Reply to  Frank from NoVA
January 3, 2023 8:53 am

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.

AndyHce
Reply to  bigoilbob
January 3, 2023 2:36 pm

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.

Last edited 24 days ago by AndyHce
Tim Gorman
Reply to  bigoilbob
January 3, 2023 9:35 am

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 Gorman
Reply to  Kip Hansen
January 4, 2023 2:13 pm

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
Reply to  Kip Hansen
January 3, 2023 10:03 am

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
Reply to  Kip Hansen
January 3, 2023 1:12 pm

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
Reply to  Kip Hansen
January 3, 2023 4:31 pm

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.

Bellman
Reply to  Bellman
January 3, 2023 4:53 pm

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.

20230104wuwt1.png
cilo
Reply to  Bellman
January 4, 2023 1:56 am

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?

bigoilbob
Reply to  cilo
January 4, 2023 6:25 am

Pseudo random numbers are perfectly useful for Bellman’s example. Unless you’re writing CIA code, it’s a bogus criticism.

Tim Gorman
Reply to  bigoilbob
January 4, 2023 9:53 am

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!

Bellman
Reply to  Tim Gorman
January 4, 2023 10:17 am

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?

bigoilbob
Reply to  Tim Gorman
January 4, 2023 3:05 pm

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.

Bellman
Reply to  cilo
January 4, 2023 7:53 am

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.

Tim Gorman
Reply to  Bellman
January 4, 2023 9:55 am

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.



Bellman
Reply to  Tim Gorman
January 4, 2023 10:18 am

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.

bigoilbob
Reply to  Bellman
January 4, 2023 3:07 pm
  1. It will happen.
  2. As with the Chimps with typewriters typing the encyclopedia, it’s going to take awhile.
Tim Gorman
Reply to  cilo
January 4, 2023 9:51 am

bellman doesn’t believe in systematic uncertainty. All dice are perfect.

Bellman
Reply to  Tim Gorman
January 4, 2023 11:21 am

Stop lying about me.

Bellman
Reply to  Tim Gorman
January 4, 2023 2:23 pm

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.

Tim Gorman
Reply to  Bellman
January 5, 2023 6:58 am

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
Reply to  Kip Hansen
January 4, 2023 8:01 am

“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.

Tim Gorman
Reply to  Bellman
January 4, 2023 10:08 am

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.

Bellman
Reply to  Tim Gorman
January 4, 2023 2:42 pm

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.

Tim Gorman
Reply to  Bellman
January 5, 2023 7:02 am

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!

Bellman
Reply to  Tim Gorman
January 5, 2023 7:57 am

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

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.

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.

Bellman
Reply to  Kip Hansen
January 4, 2023 8:15 am

You ignored the question. How would your understanding of uncertainty allow you to say if the bet was good or not?

Tim Gorman
Reply to  Bellman
January 4, 2023 10:08 am

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!

Last edited 23 days ago by Tim Gorman
bigoilbob
Reply to  Tim Gorman
January 4, 2023 3:18 pm

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:

  1. Do I know exactly when my parts will be ready and assembled?
  2. Do I know more about my delivery than I did before I turned around?

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.

Tim Gorman
Reply to  bigoilbob
January 5, 2023 7:05 am

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!

karlomonte
Reply to  Tim Gorman
January 5, 2023 8:18 am

blob is completely off in the weeds this week.

Tim Gorman
Reply to  Bellman
January 6, 2023 8:58 am

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?

Tim Gorman
Reply to  Kip Hansen
January 4, 2023 9:57 am

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.

Bellman
Reply to  Tim Gorman
January 4, 2023 2:45 pm

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.

Tim Gorman
Reply to  Bellman
January 5, 2023 7:22 am

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!

Bellman
Reply to  Tim Gorman
January 5, 2023 10:16 am

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.

ThomasH
Reply to  Kip Hansen
January 7, 2023 2:26 pm

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?

Tim Gorman
Reply to  Bellman
January 4, 2023 9:50 am

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!

Tim Gorman
Reply to  Bellman
January 4, 2023 9:46 am

 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!

Tim Gorman
Reply to  Bellman
January 4, 2023 9:42 am

Kip is correct. You can’t even admit that the average uncertainty is not the uncertainty of the average.

Bellman
Reply to  Tim Gorman
January 4, 2023 11:24 am

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.

Tim Gorman
Reply to  Bellman
January 4, 2023 2:30 pm

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.

Bellman
Reply to  Tim Gorman
January 4, 2023 2:58 pm

…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.

Tim Gorman
Reply to  Bellman
January 5, 2023 7:45 am

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!

karlomonte
Reply to  Tim Gorman
January 5, 2023 8:19 am

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…

Tim Gorman
Reply to  karlomonte
January 6, 2023 9:00 am

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.

karlomonte
Reply to  Tim Gorman
January 6, 2023 9:15 am

Looks to me like he doesn’t even realize it, must be psychological.

Bellman
Reply to  Tim Gorman
January 5, 2023 10:34 am

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.

Tim Gorman
Reply to  Bellman
January 6, 2023 9:03 am

 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!

Bellman
Reply to  Tim Gorman
January 6, 2023 9:58 am

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?

Tim Gorman
Reply to  Bellman
January 6, 2023 4:30 pm

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!

Bellman
Reply to  Tim Gorman
January 6, 2023 5:27 pm

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

Arithmetic Mean = ((x1 + x2 + …. + xn) / n) +/- (summed uncertainty / n)

The significance of this is the average of many added rounded values (say Temperatures Values, rounded to whole degrees, to be X° +/- 0.5°C) will carry the same original measurement uncertainty value.

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.

Tim Gorman
Reply to  Bellman
January 8, 2023 5:53 am

 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?

Bellman
Reply to  Tim Gorman
January 9, 2023 5:36 am

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.

Tim Gorman
Reply to  Bellman
January 8, 2023 5:56 am

 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.

Bellman
Reply to  Tim Gorman
January 8, 2023 8:54 am

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.

Screenshot 2023-01-08 165346.png
Tim Gorman
Reply to  Bellman
January 10, 2023 1:12 pm

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.

Bellman
Reply to  Tim Gorman
January 8, 2023 9:14 am

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.

Tim Gorman
Reply to  Bellman
January 10, 2023 1:14 pm

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!

Bellman
Reply to  Tim Gorman
January 11, 2023 5:26 am

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.

old cocky
Reply to  Kip Hansen
January 3, 2023 3:16 pm

Richard Feynman rather than Einstein.

Tim Gorman
Reply to  Bellman
January 4, 2023 9:41 am

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!

Bellman
Reply to  Tim Gorman
January 4, 2023 3:05 pm

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.

Last edited 23 days ago by Bellman
Tim Gorman
Reply to  Bellman
January 5, 2023 8:17 am

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?

Bellman
Reply to  Tim Gorman
January 5, 2023 10:45 am

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?

Bellman
Reply to  Tim Gorman
January 5, 2023 11:06 am

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.

karlomonte
Reply to  Bellman
January 3, 2023 8:36 am

The same old lies pop out again from da bellcurveman.

Tim Gorman
Reply to  Bellman
January 3, 2023 9:33 am

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.

Kevin Kilty
January 3, 2023 7:58 am

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.

“just one number is best”. This is a lousy approach to almost every research question.

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.

Tim Gorman
Reply to  Kevin Kilty
January 3, 2023 9:39 am

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.

Last edited 24 days ago by Tim Gorman
Clyde Spencer
Reply to  Kevin Kilty
January 3, 2023 9:09 pm

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.

Kevin Kilty
Reply to  Clyde Spencer
January 4, 2023 6:02 am

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.

bigoilbob
Reply to  Clyde Spencer
January 4, 2023 6:21 am

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.

karlomonte
January 3, 2023 7:59 am

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.

Kevin Kilty
Reply to  karlomonte
January 3, 2023 8:04 am

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.

karlomonte
Reply to  Kevin Kilty
January 3, 2023 8:39 am

Thanks, Kevin.

Tim Gorman
Reply to  Kevin Kilty
January 3, 2023 9:41 am

Even an “expanded” coverage won’t reach *all* values.

Kevin Kilty
Reply to  Tim Gorman
January 4, 2023 6:04 am

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.

karlomonte
Reply to  karlomonte
January 3, 2023 8:17 am

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.

Bellman
Reply to  karlomonte
January 3, 2023 8:20 am

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

The standard uncertainty u(y) of a measurement result y is the estimated standard deviation of y.

Then they define uncertainty

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, then the interval y u(y) to y + u(y) is expected to encompass approximately 68 % of the distribution of values that could reasonably be attributed to the value of the quantity Y of which y is an estimate. This implies that it is believed with an approximate level of confidence of 68 % that Y is greater than or equal to yu(y), and is less than or equal to y + u(y), which is commonly written as Y= y ± u(y).

https://www.physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html

karlomonte
Reply to  Bellman
January 3, 2023 8:30 am

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).

Bellman
Reply to  karlomonte
January 3, 2023 8:57 am

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.

karlomonte
Reply to  Bellman
January 3, 2023 9:14 am

You seem to think it is some sort of an insult.

No I think it is highly amusing given that you pound everything into a Gaussian curve.

Bellman
Reply to  karlomonte
January 3, 2023 9:46 am

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.

Tim Gorman
Reply to  Bellman
January 3, 2023 10:29 am

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!

Bellman
Reply to  Tim Gorman
January 3, 2023 2:43 pm

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”.

Tim Gorman
Reply to  Bellman
January 4, 2023 8:38 am

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?

karlomonte
Reply to  Tim Gorman
January 4, 2023 9:01 am

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?

Here is what the UAH has for this question:

UAH 1991-2020 sd of Mean.jpeg
Jim Gorman
Reply to  Bellman
January 4, 2023 11:49 am

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.

Bellman
Reply to  Jim Gorman
January 4, 2023 4:26 pm

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.

Tim Gorman
Reply to  Bellman
January 5, 2023 8:46 am

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.

Bellman
Reply to  Tim Gorman
January 5, 2023 11:12 am

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.

Jim Gorman
Reply to  Bellman
January 5, 2023 10:23 am

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.

Bellman
Reply to  Jim Gorman
January 5, 2023 11:14 am

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.

Thomas
Reply to  Bellman
January 3, 2023 12:26 pm

 Nature tends to produce Gaussian distributions, 

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.

bigoilbob
Reply to  Thomas
January 4, 2023 6:14 am

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.

Tim Gorman
Reply to  karlomonte
January 3, 2023 10:26 am

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.

Bellman
Reply to  Tim Gorman
January 3, 2023 10:34 am

Will you ever stop lying about me.

Tim Gorman
Reply to  Bellman
January 4, 2023 1:19 pm

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.

karlomonte
Reply to  Tim Gorman
January 3, 2023 10:34 am

Exactly right.

strativarius
Reply to  Bellman
January 3, 2023 9:50 am

You’re right it should be bellcurvedballman

Bellman
Reply to  strativarius
January 3, 2023 10:17 am

Just accept you are never going to do better than Bellend. I chose my pseudonym expecting someone would use that.

Tim Gorman
Reply to  Bellman
January 3, 2023 9:50 am

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?

Bellman
Reply to  Tim Gorman
January 3, 2023 10:30 am

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.

karlomonte
Reply to  Bellman
January 3, 2023 10:48 am

See above.

And what is the probability distribution of an error band spec for a digital voltmeter?

bigoilbob
Reply to  karlomonte
January 3, 2023 11:50 am

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

Tim Gorman
Reply to  bigoilbob
January 4, 2023 1:17 pm

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?

bigoilbob
Reply to  Tim Gorman
January 5, 2023 6:23 am

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.

Last edited 23 days ago by bigoilbob
karlomonte
Reply to  bigoilbob
January 5, 2023 6:57 am

From the GUM, blob:

6.3.2 Ideally, one would like to be able to choose a specific value of the coverage factor k that would provide an interval Y = y ± U = y ± kuc(y) corresponding to a particular level of confidence p, such as 95 or 99 percent; equivalently, for a given value of k, one would like to be able to state unequivocally the level of confidence associated with that interval. However, this is not easy to do in practice because it requires extensive knowledge of the probability distribution characterized by the measurement result y and its combined standard uncertainty uc(y). Although these parameters are of critical importance, they are by themselves insufficient for the purpose of establishing intervals having exactly known levels of confidence.

Standard uncertainty does NOT tell you a probability distribution!

bigoilbob
Reply to  karlomonte
January 5, 2023 7:10 am

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.

karlomonte
Reply to  bigoilbob
January 5, 2023 8:20 am

What happens 3, 6, 12 + months after calibration, blob?

bigoilbob
Reply to  karlomonte
January 5, 2023 8:27 am

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.

karlomonte
Reply to  bigoilbob
January 5, 2023 9:08 am

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.

A recalibration, based on use and/or Fluke experience? I don’t really know.

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.

bigoilbob
Reply to  karlomonte
January 5, 2023 9:44 am

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.

karlomonte
Reply to  bigoilbob
January 5, 2023 10:16 am

Deny reality, good job, blob.

Please continue in your delusions.

Tim Gorman
Reply to  bigoilbob
January 6, 2023 5:54 am

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?

karlomonte
Reply to  Tim Gorman
January 6, 2023 6:45 am

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.

Last edited 22 days ago by karlomonte
Tim Gorman
Reply to  karlomonte
January 6, 2023 3:35 pm

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.

Tim Gorman
Reply to  bigoilbob
January 6, 2023 9:05 am

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?

bigoilbob
Reply to  Tim Gorman
January 6, 2023 9:15 am

I agree with everything you said. Please show me where I ever said otherwise.

Tim Gorman
Reply to  bigoilbob
January 6, 2023 3:37 pm

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.

karlomonte
Reply to  Tim Gorman
January 6, 2023 3:47 pm

Yep, a lame backpedal attempt.

bigoilbob
Reply to  Tim Gorman
January 7, 2023 5:51 am

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.

Last edited 21 days ago by bigoilbob
karlomonte
Reply to  bigoilbob
January 7, 2023 6:11 am

Another noisy whiner, ask me if I care what you think.

karlomonte
Reply to  bigoilbob
January 7, 2023 7:00 am

What was my assertion? Answer: that digital voltmeters have distributed error. They do, as the Fluke engineers showed us. That’s all.

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!

Tim Gorman
Reply to  karlomonte
January 8, 2023 3:07 pm

He won’t. He’s back-pedaling so fast he’s going to wind up on his butt. Actually he already has.

Tim Gorman
Reply to  bigoilbob
January 7, 2023 7:23 am

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!

bigoilbob
Reply to  Tim Gorman
January 7, 2023 7:35 am

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.

karlomonte
Reply to  bigoilbob
January 7, 2023 7:57 am

Where is YOUR air temperature uncertainty analysis, blob. I don’t see it.

Tim Gorman
Reply to  bigoilbob
January 8, 2023 3:16 pm

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.

bdgwx
Reply to  Tim Gorman
January 7, 2023 1:43 pm

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.

Tim Gorman
Reply to  bigoilbob
January 8, 2023 3:06 pm
  1. Keep backpedaling. You’ll trip up again.
  2. The average uncertainty is *NOT* the uncertainty of the average. I showed you that with the bi-modal pile of boards where the average uncertainty is *NOT* the uncertainty of the average. And you just blew it off.

Tell me again how the average uncertainty is the uncertainty of the average and that is why Frank is wrong!

bigoilbob
Reply to  Tim Gorman
January 8, 2023 3:26 pm

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.

karlomonte
Reply to  bigoilbob
January 8, 2023 8:34 pm

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.

bigoilbob
Reply to  karlomonte
January 9, 2023 6:41 am

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.

karlomonte
Reply to  bigoilbob
January 9, 2023 7:11 am

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.

karlomonte
Reply to  Tim Gorman
January 6, 2023 9:17 am

Are the Fluke engineers psychic and know the use conditions ahead of time of all the instruments they manufacture? Must be…

Jim Gorman
Reply to  Tim Gorman
January 5, 2023 7:02 am

That multiple measurements of the same thing with the same device is needed!

Tim Gorman
Reply to  Bellman
January 4, 2023 1:00 pm

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.

Bellman
Reply to  Tim Gorman
January 4, 2023 1:47 pm

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.

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, then the interval y u(y) to y + u(y) is expected to encompass approximately 68 % of the distribution of values that could reasonably be attributed to the value of the quantity Y of which y is an estimate. This implies that it is believed with an approximate level of confidence of 68 % that Y is greater than or equal to yu(y), and is less than or equal to y + u(y), which is commonly written as Y= y ± u(y).

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.

Jim Gorman
Reply to  Bellman
January 5, 2023 4:44 am

If the probability distribution 

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.

Bellman