Plus or Minus Isn’t a Question

Guest Essay by Kip Hansen —10 December 2022

“In mathematics, the ± sign [or more easily, +/-] is used when we have to show the two possibilities of the desired value, one that can be obtained by addition and the other by subtraction. [It] means there are two possible answers of the initial value. In science it is significantly used to show the standard deviation, experimental errors and measurement errors.”  [ source ]  While this is a good explanation, it is not entirely correct.  It isn’t that there are two possible answers, it is that the answer could be as much as or as little as the “two possible values of the initial value” – between the one with the absolute uncertainty added and the one with the absolute uncertainty subtracted.

[ Long Essay Warning:  This is 3300 words – you might save it for when you have time to read it in its entirety – with a comforting beverage in your favorite chair in front of the fireplace or heater.]

When it appears as “2.5 +/- 0.5 cm”, it is used to indicate that the central value “2.5” is not necessarily the actually the value, but rather that the value  (the true or correct value) lies between the values “2.5 + 0.5” and “2.5 – 0.5”, or fully stated calculated “The value lies between 3 cm  and 2 cm”.  This is often noted to be true to a certain percentage of probability, such as 90% or 95% (90% or 95% confidence intervals).  The rub is that the actual accurate precise value is not known, it is uncertain; we can only correctly state that the value lies somewhere in that range — but only “most of the time”. If the answer is to 95% probability, then 1 out of 20 times, the value might not lie within the range of the upper and lower limits of the range, and if 90% certainty, then 1 out of ten times the true value may well lie outside the range. 

This is important.  When dealing with measurements in the physical world,  the moment the word “uncertainty” is used, and especially in science, a vast topic has been condensed into a single word.  And, a lot of confusion. 

Many of the metrics presented in many scientific fields are offered as averages, as the arithmetic  or probabilistic averages (usually ‘means’).  And thus, when any indication of uncertainty or error is included, it is many times not the uncertainty of the mean value of the metric, but the uncertainty of the mean of the values.    This oddity alone is responsible for a lot of the confusion in science. 

That sounds funny, doesn’t it.  But there is a difference that becomes important.  The mean value of a set of measurements is given in the formula: 

So, the average—the arithmetic mean—by that formula itself carries with it the uncertainty of the original measurements (observations).  If the original observations look like this:  2 cm +/- 0.5 cm then the value of the mean will have the same form:  1.7 cm +/- the uncertainty. We’ll see how this is properly calculated below.

In modern science, there has developed a tendency to substitute instead of that, the “uncertainty of the mean” – with a differing definition that is something like “how certain are we that that value IS the mean?”.  Again, more on this later.

Example: Measurements of high school football fields, made rather roughly to the nearest foot or two (0.3 to 0.6 meters),  say by counting the yardline tick marks on the field’s edge, give a real measurement uncertainty of +/- 24 inches.  By some, this could be averaged to produce a mean of measurements of many high school football fields by a similar process with the uncertainty of the mean reportedly reduced to a few inches.   This may seem trivial but it is not.  And it is not rare, but more often the standard.   The pretense that the measurement uncertainty (sometimes stated as original measurement error) can be reduced by an entire order of magnitude by stating it as the “uncertainty of the mean” is a poor excuse for science.  If one needs to know how certain we are about the sizes of those football fields, then we need to know the real original measurement uncertainty.   

The trick here is switching from stating the mean with its actual original measurement uncertainty (original measurement error) replacing it with the uncertainty of the mean.  The new much smaller uncertainty of the mean is a result of one of two things:  1)  it is the Product of Division or 2) Probability (Central Limit Theory).

Case #1, the football field example is an instance of:  a product of division.  In this case, the uncertainty is no longer about the length of the, and any of the, football fields.  It is only how certain we are of the arithmetic mean, which is usually only a function of how many football fields were included in the calculation.  The original measurement uncertainty has been divided by the number of fields measured in a mockery of the Central Limit Theory. 

 Case#2: Probability and Central Limit Theorem.  I’ll have to leave that topic for the another part in this series – so, have patience and stay tuned.

Now, if arithmetical means are all you are concerned about – maybe you are not doing anything practical or just want to know, in general, how long and wide high school football fields are because you aren’t going to actually order astro-turf to cover the field at the local high school, you just want a ball-park figure (sorry…).  So, in that case, you can go with the mean of field sizes which is about  57,600 sq.ft (about 5351 sq. meters), unconcerned with the original measurement uncertainty.    And then onto the mean of the cost of Astro-turfing a field.  But, since “Installation of an artificial turf football field costs between $750,000 to $1,350,000” [ source ], it is obvious that you’d better get out there with surveying-quality measurement tools and measure your desired field’s exact dimensions, including all the area around the playing field itself you need to cover.  As you can see, the cost estimates have a range of over half a million dollars.      

We’d write that cost estimate as a mean with an absolute uncertainty  — $1,050,000 (+/- $300,000).  How much your real cost would be would depends on a lot of factors.  At the moment, with no further information and details, that’s what we have….the best estimate of cost is in there somewhere —> between $750,000 and $1,350,000 – but we don’t know where.  The mean $1,050,000 is not “more accurate” or “less uncertain”.    The correct answer, with available data, is the RANGE.

Visually, this idea is easily illustrated with regards to GISTEMPv4

The absolute uncertainty in GISTEMPv4 was supplied by Gavin Schmidt.  The black trace, which is a mean value,  is not the real value.  The real value for the year 1880 is a range—about 287.25° +/- 0.5°.  Spelled out properly, the GISTEMP in 1880 was somewhere between 286.75°C and 287.75°C.    That’s all we can say.  GISTEMPv4 mean for 1980, one hundred years later, still fits inside that range with the uncertainty ranges of both years overlapping by about 0.3°C; meaning it is possible that the mean temperature had not risen at all.  In fact, uncertainty ranges for Global Temperature overlap until about 2014/2015. 

(Correction: Embarrassingly, I have inadvertently used degrees C in the above paragraph when it should be K — which in proper notation doesn’t require a degree symbol. The values are eyeballed from the graph. Some mitigation is Gavin uses K and C in his original quote below. The graph should also be mentally adjusted to K. h/t to oldcocky! )

The quote from Gavin Schmidt on this exact point: 

“But think about what happens when we try and estimate the absolute global mean temperature for, say, 2016. The climatology for 1981-2010 is 287.4±0.5K, and the anomaly for 2016 is (from GISTEMP w.r.t. that baseline) 0.56±0.05ºC. So our estimate for the absolute value is (using the first rule shown above) is 287.96±0.502K, and then using the second, that reduces to 288.0±0.5K. The same approach for 2015 gives 287.8±0.5K, and for 2014 it is 287.7±0.5K. All of which appear to be the same within the uncertainty. Thus we lose the ability to judge which year was the warmest if we only look at the absolute numbers.” [ source – repeating the link ]

To be absolutely correct, the global annual mean temperatures have far more uncertainty than is shown or admitted by Gavin Schmidt, but at least he included the known original measurement error (uncertainty) of the thermometer-based temperature record.  Why is that?  Why is it greater than that? ….  because the uncertainty of a value is the cumulative uncertainties of the factors that have gone into calculating it, as we will see below  (and +/- 0.5°C is only one of them).

Averaging Values that have Absolute Uncertainties

Absolute uncertainty. The uncertainty in a measured quantity is due to inherent variations in the measurement process itself. The uncertainty in a result is due to the combined and accumulated effects of these measurement uncertainties which were used in the calculation of that result. When these uncertainties are expressed in the same units as the quantity itself they are called absolute uncertainties. Uncertainty values are usually attached to the quoted value of an experimental measurement or result, one common format being: (quantity) ± (absolute uncertainty in that quantity).  [ source ]

Per the formula for calculating a arithmetic mean above, first we add all the observations (measurements) and then we divide the total by the number of observations.

How do we then ADD two or more uncertain values, each with its own absolute uncertainty? 

The rule is:

When you add or subtract the two (or more) values to get a final value, the absolute uncertainty [given as “+/- a numerical value”] attached to the final value is the sum of the uncertainties. [ many sources: here or here]
For example:
5.0 ± 0.1 mm + 2.0 ± 0.1 mm = 7.0 ± 0.2 mm
5.0 ± 0.1 mm – 2.0 ± 0.1 mm = 3.0 ± 0.2 mm

You see, it doesn’t matter if you add or subtract them, the absolute uncertainties are added.   This applies no matter how many items are being added or subtracted.  In the above example, if 100 items (say sea level rise at various locations) each with its own absolute measurement uncertainty of 0.1 mm, then the final value would have an uncertainty of +/- 10 mm (or 1 cm).

This is principle easily illustrated in a graphic:

In words:  ten plus or minus one PLUS twelve plus or minus one EQUALS twenty-two plus or minus two.  Ten plus or minus 1 really signifies the range eleven down to nine and twelve plus or minus one signifies the range thirteen down to eleven.  Adding the two higher values of the ranges, eleven and thirteen,  gives twenty-four which is twenty-two (the sum of ten and twelve on the left) plus two, and adding the too lower values of the ranges, nine and eleven, gives the sum of twenty which is twenty-two minus two.  Thus our correct sum is twenty-two plus or minus two, shown at the top right. 

Somewhat counter-intuitively, the same is true if one subtracts one uncertain number from another, the uncertainties (the +/-es) are added, not subtracted, giving a result (the difference) more uncertain than either the minuend (the top number) or the subtrahend (the number being subtracted from the top number).  If you are not convinced, sketch out your own diagram as above for a subtraction example.

What are the implications of this simple mathematical fact? 

When one adds (or subtracts) two values with uncertainty, one adds (or subtracts) the main values and adds the two uncertainties (the +/-es) in either case (addition or subtraction) – the uncertainty of the total (or difference) is always higher than the uncertainty of either original values.

How about if we multiply? And what if we divide?

If you multiply one value with absolute uncertainty by a constant (a number with no uncertainty)

The absolute uncertainty is also multiplied by the same constant.
eg.   2 x (5.0 ± 0.1 mm ) = 10.0 ± 0.2 mm

Likewise, if you wish to divide a value that has an absolute uncertainty by a constant (a number with no uncertainty), the absolute uncertainty is divided by the same amount.  [ source ]

So, 10.0 mm +/- 0.2mm divided by 2 = 5.0 +/- 0.1 mm. 

Thus we see that the arithmetical mean of the two added measurements (here we multiplied but it is the same as adding two–or two hundred–measurements of 5.0 +/- 0.1 mm) is the same as the uncertainty in the original values, because, in this case, the uncertainty of all (both) of the measurement is the same (+/- 0.1).  We need this to evaluate averaging – the finding of a arithmetical mean.

So, now let’s see what happens when we find a mean value of some metric.  I’ll use a tide gauge record as tide gauge measurements are given in meters – they are addable (extensive property) quantities.  As of October 2022, the Mean Sea Level at The Battery was 0.182 meters (182 mm, relative to the most recent Mean Sea Level datum established by NOAA CO-OPS.)  Notice that here is no uncertainty attached to the value.  Yet, even mean sea levels relative to the Sea Level datum must be uncertain to some degree.  Tide gauge individual measurements have a specified uncertainty of +/- 2 cm (20 mm).   (Yes, really.  Feel free to read the specifications at the link). 

And yet the same specifications claim an uncertainty of only  +/- 0.005 m (5 mm) for monthly means.  How can this be?  We just showed that adding all of the individual measurements for the month would add all the uncertainties (all the 2 cms) and then the total AND the combined uncertainty would both be divided by the number of measurements – leaving again the same 2 cm as the uncertainty attached to the mean value.

The uncertainty of the mean would not and could not be mathematically less than the uncertainty of the measurements of which it is comprised.

How have they managed to reduce the uncertainty to 25% of its real value?  The clue is in the definition:  they correctly label it the “uncertainty of the mean” — as in “how certain are we about the value of the arithmetical mean?”    Here’s how they calculate it: [same source]

“181 one-second water level samples centered on each tenth of an hour are averaged, a three standard deviation outlier rejection test applied, the mean and standard deviation are recalculated and reported along with the number of outliers.  (3 minute water level average)”    

Now you see, they have ‘moved the goalposts’ and are now giving not the uncertainty of the value of mean at all, but the “standard deviation of the mean” where “Standard deviation is a measure of spread of numbers in a set of data from its mean value.” [ source or here ].  It is not the uncertainty of the mean.  In the formula given for arithmetic mean (image a bit above), the mean is determined by a simple addition and division process.  The numerical result of the formula for the absolute value (the numerical part not including the +/-) is certain—addition and division produce absolute numeric values —  there is no uncertainty about that value.  Neither is there any uncertainty about the numeric value of the summed uncertainties divided by the number of observations. 

Let me be clear here:  When one finds the mean of measurements with known absolute uncertainties, there is no uncertainty about the mean value or its absolute uncertainty.  It is a simple arithmetic process.

The mean is certain.  The value of the absolute uncertainty is certain. We get a result such as:

3 mm +/- 0.5 mm

Which tells us that the numeric value of the mean is a range from 3 mm plus 0.5 mm to 3 mm minus 0.5 mm or the 1 mm range:  3.5 mm to 2.5 mm.

The range cannot be further reduced to a single value with less uncertainty.

And it really is no more complex than that.

# # # # #

Author’s Comment:

I heard some sputtering and protest…But…but…but…what about the (absolutely universally applicable) Central Limit Theorem?  Yes, what about it?  Have you been taught that it can be applied every time one is seeking a mean and its uncertainty?  Do you think that is true?  

In simple pragmatic terms, I have showed above the rules for determining the mean of a value with absolute uncertainty — and shown that the correct method produces certain (not uncertain) values for both the overall value and its absolute uncertainty.  And that these results represent a range.

Further along in this series, I will discuss why and under what circumstances the Central Limit Theorem shouldn’t be used at all. 

Next, in Part 2, we’ll look at the cascading uncertainties of uncertainties expressed as probabilities, such as “40% chance of”.

Remember to say “to whom you are speaking”, starting your comment with their commenting handle,  when addressing another commenter (or, myself). Use something like “OldDude – I think you are right….”. 

Thanks for reading.

# # # # #

Epilogue and Post Script:

Readers who have tortured themselves by following all the 400+ comments below — and I assure you, I have read every single one and replied to many — can see that there has been a lot of pushback to this simplest of concepts, examples, and simple illustrations.

Much of the problem stems from what I classify as “hammer-ticians”. Folks with a fine array of hammers and a hammer for every situation. And, by gosh, if we had needed a hammer, we’d have gotten not just a hammer but a specialized hammer.

But we didn’t need a hammer for this simple work. Just a pencil, paper and a ruler.

The hammer-ticians have argued among themselves as to which hammer should have been applied to this job and how exactly to apply it.

Others have fought back against the hammer-ticians — challenging definitions taken only from the hammer-tician’s “Dictionary of Hammer-stitics” and instead suggesting using normal definitions of daily language, arithmetic and mathematics.

But this task requires no specialist definitions not given in the essay — they might as well have been arguing over what some word used in the essay would mean in Klingon and then using the Klingon definition to refute the essay’s premise.

In the end, I can’t blame them — in my youth I was indoctrinated in a very specialist professional field with very narrow views and specialized approaches to nearly everything (intelligence, threat assessment and security, if you must ask) — ruined me for life (ask my wife). It is hard for me even now to break out of that mind-set.

So, those who have heavily invested in learning formal statistics, statistical terminology and statistical procedures might be unable to break out of that narrow canyon of thinking and unable to look at a simple presentation of a pragmatic truth.

But hope waxes eternal….

Many have clung to the Central Limit Theorem — hoping that it will recover unknown information from the past and resolve uncertainties — that I will address that in the next part of this series.

Don’t hold your breathe — it may take a while.

# # # # #

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JCM
December 9, 2022 10:16 pm

There is no need for averages, a notion of certainty, absolutes, or abstraction. Only reality.

The data is clear that there is no such thing that the outgoing radiation diminishes or remains constant and the surface warms up due to some incorrect greenhouse gas enhancement hypothesis, or because of the outcomes of CO2 doubling experiments conducted with never validated models.

The faux environmentalists ignore the fact that the outgoing radiation is governed by the upper tropospheric humidity field which cannot be modeled by any (deterministic) global climate model.

The nature of environmental systems, down to miniscule anomalies, will not be determined by finite computation or mechanistic interpretations. Rather, the system can only be understood at such a level by consciousness and intuition.

The perceived problems are rooted in nature, and it is in nature – with our innate connection thereto – that the solutions will be found.

It is for certain that Earth system intervention will not be successful by our puny technological interventions, statistical sophistry, or ideological standpoints.

JCM
Reply to  JCM
December 9, 2022 11:24 pm

the notion of absolute uncertainty when it comes earth system spatial sampling is boundless and unknowable. Whatever one thinks it is, multiply by a factor n based on one’s philosophy. Common methods do not apply to the infinitesimal anomalies for which most of these debates involve. Furthermore, in typical environmental variables, a geometric mean is more appropriate. But even then, still nonsense when it comes to infinitesimal residuals subtracted from a baseline. It’s wacky stats – subject to judgement by the analyst. There is no rule or convention that will give the answer. It comes down to a reasonable sense of the phenomenon; quantification, a subject which cannot be adequately regulated by rule.

Last edited 1 month ago by JCM
JCM
Reply to  JCM
December 10, 2022 1:16 am

Whatever the downvoters are thinking is moot. The absolute uncertainty of temperature anomaly is infinite (which is what this is all about). The bounds one chooses to impose, and the implications drawn thereupon, are judgements. The limit of infinity is up to you. You will not find the answers in methodical tricks. One can only establish a lower bound – full stop.

Richard M
Reply to  JCM
December 10, 2022 2:48 pm

Exactly. As discovered by Dr. F Miskolczi, the opacity of the atmosphere remains nearly constant. Water vapor and CO2 work together in a complex manner to achieve this balance. When CO2 is increased much of the DWIR leads to increased evaporation. As Dr. William Gray pointed out, this accelerates convection which pushes the air higher into a colder part of the atmosphere. The result is more condensation reducing the high atmosphere water vapor. The two main components of upward radiation through the atmosphere work together to keep the energy flow consistent.

Yes, we get more rainfall as CO2 increases. Exactly what is needed to balance the growing of plants and arable land.

There is no warming effect.

Nick Stokes
December 9, 2022 10:51 pm

“When it appears as “2.5 +/- 0.5 cm”, it is used to indicate that the central value “2.5” is not necessarily the actually the value, but rather that the value (the true or correct value) lies between the values “2.5 + 0.5” and “2.5 – 0.5”, or fully stated calculated “The value lies between 3 cm and 2 cm”.”

It actually doesn’t mean that in statistical usage. It means there is a distribution of values about 2.5 with standard deviation 0.5. That means that there is about a 2/3 probability that the value lies in the range. It is rare that you can be certain that a value lies in a range like 2 to 3. 

So the essay drifts into nonsense about absolute uncertainty. The notion of absolute temperature (as opposed to anomaly) is a red herring here. 

I’d strongly recommend, if you want to talk about “absolute uncertainty”, find a reference to where some more authoritative person talks about it, and be rather careful in quoting what they say. Otherwise you’ll only have a straw man.

Nick Stokes
Reply to  Nick Stokes
December 10, 2022 12:04 am

 find a reference to where some more authoritative person talks about it”

I spoke too hastily here; I see that you have done that. But I don’t think your references are very authoritative. The definition of “absolute uncertainty” that you quote is just distinguishing from relative uncertainty, given as a % of the mean. But the arithmetic that you quote comes from dubious sources.

In fact error should add in quadrature. So adding 5.0 ± 0.1 mm + 2.0 ± 0.1 mm should give
7.0 ± 0.1414mm ie sqrt(.01+.01).


Nick Stokes
Reply to  Nick Stokes
December 10, 2022 12:22 am

I should follow my own prescription there and give authority. One commonly quoted, here and elsewhere, is the GUM; “Guide to the expression of uncertainty in measurement”

In sec 2, it says:
“2.3.1 standard uncertainty
uncertainty of the result of a measurement expressed as a standard deviation “

“2.3.4 combined standard uncertainty
standard uncertainty of the result of a measurement when that result is obtained from the values of a number of other quantities, equal to the positive square root of a sum of terms, the terms being the variances or covariances of these” (ie quadrature)

Tim Gorman
Reply to  Nick Stokes
December 10, 2022 5:03 am

You *always* ignore the fact that the GUM assumes multiple measurements of the same thing which generates a normal curve where the random errors cancel. Thus the mean becomes the average of the stated values whose uncertainty is the standard deviation of the stated values.

Note carefully the phrase “the positive square root of a sum of terms, the terms being the variances or covariances of these”. A single measurement cannot have a variance since there is only one data point. What that single measurement can have is an uncertainty interval which one can treat as a variance.

If the result of a measurement is a result of the value of a number of other quantities (i.e. multiple single measurements of multiple different things) then the standard uncertainty is the SUM of the uncertainty intervals for each single measurement (done using the root-sum-square addition method). This is *NOT* the same thing as the standard deviation of the stated values, it is a sum of the uncertainties of the stated values.



karlomonte
Reply to  Tim Gorman
December 10, 2022 7:38 am

Stokes is just another GUM cherry picker.

KB
Reply to  Tim Gorman
December 10, 2022 1:07 pm

Incorrect. You first evaluate the distribution shape of the uncertainties in the calculation.
Often, but not always, they will have approximately normal distributions.
When that is the case, the uncertainties add IN QUADRATURE.

For example, (25.30+/- 0.20) + (25.10 +/- 0.30)

= 50.40 +/- SQRT(0.20^2 + 0.30^2) = +/-0.36

You would report the result as 50.40 +/- 0.36

Tim Gorman
Reply to  KB
December 10, 2022 2:49 pm

KB,

First, you didn’t assume any specific distribution for the uncertainties you give. +/- 0.2 and +/- 0.3 are intervals and you have not assigned any specific probabilities to any specific values in those intervals.

Individual, single measurements of different things have *NO* distribution of uncertainties – e.g. Tmax and Tmin.

Thus there is no distribution shape to evaluate.

You are, like most of the climate astrologers on here, only trained in distributions that are random, i.e. multiple measurements of the same thing which CAN, but not always, result in a random measurement distribution.

How do you evaluate a distribution for one single value of temperature, e.g. Tmax? The only thing you can assume is that there is ONE value in the uncertainty interval which is the true value, meaning the probability of that value being the true value is 1. All the other values have a probability of zero of being the true value.

That is, in essence, what you have followed in your example. If there was a distribution involved with the uncertainty interval values then what is it? Would 0.1 be a higher probability than 0.2 for the first measurement?

KB
Reply to  Tim Gorman
December 11, 2022 8:36 am

I said that you assign the distribution shapes at the outset. You are correct that the rest of my example assumes an approximately normal distribution for both uncertainties.

But they don’t need to be. There are established ways to obtain the standard deviation of a rectangular distribution, a triangular distribution or even a U-shaped distribution.

Having done that, they are added in quadrature just like the normally distributed uncertainties.

Assigning an uncertainty to one individual thermometer reading is easy. Look at its calibration certificate and it should tell you the uncertainty. It should also tell you the effective degrees of freedom.

That calibration uncertainty will take into account the uncertainty on the standard against which it was measured and a number of other individual uncertainties. Because it is a combination of several individual uncertainties it is usually safe to assume the uncertainty is approximately normally distributed.

karlomonte
Reply to  KB
December 11, 2022 8:43 am

Because it is a combination of several individual uncertainties it is usually safe to assume the uncertainty is approximately normally distributed.

A does not imply B here. A single air temperature measurement has no distribution because the sample size is always and exactly equal to one. Tim raised this point and you ignored it.

KB
Reply to  karlomonte
December 11, 2022 1:47 pm

A single air temperature measurement does have a distribution. Every measurement has an associated uncertainty as we well know.
I’ve repeatedly said the same thing in other posts so I have not ignored the point at all.

karlomonte
Reply to  KB
December 11, 2022 6:07 pm

And it is still nonsense.

Clyde Spencer
Reply to  KB
December 10, 2022 3:35 pm

It is generally recommended that the “.40” be truncated to “.4” and the “0.36” be rounded up to “0.4” when the uncertainty is so large as to impact the “4” in all cases. Thus, despite starting with 4 significant figures in the addends, the sum is best characterized with 3 significant figures because of the impact of the large uncertainty.

It can be impossible to “evaluate the distribution shape of the uncertainties” for a small number of samples, despite being able to calculate the SD. However, at the least, one should note that one is assuming the distribution is close enough to being normal that one is justified in the assumption. There is a theorem for calculating a lower-bound of the SD even when the samples are not normally distributed.

Last edited 1 month ago by Clyde Spencer
KB
Reply to  Clyde Spencer
December 11, 2022 8:39 am

Yes there are rounding conventions also, but I left out that discussion in the interests of brevity.

Frankly there is so much wrong with this article on even basic levels that I thought it best to leave that out for the time being.

Clyde Spencer
Reply to  Tim Gorman
December 10, 2022 3:21 pm

What that single measurement can have is an uncertainty interval which one can treat as a variance.

Which, by convention, is displayed with the last significant figure implying an uncertainty of ±5 in the next position to the right.

Nick Stokes
Reply to  Kip Hansen
December 10, 2022 10:20 am

Kip,
You’re misreading the notion of absolute uncertainty, at least in the definition you quote. It isn’t a definition of a different kind of uncertainty. It is just making the distinction between the ±  quantification of uncertainty, and relative uncertainty, which is expressed as a % or fraction.

“Temperature records, rounded to the nearest whole degree, have a known absolute uncertainty of +/- 0.5°C.”
That is not a useful notion of uncertainty, which is actually greater. If you see a temperature quoted as 13°C, then certainly whoever wrote it down thought 13 was the closest. But you can’t be certain that it wasn’t, as a matter of measurement accuracy, actually closer to 14, or even 15. That is why meaningful physical uncertainty is expressed via a distribution, with a standard deviation.

Tim Gorman
Reply to  Nick Stokes
December 10, 2022 2:40 pm

I’m not sure where you got these definitions from. In my real world experience absolute uncertainty is that which you can define for any measurement of a physical attribute and does not depend on the size of the measurand.. Relative uncertainty is that uncertainty which depends on the size of the measurand and is not typically measured directly.

The uncertainty in the length of a board is directly x +/- u. If the board was 10′ long the uncertainty would still be the same -> u.

The uncertainty of the volume of a cylinder is (2 x u(R)/R) + (u(H)/H) where u(R)/R and u(H)/H are “relative uncertainties. The larger the volume of the cylinder the large the uncertainty will be.

bdgwx
Reply to  Kip Hansen
December 10, 2022 7:56 pm

Kip Hansen said: “It is called an “absolute uncertainty” because we know the range — it must be at least 0.5°”

It is called an “absolute uncertainty” at least in so far as the definition you provided because the units are in degrees (ie. K or C) and not %.

If the the 0.5 uncertainty you speak of here is bounded at -0.5 and +0.5 it is called a uniform or rectangular distribution. It an be converted into a standard uncertainty via the canonical 1/sqrt(3) multiplier thus yielding 0.289 K. You can then plug this standard uncertainty u = 0.289 K into the law of propagation of uncertainty formula or one of its convenient derivations.

Last edited 1 month ago by bdgwx
Tim Gorman
Reply to  bdgwx
December 11, 2022 8:20 am

bdgwx,

How many true values are there inside an uncertainty interval? If there is systematic uncertainty how does that impact the probability distribution inside the uncertainty interval?

bdgwx
Reply to  Kip Hansen
December 11, 2022 3:42 pm

KH said: “Why oh Why would we want to convert a known numerical perfectly correct and accurate numerical absolute uncertain of a measurement to the statistical animal called “standard uncertainty”?”

Because it is required to propagate it through measurement models (functions or equations) that depend on it.

karlomonte
Reply to  bdgwx
December 11, 2022 6:08 pm

And more nonsense here.

AndyHce
Reply to  Kip Hansen
December 10, 2022 8:31 pm

Kip,
In regard to higher uncertainties:
This may be outside the considerations of the topic but there are one or more other uncertainties in many measurements, not generally considered as far as I can see, A simple example is a measuring tape for length of objects. I believe a thermometer would be similar. Perhaps, for practical purposes these uncertainties are considered too small to be relevant but they must be real?

As example, one measures a board of lumber with a tape measure. Perhaps it is reasonable to read the measurement to 1/8″, thus it would be quoted as x and y/8 inches +/- 1/16 inch. There are three assumptions in such a quote, none of which is true.

No real board is a perfect geometric figure. Just where one places the tape on the board can determine what one reads. If one is measuring the length of a 6″ wide board, how many different readings might one get that depend on just where along the 6 inch ends one places the tape? Further, from any of those possible placements, what is the variation in angle along the board necessary to change the length read by 1/8″?

The tape itself cannot be perfect. There is some difference, probably too small to see on a good tape, but still there, between every 1/8″, 1/4″, … 1″. On a long enough board those might mean 1/8 “ error or more. Measured with another tape the reading might be 1/8″ or more different. For lumber this might be irrelevant nonsense but what about sea level quoted in mm to three decimal places?

I used “error” in the last paragraph because it seems to me the instrument itself (tape measure, thermometer) is in error rather than “uncertain” relative to the standard, which is itself most likely not absolute to the quantum uncertainty level, but may be the best currently possible.

ThinkingScientist
Reply to  Kip Hansen
December 11, 2022 12:51 pm

Kip, why do we need to have absolute uncertainty when we are dealing with relative changes in temp over time?

Clyde Spencer
Reply to  Nick Stokes
December 10, 2022 3:57 pm

While what you say is true, the accepted procedure is to record what it appears to be closest to, and the implication being that unless a gross error was made, it will have an uncertainty of ±0.5 units. Of course we can never be absolutely certain that the numbers weren’t transposed, or invented because the measurer didn’t want to go out in the rain, snow, dust storm, heat, you name it. Thus, all uncertainties should be considered a lower bound, with the small, but finite probability the uncertainties could possibly be larger. However, under best practices, a temperature recorded to the nearest degree implies an uncertainty of ±0.5 degree.

That is why meaningful physical uncertainty is expressed via a distribution, with a standard deviation.

A single measurement does not have a standard deviation! Remember the old saying: “You only have one chance to make a first good impression.” Similarly, you only have one chance to record a temperature in a time-series. After the first reading, it is a different parcel of air that gets measured. Or, as Heraclitus said, “You cannot step into the same river twice, for other waters are continually flowing on.”

Tim Gorman
Reply to  Clyde Spencer
December 11, 2022 8:24 am

Clyde:

However, under best practices, a temperature recorded to the nearest degree implies an uncertainty of ±0.5 degree.”

I assume that by this you mean any systematic bias has been reduced to a value smaller than the resolution of the measurement.

This is *not* a good assumption for field sited temperature measuring stations where the systematic bias (e.g. calibration drift, etc) is highly likely to be greater than the actual resolution of the instrument itself.

A single measurement does not have a standard deviation! “

Hallelujah!

Clyde Spencer
Reply to  Tim Gorman
December 11, 2022 12:34 pm

This is *not* a good assumption for field sited temperature measuring stations where the systematic bias (e.g. calibration drift, etc) …

Strictly speaking, you are right. However, the unstated assumption for those claiming they can improve precision with many measurements is that those systematic biases don’t exist.

KB
Reply to  Kip Hansen
December 10, 2022 1:00 pm

This is a rectangular distribution and there is a standard way of dealing with it. imagine we have a temperature reading of 280K, reported to a precision of 1K.

We say that any value between 279.5 and 280.5K is equally likely. It is a rectangular distribution, not a normal distribution.

The standard uncertainty is the semi-range divided by SQRT(3). In this case it would be 0.5/SQRT(3) = 0.289.

Let’s say we want to find the difference between that and another temperature reading of 270K, again reported with a precision of 1K.

The estimate of the difference in temperature is of course 280-270 = 10K

The uncertainty is SQRT(0.289^2 + 0.289^2) = +/- 0.408.

That 0.408 is the one-standard deviation uncertainty. To obtain the approximate 95% confidence range you would multiply by 2.

0.408 x 2 = 0.816

You should then round this value and obtain the result

10.0 +/- 0.8 K

Tim Gorman
Reply to  KB
December 10, 2022 2:54 pm

Uncertainty is not typically a rectangular distribution. Not all values are equally probable for being the “true value”, especially when you have only one measurement to look at. There will be one value that has a probability of 1 and all the others will have a probability of zero. The issue is that you don’t know which value has a probability of 1.

You are still stuck in the box of assuming you have multiple values in a distribution from which you can evaluate what the distribution might be. With just one measurement and one uncertainty interval you do not have multiple values upon which to assess a probability distribution for the uncertainty. All you can do is assume that it is unknown.

KB
Reply to  Tim Gorman
December 11, 2022 8:52 am

I didn’t say uncertainty is typically a rectangular distribution. I said in this particular case of digitally rounding a thermometer reading it could be treated like that.

If the reading is (say) 50 degrees, the distribution of the uncertainty component due to digital rounding is rectangular. It can be anywhere between 49.5 and 50.5 with equal probability across the range.

This is a separate uncertainty component to the bias on the thermometer.

bdgwx
Reply to  Kip Hansen
December 11, 2022 5:16 am

KP said: “However, we are looking for an arithmetic mean which is found using simple arithmetic. It does not involve statistical processes or any such — its results are precise.”

The measurement model Y is found using simple arithmetic.

The uncertainty of the measurement model u(Y) is found using statistical processes or established propagation procedures like what is found in JCGM 100:2008 section 5.

This is true even when the measurement model Y is but a trivial sum or average.

Last edited 1 month ago by bdgwx
karlomonte
Reply to  bdgwx
December 11, 2022 7:05 am

Did you even read the main article? Apparently not.

Tim Gorman
Reply to  bdgwx
December 11, 2022 8:29 am

bdgwx:

The uncertainty of the measurement model u(Y) is found using statistical processes”

What do you statistically analyze when the data set consists of one value?

Even the GUM, see annex H, assumes that you have an SEM – which requires multiple measurements of the same thing which can generate a standard deviation.

If you have one measurement then there is no standard deviation or SEM, only a u_c, an estimate of the uncertainty. When you have multiple single measurements those u_c’s add, either directly or by root-sum-square.

TimTheToolMan
Reply to  Kip Hansen
December 10, 2022 6:34 pm

You are still stuck in the box of assuming you have multiple values in a distribution from which you can evaluate what the distribution might be.

This is the most important idea that neither Nick nor KB have attempted to address.

Tim Gorman
Reply to  TimTheToolMan
December 11, 2022 8:39 am

They can’t address it. It’s outside their little box that all measurements have only random error that cancels and all measurements are of the same thing.

KB
Reply to  TimTheToolMan
December 11, 2022 8:53 am

I’ve posted multiple times to address this very issue.

karlomonte
Reply to  KB
December 11, 2022 9:03 am

How do you manufacture a sampling population from exactly one measurement?

KB
Reply to  Kip Hansen
December 11, 2022 8:41 am

What you are describing is a simple digital rounding uncertainty. There is an established way of dealing with those, but it is nowhere near what you describe.

karlomonte
Reply to  KB
December 11, 2022 9:04 am

This is only one component of combined uncertainty, there will be others. It does not tell you the “error”.

KB
Reply to  karlomonte
December 11, 2022 1:52 pm

Correct, you should have a full uncertainty budget for the temperature measurement. The digital rounding uncertainty is but one component of this uncertainty budget.
However I am trying to deal with things one step at a time.

karlomonte
Reply to  KB
December 11, 2022 6:11 pm

Kip is using combined uncertainties that are already known, not doing a UA.

bdgwx
Reply to  Kip Hansen
December 11, 2022 1:50 pm

Kip,

It doesn’t matter that much or at least as much as you think. The reason is because the focus is on the monthly global average temperature and the uncertainty on that value. The formula for the propagation of uncertainty based on the 0.5 rounding uncertainty through a measurement model that computes an average is u(avg) = u(x)/(sqrt(N)*sqrt(3)). A typical global monthly average can incorporate 50,000 values or more. If those values are uncorrelated then that 0.5 C rounding uncertainty propagates as 0.5/(sqrt(3) * sqrt(50000)) = 0.001 C when averaging them. This is just the way the math works out whether some people refuse to accept it or not. My sources here are JCGM 100:2008, NIST TN 1297, and others and the NIST uncertainty machine.

And don’t hear what hasn’t been said. No one is saying the uncertainty of individual observations doesn’t matter. They do. No one is saying the individual observations themselves doesn’t matter. They do. No one is saying that the 0.5 rounding uncertainty is the only uncertainty that needs to be considered. It’s not. Nor is anyone saying the monthly global average temperature is computed with a trivial average of individual temperature observations. It’s not. Nor is anyone saying that the uncertainties are uncorrelated. They aren’t. Nor is anyone saying that the total combined uncertainty is 0.001 C or millikelvin or any of the other nonsense the contrarians are creating as strawmen. It’s not. So when the contrarians start trolling this post, and they will, I want you know what wasn’t said or claimed.

Last edited 1 month ago by bdgwx
karlomonte
Reply to  bdgwx
December 11, 2022 6:12 pm

And another spin of the trendology hamster wheel…

bnice2000
Reply to  Nick Stokes
December 10, 2022 2:44 am

Only in certain circumstances, when you are measuring the same thing with the same instrument.

Seem Nick skips some courses on basic statistics. !

Bellman
Reply to  bnice2000
December 10, 2022 2:56 am

Where do people here come up with that. There is no rule I’ve ever been showen that says there is a difference in propagating uncertainty when measuring the same thing rather than different things. The rules for propagating uncertainties in quadrature are normally used when you are adding or subtracting different things.

Tim Gorman
Reply to  Bellman
December 10, 2022 7:13 am

You didn’t even bother to read what Stokes posted did you?

“2.3.4 combined standard uncertainty
standard uncertainty of the result of a measurement when that result is obtained from the values of a number of other quantities, equal to the positive square root of a sum of terms, the terms being the variances or covariances of these” (ie quadrature) (bolding mine, tpg)

When you have single measurements of different things none of those measurements have a variance since there is only one data point. You need to have multiple data points in order to have a variance!

In this case the uncertainty of those single measurements becomes the variance. Those single measurements become the “number of other quantities” and their uncertainties (variances) add by root-sum-square.

You add by root-sum-square because there will probably be *some* cancellation. If this cannot be justified then a direct addition of the uncertainties should be done.

Please note that 2.25 does *NOT* say the standard deviation of the sample means becomes the uncertainty but, instead, the root-sum-square of the uncertainty of the terms becomes the standard uncertainty.

Bellman
Reply to  Tim Gorman
December 10, 2022 8:27 am

What do you think “other quantities” means? The rules for propagating uncertainties or errors are same regardless of if the different quantities are different measurements if the same thing or different measurements if different things.

Tim Gorman
Reply to  Bellman
December 10, 2022 2:56 pm

What do *YOU* think “ sum of terms” means? If there are multiple terms, i.e. other quantities, you *still* add the uncertainties by root-sum-square!

Bellman
Reply to  Tim Gorman
December 10, 2022 3:15 pm

I’m getting lost in your argument. My point is it doesn’t matter if the things being added are different measurements of the same thing or multiple different things. I’m really not sure what point you think you are making.

Tim Gorman
Reply to  Bellman
December 11, 2022 8:52 am

My point is it doesn’t matter if the things being added are different measurements of the same thing or multiple different things. I’m really not sure what point you think you are making.”

This statement only highlights that you haven’t learned anything about measurements no matter how much you cherry pick bits from different places.

Multiple measurements of the same thing *usually* (but not always) forms a probability distribution that approaches a Gaussian. Thus you can assume that positive errors are cancelled by negative errors. This allows you to estimate the true value by averaging the stated values and calculate an estimate of the uncertainty using the standard deviation of stated values.

Multiple measurements of different things probably do *NOT* generate a distribution amenable to statistical analyse where uncertainties cancel. I’ll give you the old example of two boards, one 2′ +/- 1″ and one 10′ +/- 0.5″. Exactly what kind of distribution do those two measurements represent? What does their mean represent? It certainly isn’t a “true value” of anything!

Bellman
Reply to  Tim Gorman
December 11, 2022 6:26 pm

Multiple measurements of the same thing *usually* (but not always) forms a probability distribution that approaches a Gaussian.

I think you have that back to front. All the measurements are coming from a probability distribution that may or may not be Gaussian. If you took a large number of measurements you may be able to estimate what that distribution is, but the measurements are not what forms the distribution. Even if you only take one or two measurements, they are still coming from the distribution, but you won;t be able to tell what it is just by looking at them.

Thus you can assume that positive errors are cancelled by negative errors.

Once again, this has nothing to do with the distribution being Gaussian, or even symmetric.

This allows you to estimate the true value by averaging the stated values and calculate an estimate of the uncertainty using the standard deviation of stated values.

That’s one way, but in the discussions about propagating the measurement uncertainties, the assumption is you already know the uncertainties, and can estimate the uncertainties of the mean from them. If you don;t you are going to have to take a large number of measurements just to get a realistic estimate of the standard deviation.

Bellman
Reply to  Bellman
December 11, 2022 6:53 pm

Continued.

Multiple measurements of different things probably do *NOT* generate a distribution amenable to statistical analyse where uncertainties cancel.

You just keep saying this as if that makes it true. Multiple measurements of different things will still form a probability distribution which for the most part will be the same as the distribution of the population. The measurement errors just add a bit of fuzziness to this and increase the variance a little.

However, the real problem here is that you keep conflating these two things. The estimate of the measurement uncertainty with the variation of the population. If, as we have been discussing, you only want the measurement uncertainty then the distribution you want is the distribution of the errors around each thing. E.g.

I’ll give you the old example of two boards, one 2′ +/- 1″ and one 10′ +/- 0.5″.

Rather than mess about with antique units I’ll convert that to 20 ± 1cm and 100 ± 0.5cm.

Here you have one measurement with an error taken from the ±1cm distribution and another taken from the ±0.5cm distribution. The actual error for each could be anything within that range, but you won’t know what it is because you don’t know the actual size of the boards. Maybe you first board has a reading of 20.8cm, and the other 99.6cm. Your rather pointless average is 60.2cm. There’s little point in worrying about the standard error of the mean given the tiny sample size, and the fact I’ve no idea what population you are trying to measure.

But we do know the measurement uncertainties, and with the usual caveats about independence, we can say the measurement uncertainty is sqrt(1^2 + 0.5^2) / 2 ~= 0.56cm. So the actual average of the two planks could be stated as 60.2 ± 0.6 cm.

Exactly what kind of distribution do those two measurements represent?

Impossible to say without an explanation of the population you are taking them from. Two values are just not enough to tell.

What does their mean represent?

Again depends on what you are trying to do. If all you want to know is what is the exact average of those two boards, that’s what it represents. If you are trying to make a statement about a population it represents the best estimate of that populations mean, but with a sample size of two the uncertainties are enormous.

It certainly isn’t a “true value” of anything!

It’s an estimate of the true value of the mean, which ever one you are looking for. If you want an exact average, the use you could put it to is to estimate what the total length of joining the two boards is. It will be the average times two, and the uncertainty will be the uncertainty of the average times two. 120 ± 1.2 cm.

Tim Gorman
Reply to  Bellman
December 12, 2022 7:38 am

 The actual error for each could be anything within that range, but you won’t know what it is because you don’t know the actual size of the boards. “

The error has nothing to do with the length of the boards, it has to do with the uncertainty of the measurement device, at least if we assume the same measuring environment.

An uncertainty interval has no distribution since the probabilities for each value in the interval are not known.

 So the actual average of the two planks could be stated as 60.2 ± 0.6 cm.”

This is EXACTLY what we’ve been trying to teach you for two years. Welcome to the right side.

“Impossible to say without an explanation of the population you are taking them from. Two values are just not enough to tell.”

The uncertainties and the measurements are two different things. How do you tell the distribution inside the uncertainty interval?

You didn’t answer the question about the mean. All you basically said in the word salad was: the mean is the mean! So I’ll ask again – What does their mean represent?

“It’s an estimate of the true value of the mean”

More word salad basically saying: the true value of the mean is the mean! But even that can’t be true if you say there is an SEM or standard deviation around the mean!

 If you want an exact average”

I want to know what use can be made of that *exact average” when you have multiple different things making up your data set.

And you just keep avoiding answering by saying the mean is the mean. So what?

Bellman
Reply to  Tim Gorman
December 12, 2022 9:47 am

The error has nothing to do with the length of the boards,

That wasn’t my point. I’m assuming the uncertainties are independent (though they might not be, in which case you have another source of bias). My point is that if you only have one measurement you cannot know what the true value is and therefore don’t know what the error is.

An uncertainty interval has no distribution since the probabilities for each value in the interval are not known.

Just because you don’t know what it is doesn’t mean it doesn’t exist. All uncertainty intervals have to have some form of a probability distribution. How else could you have a “standard uncertainty” if there was no distribution. You need a distribution to have a standard deviation, you need a standard deviation to have a standard uncertainty.

This is EXACTLY what we’ve been trying to teach you for two years. Welcome to the right side.”

If you didn’t spend all your time ignoring everything I say and arguing against your own strawmen, maybe it wouldn’t come as a surprise when you finally see something you agree with.

The uncertainties and the measurements are two different things. How do you tell the distribution inside the uncertainty interval?

That’s my point. You can’t tell what the uncertainties are just from two measurements.

“So I’ll ask again – “What does their mean represent?””

This style of argument is so tedious. All you keep doing is coming up with these pointless examples and then demanding I explain what the point of your example is. I don’t know what you want me to tell you about the mean of a sample of two planks of wood. You’re the one who wanted to know what the mean is, you say what the purpose was.

All you are saying is “I can’t figure out why I’d ever want to know what the average of two very different planks of wood is, therefore that proves that all averages must be useless.”

More word salad

sorry. Is “It’s an estimate of the true value of the mean” too complicated for you. I’m not sure how to make it any simpler. You keep claiming that means are meaningless because they don’t have a true value, by which I assume you think there has to be one specific example that is the same as the mean. And I’m saying that is wrong. The mean is the mean, the true value of the mean is it’s true value.

But even that can’t be true if you say there is an SEM or standard deviation around the mean!

[Takes of glasses. Pinches nose]. The standard error is around the sample mean. It is telling you that the sample mean may not be the same as the population mean (i.e. the true mean). It is telling you how much of difference there is likely to be between your sample and the population.

(And yes, before we go down another rabbit hole – that’s dependent on a lot of caveats about biases, systematic errors, calculating mistakes, badly defined populations, and numerous other real world issues.)

I want to know what use can be made of that *exact average” when you have multiple different things making up your data set.

Then take a course in statistics or read any of the numerous posts here which use the global averages to make claims about pauses or whatnot.

karlomonte
Reply to  Bellman
December 12, 2022 11:40 am

Just because you don’t know what it is doesn’t mean it doesn’t exist. All uncertainty intervals have to have some form of a probability distribution. How else could you have a “standard uncertainty” if there was no distribution. You need a distribution to have a standard deviation, you need a standard deviation to have a standard uncertainty.

You don’t understand any of this, and I’m not going to waste any time time explaining what you refuse learn.

Tim Gorman
Reply to  Bellman
December 12, 2022 6:46 am

All the measurements are coming from a probability distribution that may or may not be Gaussian.”

A probability distribution provides an expectation for the next possible value.

How do you get an expectation for the next value of a different thing? Tmin is *NOT* a predictor of Tmax so how do you get an expectation for Tmax from a probability distribution?

“the measurements are not what forms the distribution.”

Jeesh! Did one of those new “chatbots” write this for you?

“Even if you only take one or two measurements, they are still coming from the distribution, but you won;t be able to tell what it is just by looking at them”

Again, what distribution does the boards used to build a house come from?

“Once again, this has nothing to do with the distribution being Gaussian, or even symmetric.”

Gaussian can be assumed to provide cancellation. You must *PROVE* that a skewed or multi-nodal distribution provides for cancellation. How do you do that?

“the assumption is you already know the uncertainties, and can estimate the uncertainties of the mean from them.”

Are yo finally coming around to understanding that the standard deviation of the mean is *NOT* the accuracy of the mean? That the uncertainty of the mean depends on the propagation of the individual member uncertainties?

Bellman
Reply to  Tim Gorman
December 12, 2022 9:00 am

Tmin is *NOT* a predictor of Tmax so how do you get an expectation for Tmax from a probability distribution?

Your really obsessed with this max min business aren’t you. As always it depends on what you are trying to do. If you want to know the distribution around TMax don;t use the distribution around TMin.

Again, what distribution does the boards used to build a house come from?

I’ve no idea because usual you are getting lost in the wood. What boards? What house? What distribution. If you have a big room full of assorted boards and you pull out one at random it will come from the distribution of all boards in that room. If you find your boards in a ditch it will come from the distribution of all boards that a dumped in ditches.

Gaussian can be assumed to provide cancellation.

As can any random distribution with a mean of zero.

You must *PROVE* that a skewed or multi-nodal distribution provides for cancellation.

It’s already been done for me.

Are yo finally coming around to understanding that the standard deviation of the mean is *NOT* the accuracy of the mean?

Do you mean, is the thing I’ve been telling you I don’t believe in still the thing I don’t believe in? Yes.

The standard error of the mean, or whatever you want to call it, is not necessarily the accuracy of the mean. That’s because there can be biases or systematic errors that will affect the mean. In metrology terms, the standard error of the mean is akin to the precision of the mean, but not its trueness.

But I’m not sure what this has go to do with the comment you were replying to, which was about whether you are deriving the measurement uncertainties from an experimental distribution, or assuming you already know them.

That the uncertainty of the mean depends on the propagation of the individual member uncertainties?

No I don’t agree with that. The measurement uncertainty is only a part, and usually very small part, of the uncertainty of the mean.

sherro01
Reply to  Bellman
December 11, 2022 4:38 pm

Bellman,
To use statistics, you need to be able to show that your numbers are samples from a population. A population is best considered to arise when extraneous variables are absent or minimal.
When discussing thermometers at weather stations, two different stations will produce two different statistical populations because of different types and intensities of extraneous variables.
Just as two people have different variables – one should not take the body temperature of person A by inserting a thermometer in person B.
Geoff S

Tim Gorman
Reply to  sherro01
December 12, 2022 4:38 am

100% correct.

karlomonte
Reply to  Bellman
December 10, 2022 7:39 am

And right on cue, bellcurveman (Stokes disciple) pops in with his usual and tedious nonsense.

Bellman
Reply to  karlomonte
December 10, 2022 8:15 am

And right on queue the troll pops up with an insult that has nothing to do with the discussion

karlomonte
Reply to  Bellman
December 10, 2022 8:44 am

Note that bellcurveman is unable to refute anything Kip wrote in a coherent fashion.

Tim Gorman
Reply to  karlomonte
December 10, 2022 2:59 pm

Of course he can’t. Neither can most on here because all they know is what they learned in Statistics 101 where *NONE* of the examples have anything but stated values with the uncertainties of the values totally ignored.

It’s amazing to me that none of them have ever measured crankshaft journals to size the bearings that need to be ordered. None of them have ever apparently designed trusses for a hip roof and ordered the proper lumber so that wastage is minimized.

All they know is random distributions of stated values and that the average of those stated values somehow helps in figuring out the bearing size and stud length.

Bellman
Reply to  Tim Gorman
December 10, 2022 3:22 pm

I’m not sure how much more coherent my points can be. Rather than making snide remarks you could actually ask for more clarity.

I’ve only said three points relating to this article. One was agreeing with Kip that you have to divide the total uncertainty by the number of measurements when taking an average. The two points I disagreed with were claiming that a mean has no uncertainty. I disagreed when the mean is of a sample being used to estimate a population. And the other was to point out that you can add uncertainties in quadrature when they are random and independent. I provided a source for that which I think is taken from your favorite authority, Taylor.

So I’ll ask again, which points do you want me to be more coherent about?

karlomonte
Reply to  Bellman
December 10, 2022 5:52 pm

Rather than making snide remarks you could actually ask for more clarity.

And take another spin on the bellcurveman hamster wheel? I’ll pass.

Tim Gorman
Reply to  Bellman
December 11, 2022 9:04 am

 One was agreeing with Kip that you have to divide the total uncertainty by the number of measurements when taking an average. “

That is the AVERAGE UNCERTAINTY. It is *NOT* the UNCERTAINTY OF THE AVERAGE!

Like the two boards of 2′ +/- 1″ and 10′ +/- 0.5″. The average uncertainty is 0.8″. Now nail those two boards together. What is the uncertainty interval you can see? It is *NOT* +/- 0.8″!

“The two points I disagreed with were claiming that a mean has no uncertainty. I disagreed when the mean is of a sample being used to estimate a population.”

If the measurements themselves have uncertainty then even if you have the entire population that you can use to calculate the average of the population (i.e. zero standard deviation) that average will still have uncertainty propagated from the individual measurements. That’s why the SEM only tells you how close you are to the population mean but it does *NOT* tell you anything about the accuracy of that average.

“And the other was to point out that you can add uncertainties in quadrature when they are random and independent. I provided a source for that which I think is taken from your favorite authority, Taylor.”

Which, as usual, you cherry picked with no understanding of what you were posting.

Not all uncertainties can or should be added in quadrature. That carries with it the assumption that some of the random error in the uncertainties can cancel. That is *NOT* always true.

Again, with the two board example there is not very much likelihood that the uncertainties of the two boards have any cancellation at all. Direct addition of the two uncertainties is probably a better estimate of what you will find in the real world.

Bellman
Reply to  Tim Gorman
December 11, 2022 10:04 am

Like the two boards of 2′ +/- 1″ and 10′ +/- 0.5″. The average uncertainty is 0.8″. Now nail those two boards together. What is the uncertainty interval you can see? It is *NOT* +/- 0.8″!

How many more times before you get it. The uncertainty of two planks nailed together is the uncertainty of the sum of the two boards, not the average. If you know the average length and uncertainty of the average, you can multiply it by 2 to get the total length of the two boards and the uncertainty of that length. If the uncertain the of average length is ±0.75″ the uncertainty of two nailed together will be ±1.5″.

That’s why the SEM only tells you how close you are to the population mean but it does *NOT* tell you anything about the accuracy of that average.

I’m not the one saying the mean has no uncertainty. But if the SEM tells you how close to the mean you are, that is what is meant by uncertainty. And yes, there can always be systematic biases either caused by the measurements or the sampling that can change how true and hence accurate the result is, but that is true about any measuring method.

Which, as usual, you cherry picked with no understanding of what you were posting.

What [part do you think I’m cherry picking? I just supplied a link to the entire document. It starts of given the simple approach that Kip uses, and then goes on to say if you can assume random measurement uncertainties, then you can use adding in quadrature. What context do you think I am ignoring?

Not all uncertainties can or should be added in quadrature.

Which is why I keep adding the qualification of random and independent.

That carries with it the assumption that some of the random error in the uncertainties can cancel. That is *NOT* always true.

Under what circumstances would you assume that random errors will not cancel? The only requirement is that the mean of any uncertainty is zero, otherwise you have a systematic error.

Again, with the two board example there is not very much likelihood that the uncertainties of the two boards have any cancellation at all.

The argument that random uncertainties will cancel is probabilistic. It isn’t saying this will always happen, just that the average will reduce. With just two boards it’s possible that both will have the same maximum error in the same direction, but it’s less likely.

Direct addition of the two uncertainties is probably a better estimate of what you will find in the real world.

It isn’t if you assume the uncertainties are random. Suppose your thought experiment says that an uncertainty of ±1cm means that every measurement is either 1cm too long or 1 cm too short, with a 50% chance of either. There is only a 25% chance that both will be +1, and a 25% chance that they will both be -1. But there’s a 50% chance that one is +1 and the other – 1, with an aggregate error of 0cm. With a different error distribution it’s even less likely that you would see both errors being the maximum possible. And of course, this becomes less likely the more samples you take.

karlomonte
Reply to  Bellman
December 11, 2022 12:24 pm

But if the SEM tells you how close to the mean you are, that is what is meant by uncertainty.

Nonsense! You STILL don’t even know what uncertainty is! Not even the vagest idea!

Clyde Spencer
Reply to  karlomonte
December 10, 2022 8:44 pm

Despite the inference one might draw from the handle he has chosen, he has acknowledged that he has no particular expertise in statistics.

karlomonte
Reply to  Clyde Spencer
December 11, 2022 8:17 am

I believe you are correct here, yet he seems to always circle around and lecture people as if he is the expert in statistics as well as uncertainty.

Bellman
Reply to  karlomonte
December 11, 2022 8:52 am

I correct people when I think they are wrong. I try to back this up with evidence, but am always prepared to accept when I’m shown to be wrong.

But when people claim that the uncertainty off an average increases with sample size and as evidence use clearly incorrect interpretation of the maths and then refuse to even consider the possibility they might be wrong, then there’s little I can do but keep explaining why they are wrong. This doesn’t require any expertise, just the ability to read an equation.

karlomonte
Reply to  Bellman
December 11, 2022 9:09 am

You still cannot comprehend that error is not uncertainty, regardless of what you write about “the maths”.

And you still ignore that a single temperature measurement cannot be “cured” with averaging.

And you ignore that a different analysis gives a different answer to your sacred averaging formula that you plug into the maths that you don’t understand.

Bellman
Reply to  karlomonte
December 11, 2022 9:29 am

“You still cannot comprehend that error is not uncertainty, regardless of what you write about “the maths”.”

Uncertainty isn’t error? Why didn’t you mention this earlier. It changes everything. Rather than using hte equations from sources like Taylor which is all about error propagation, I’ll instead use the ones from the GUM which only talks about uncertainty. Oh wait, they’re identical.

And you still ignore that a single temperature measurement cannot be “cured” with averaging.

No idea how you would average a single temperature measurement.

And you ignore that a different analysis gives a different answer to your sacred averaging formula that you plug into the maths that you don’t understand.

What different analysis? I’ve gone through this using the standard rules for error/uncertainty propagation, I’ve used the general equation using partial differential equations, I’ve used the rules for combining random variables, and I’ve made many common sense logical arguments for why what is being claimed cannot happen. You cannot get a bigger uncertainty in the average than the biggest uncertainty of a single element. In the worst case you assume all uncertainties are systematic, and the uncertainty of the mean is just the uncertainty of the individual element, or you assume some randomness in the uncertainties, in which case the uncertainty of the mean is reduced.

karlomonte
Reply to  Bellman
December 11, 2022 12:30 pm

Uncertainty isn’t error? Why didn’t you mention this earlier. It changes everything.

Duh! How many times have you been told this?

Innumerable.

What different analysis?

Duh #2!

1) There is your beloved GUM eq. 10 (which you misapply to averaging).

2) Apply GUM 10 separately to the sum and N, then to the ratio.

3) Kip’s analysis, in which N cancels, and which you obviously have not read.

You pick the one that gets you the tiny numbers.

This is what passes for science in climastrology.

Bellman
Reply to  karlomonte
December 11, 2022 3:34 pm

Duh! How many times have you been told this?

I think you need to check the battery in your sarcasm detector.

1) There is your beloved GUM eq. 10 (which you misapply to averaging).

This is the equation you spent ages insisting I had to follow in order to get the uncertainty of the average. I have no special feeling for it, but you can derive all the other rules from it. Applying it to an average means that as the partial differential for the mean is 1/N for each term results in u(average)^2 = (u(x1)/N)^2 + (u(x2)/N)^2 + … + (u(xn)/N)^2, or

u(average) = sqrt[u(x1)^2 + u(x2)^2 + … + u(x2n)^2] / N = u(sum) / N

If you think it’s misapplying it to use for the uncertainty of an average then why bring it up?

2) Apply GUM 10 separately to the sum and N, then to the ratio.

Same thing.

For the sum all the partial. derivatives are 1, so

u(sum)^2 = u(x1)^2 + u(x2)^2 + … + u(x2n)^2

For N the uncertainty is zero

u(N)^2 = 0

For the ratio, mean = sum / N, partial derivative of mean with respect to sum is 1/N, so

u(mean)^2 = (u(sum)/N)^2 + (sum * u(N))^2 = (u(sum)/N)^2 + 0

so u(mean) = u(sum) / N

3) Kip’s analysis, in which N cancels, and which you obviously have not read.

Kip gets a different result for the uncertainty of the sum because he’s not adding in quadrature. But the principle he states is still the same

Likewise, if you wish to divide a value that has an absolute uncertainty by a constant (a number with no uncertainty), the absolute uncertainty is divided by the same amount.

So for an average he is saying

u(mean) = u(sum) / N

If you assume all uncertainties are equal u(x), than the first two methods resolve to

u(mean) = u(x) / sqrt(N)

whilst Kip gets

u(mean) = u(x)

The difference being in the assumptions made about the uncertainties. Equation 10 is for independent random uncertainties, whilst Kip’s is for non-indpendent uncertainties. But the calculation of the average is the same (divide the uncertainty of the sum by N) and in neither case do you get an increase in uncertainty when taking an average.

karlomonte
Reply to  Bellman
December 11, 2022 6:16 pm

And here is YASHW**, usual trendology nonphysical nonsense ignored…

**yet another spin of the hamster wheel

Bellman
Reply to  karlomonte
December 11, 2022 6:55 pm

And to no ones surprise, Carlo ignores the answer and just responds with another troll-ish insult.

Bellman
Reply to  karlomonte
December 11, 2022 6:58 pm

And I see that since this comment, Mr “How dare you call me a Troll” has posted 15 other one line insults directed at others.

Last edited 1 month ago by Bellman
Bellman
Reply to  Clyde Spencer
December 11, 2022 8:46 am

I’m not sure what inference you want to draw from my psudonym. It was meant to be a self-derogatory nod to the bellman from the Hunting of the Snark. Maybe I should add “the” to it as some seem to think it’s a surname.

ThinkingScientist
Reply to  Kip Hansen
December 11, 2022 12:54 pm

Sorry Kip, some of us on both sides of the debate have good reason to choose pseudonyms. When I retire I will switch to using my real name but in the meantime like many others I have good reason related to my employer why I do not.

Last edited 1 month ago by ThinkingScientist
Bellman
Reply to  Kip Hansen
December 12, 2022 4:41 am

I could, and I’ve considered it. But I don’t see what purpose it would serve, beyond self promotion. Why should my real name have any bearing on the arguments. If I was claiming any sort of expertise, fair enough, I would need to provide my name in order to justify it – but I really don’t and my name isn’t going to have any meaning to anyone.

And then there’s the issue that I tend to provoke very strong reactions here. Many really seem to hate me, and at present I don’t mind that because they only hate the Bellman, not the real me.

karlomonte
Reply to  Bellman
December 12, 2022 6:34 am

I’ll try again—stop posting nonphysical nonsense.

Bellman
Reply to  karlomonte
December 12, 2022 7:59 am

Oh the irony. A pseudonymous troll barges in to a discussion about whether you should use your real name.

karlomonte
Reply to  Bellman
December 12, 2022 8:26 am

You mistake “hate” for pushback against the nonsense you post.

Bellman
Reply to  karlomonte
December 12, 2022 8:37 am

Egotist. I wasn’t talking about you when I said “hate”. I think you’re just desperate for attention.

But in either case, if you want to “pushback” you need to engage in the argument and not simply post 50 variations of “you’re an idiot”.

karlomonte
Reply to  Bellman
December 12, 2022 8:39 am

I think you’re just desperate for attention.

Projection time again?

Phil.
Reply to  Kip Hansen
December 13, 2022 8:29 am

I learned a long time ago that I should not use my full name after my email was completely blocked by trolls. For a week my students and colleagues were unable to communicate with me. I’m sure others have similar reasons.

bigoilbob
Reply to  Kip Hansen
December 12, 2022 6:35 am

bdgwx let’s me take the easy ones. He’s provided it more than once. And I have provided my Oklahoma Professional Petroleum Engineering Registration number, which would take 45 seconds to trace back to my name.

Nom’s de WUWT, and just about everywhere else are common. Abraham Lincoln used one.

Nick Stokes
Reply to  Bellman
December 12, 2022 12:05 pm

I think Kip is the Bellman here:

He had bought a large map representing the sea,
Without the least vestige of land:
And the crew were much pleased when they found it to be
A map they could all understand.

“What’s the good of Mercator’s North Poles and Equators,
Tropics, Zones, and Meridian Lines?”
So the Bellman would cry: and the crew would reply
“They are merely conventional signs!

“Other maps are such shapes, with their islands and capes!
But we’ve got our brave Captain to thank:
(So the crew would protest) “that he’s bought us the best —
A perfect and absolute blank!”

bdgwx
Reply to  Clyde Spencer
December 11, 2022 2:42 pm

For the record, I don’t consider myself an expert in statistics or uncertainty either. That is why I cite so much literature.

karlomonte
Reply to  bdgwx
December 11, 2022 6:17 pm

Yet you beat people (“contrarians”) over the head with your nonsense if they don’t hoe to your propaganda.

Clyde Spencer
Reply to  Bellman
December 10, 2022 8:40 pm

There is no rule I’ve ever been showen [sic] …

That isn’t a defense to dismiss bnice2000.

You probably also are unaware that the purpose for calculating a mean, and whether the data are stationary or non-stationary determine the appropriate treatment.

If you are only interested in an approximate value of a bounded time-series, then a simple arithmetic mean or even a mid-range value may suffice. In any event, high precision is hardly warranted for that purpose.

A bigger problem is that non-stationary data — a time-series where the mean and variance change with time — does not have a normal distribution. It will be strongly skewed, if there is a trend. The requirements of the same thing (e.g. the diameter of a ball bearing) being measured multiple times with the same instrument is not met. Otherwise, we might as well be averaging the diameter of black berries and watermelons. One can calculate an average, but of what utility is it? All one can say is that the average diameter of those two varieties of fruit is X.

The rationale behind taking multiple readings of something with the same instrument is that most of the random errors have to cancel to improve the precision and hence the estimate of the mean. One only has high probability of that happening with a fixed value (stationarity) and the same instrument, thereby creating a symmetric probability distribution of all the sample measurements.

bdgwx
Reply to  Clyde Spencer
December 11, 2022 5:12 am

CS said: “That isn’t a defense to dismiss bnice2000.”

This “measuring the same thing with the same instrument” rule for propagating uncertainty doesn’t actually exist. You can prove this out for yourself by reading JCGM 100:2008, JCGM 6:2008, NIST 1297, and NIST 1900 and seeing that many (maybe even most) of the examples are actually have measuring different things with different instruments.

Last edited 1 month ago by bdgwx
karlomonte
Reply to  bdgwx
December 11, 2022 7:08 am

How do you know that Clyde has not done so?

Are you psychic?

Tim Gorman
Reply to  bdgwx
December 11, 2022 9:19 am

bdgwx:

This “measuring the same thing with the same instrument” rule for propagating uncertainty doesn’t actually exist. “

Malarky. Total and utter BS. Go look at annex H in the GUM. Specifically H.6.3.2 where you have two different machines making measurements. The total uncertainty is the sum of the SEM for each machine. If you only have one measurement per machine then the SEM is actually the uncertainty of that single measurement. Since you only have one measurement the uncertainty of each measurement gets divided by 1.

Thus for a multiplicity of measurements of different machines measuring different things you wind up with a root-sum-square of all the uncertainties.



Clyde Spencer
Reply to  bdgwx
December 11, 2022 12:45 pm

This rule for propagating uncertainty doesn’t actually exist.

Just because it isn’t explicitly mentioned in the documents you list doesn’t mean it can be ignored. If calibration of the instruments reveals an uncertainty or bias that is non-negligible, then it shouldn’t be ignored. Using the same instrument eliminates the possibility of variations in the uncertainty and bias. I doubt that any of the data processing uses a rigorous propagation of error where every instrument has its unique precision and bias taken into account.

bdgwx
Reply to  Clyde Spencer
December 11, 2022 1:35 pm

CS said: “Just because it isn’t explicitly mentioned in the documents you list doesn’t mean it can be ignored.”

Yes it does. And just be clear I’m talking about the mythical “measuring the same thing with the same instrument” rule for JCGM 100:2008 (and others) and implemented by the NIST uncertainty machine which isn’t a thing. If someone makes up a fake rule you can and should ignore it.

CS said: “If calibration of the instruments reveals an uncertainty or bias that is non-negligible, then it shouldn’t be ignored.”

Nobody is ignoring uncertainty or bias here. The discussion isn’t whether the uncertainty or bias should be ignored. The discussion is about ignoring fake rules for its propagation that don’t actually exist.

CS said: “Using the same instrument eliminates the possibility of variations in the uncertainty and bias.”

You can’t always use the same instrument such as the case of section 7 in the NIST uncertainty machine manual in which the measurement model is A = (L1 − L0 ) / (L0 * (T1 − T0)). You might be able to measure T1/T0 or L1/L0 with the same instrument, but you can’t measure all 4 with the same instrument.

CS said: ” I doubt that any of the data processing uses a rigorous propagation of error where every instrument has its unique precision and bias taken into account.”

The law of propagation of uncertainty documented in the JCGM 100:2008 (and others) does just that. The NIST uncertainty machine will perform both the deterministic and monte carlo methods for you as a convenience. You are certainly free to do the calculations by hand if you wish though.

karlomonte
Reply to  bdgwx
December 11, 2022 6:19 pm

Someone hit the Big Red Switch on the bgwxyz bot, it is stuck in another infinite loop.

Tim Gorman
Reply to  bdgwx
December 12, 2022 6:02 am

The discussion is about ignoring fake rules for its propagation that don’t actually exist.”

There are no fake rules. There is only an fake assumption that all measurement error cancels in every case and the standard deviation of the stated values is the uncertainty.

The *ONLY* way to get uncertainties in the hundredths digit from measuring different things is to assume the uncertainties of the individual measurements always cancel. It is truly just that simple – and incorrect.

Clyde Spencer
Reply to  bdgwx
December 12, 2022 5:05 pm

If someone makes up a fake rule you can and should ignore it.

That cuts both ways! That also means if something is left out, and a case can be made that it shouldn’t have been, then you are obligated to do the right thing if you are intellectually honest.

Jim Gorman
Reply to  bdgwx
December 14, 2022 6:33 am

You are full of crap. Do you need more references than I gave you above?

Jim Gorman
Reply to  bdgwx
December 14, 2022 6:31 am

Several relevant definitions from JCGM 100:2008. I can give you several more from Dr. Taylor book if you like.

“successive measurements of the same measurand” from B.2.15 and

“closeness of the agreement between the results of measurements of the same measurand” from B.2.16 and

“infinite number of measurements of the same measurand carried out under repeatability conditions” from B.2.21.

You obviously have no training or experience dealing with these issues in sufficient detail to be making wild assertions like not needing multiple measurements of the same thing with the same device.

bdgwx
Reply to  Jim Gorman
December 14, 2022 8:34 am

Yep. And I agree with JCGM 100:2008.

Repeatability is for the same measurand.

Reproducibility is for the same measurand.

Random error is for the same measurand.

But, just because those terms apply to the same measurand does not invalidate anything in section 4 or 5 including the measurement model Y which has “input quantities X1, X2, …, XN upon which the output quantity Y depends may themselves be viewed as measurands and may themselves depend on other quantities, including corrections and correction factors for systematic effects, thereby leading to a complicated functional relationship f that may never be written down explicitly”

It is unequivocal. The measurement model Y may depend on other measurands. It says so in no uncertain terms.

bdgwx
Reply to  Clyde Spencer
December 11, 2022 12:00 pm

Clyde,

The example H.6 from JCGM 100:2008 that Tim mentions below proves my point. The measurement model is Y = f(d, Δc, Δb, Δs) where d, Δc, Δb, and Δs are all different things. Furthermore Δc = y’s – y’ where y’s is from one instrument and y’ is from another. H.6 is example where the uncertainty was propagated from different things measured by different instruments.

Furthermore, the manual for the NIST uncertainty machine make no mention of the mythic “measuring the same thing with the same instrument” rule. In fact, every single example (bar none) is of using the tool with different things measured by different instruments.

This “measuring the same thing with the same instrument” is completely bogus. It is absolutely defensible to dismiss bnice200, Tim Gorman, Jim Gorman, and karlomonte on this point.

karlomonte
Reply to  bdgwx
December 11, 2022 12:33 pm

The expert speaks, y’all better listen up!

Clyde Spencer
Reply to  bdgwx
December 11, 2022 1:15 pm

This “measuring the same thing with the same instrument” is completely bogus.

“3-3 Repeated Measurements. It is possible to increase the accuracy of a measurement by making repeated measurements of the same quantity and taking the average of the results. The method of repeated measurements is used in all cases when precision of the instrument is lower than the prescribed accuracy. … In those instances when the readings cannot be accumulated in repeated measurements, the prerequisite condition for improving the accuracy is that measurements must be of such an order of precision that there will be some variations in the recorded values. … In this connection it must be pointed out that the increased accuracy of the mean value of the repeated single measurements is possible only if the discrepancies in measurements are entirely due to so-called accidental errors … In other words, at a low order of precision no increase in accuracy will result from repeated measurements.”

[Smirnoff, Michael V., (1961), Measurements for engineering and other surveys, Prentice Hall, p. 29]

It should be obvious that the description above applies to a single instrument (particularly accumulations), not a conflation of measurements from many instruments because the uncertainty will grow and the precision will decrease when the uncertainty from many instruments is propagated rigorously.

bdgwx
Reply to  Clyde Spencer
December 11, 2022 2:38 pm

Yes. It does apply to a single measurand and presumably from the same instrument. No one (even contrarians) is challenging that fact that the uncertainty of an average of measurements of the same thing will improve (with limits) as the sample size increases. But notice what wasn’t said there. It never said that the uncertainty of the average of different measurands cannot also be improved (with limits) as the number of measurands increases. That is the salient point. We aren’t discussing repeated measurements of the same thing here so the Smirnoff 1961 verbiage, which is consistent with all of the other sources I’ve cited in the comments here and which I happily accept, is irrelevant to the discussion at hand.

Last edited 1 month ago by bdgwx
karlomonte
Reply to  bdgwx
December 11, 2022 6:21 pm

 No one (even contrarians)

Idiot.

Tim Gorman
Reply to  bdgwx
December 12, 2022 6:15 am

But notice what wasn’t said there. It never said that the uncertainty of the average of different measurands cannot also be improved (with limits) as the number of measurands increases.”

In the case of multiple measurements of the same thing you are identifying a TRUE VALUE.

Multiple measurements of different things do *NOT* identify a TRUE VALUE.

One average is useful and the other is not.

The average of 1 unit, 5 units, 7 units and 9 units is 5.5 units. That is *NOT* a TRUE VALUE of anything. If you are measuring different things you will *never* find a true value, only an average. So of what use is an “improved” average in the real world?

Do you build a roof truss using the average length of all the boards the lumber yard delivered ranging from 3′ to 20′? Do you just assume that all the boards you take from the pile is of average length? You might be able to figure out the board feet you are charged for by the lumberyard assuming all of the boards are of the same width and height but how many of the boards will meet that restriction?

karlomonte
Reply to  Tim Gorman
December 12, 2022 6:37 am

Multiple measurements of different things do *NOT* identify a TRUE VALUE.

They will never acknowledge the truth here because it threatens their entire worldview.

Clyde Spencer
Reply to  bdgwx
December 12, 2022 5:13 pm

It never said that the uncertainty of the average of different measurands cannot also be improved (with limits) as the number of measurands increases.

It is those “limits” that are the gotcha that you and others conveniently ignore.

To paraphrase Einstein: “There are only two things that are infinite, the universe and the ability of humans to rationalize. And, I’m not sure about the universe.”

bdgwx
Reply to  Clyde Spencer
December 13, 2022 6:33 am

CS said: “It is those “limits” that are the gotcha that you and others conveniently ignore.”

Then you are ignoring Bellman, Nick, bigoilbob, my, etc. posts because we talk about those limits all of the time.

Tim Gorman
Reply to  bdgwx
December 13, 2022 5:40 am

You ignore the implications of H.6! I’ve showed you how the equation applies when you have single measurements of an multiple objects.

In the example, you have two machines making 5 readings each of the same thing. Thus you can get a standard deviation for each machine. And you know the number of measurements made.

When you have single measurements of different things then you have no standard deviation of stated values and must use the uncertainty interval as the standard deviation. The number of measurements equals 1. So each term in the equation becomes nothing more than u(i)^2 and the equation defaults to the typical root-sum-square of the individual uncertainties.

Furthermore, the manual for the NIST uncertainty machine make no mention of the mythic “measuring the same thing with the same instrument” rule. “

It’s not mythic. As I point out GUM H.6 shows how to handle measurement uncertainty when you have multiple measurements of different things.

If you have only ONE measurement then each term in Eq H.36 ( s_av^2(x_i)/ N) becomes nothing more than

u^2(x_i)/ 1. Which means you get a root-sum-square addition of the uncertainties associated with the individual measurements.

THERE IS NOTHING MYTHIC ABOUT THAT.

All this demonstrates that you simply can’t think outside the box you are stuck in where you assume all measurement uncertainty cancels.

The NIST machine makes *NO* attempt to handle single measurements of different things. It just calculates the standard deviations of the stated values.

Break out of your box and join the rest of us in the real world!

Jim Gorman
Reply to  bdgwx
December 14, 2022 6:46 am

You did not read this example very well did you? There are FIVE REPEATED INDENTATIONS IN THE SAME TRANSFER BLOCK. The hardness is derived from the measured depths of penetration. Additional uncertainty is derived from the difference between the sample machine and the national standard machine.

Do you even know the purpose of the transfer block? I’ll bet not if you’ve no experience in a machine shop.

Clyde Spencer
Reply to  Nick Stokes
December 10, 2022 3:15 pm

So adding 5.0 ± 0.1 mm + 2.0 ± 0.1 mm should give

7.0 ± 0.1414mm ie sqrt(.01+.01).

Strictly speaking, 4 significant figures in the uncertainty are not justified. The rule for adding (or subtracting) two or more numbers is to retain no more significant figures than are in the least precise number(s), 7.0. That is, it should give 7.0 ± 0.1, which is larger than the implied precision of ± 0.05, for the number 7.0.

May Contain Traces of Seafood
Reply to  Clyde Spencer
December 11, 2022 10:30 pm

You really need to define what you are trying to calculate.

Root Mean Squared as a method for calculating tolerance stack is valid in practical terms. The idea is while a single part may be anywhere within the allowed tolerance, the odds of two parts both being at the extremes are low.

So RMS when checking tolerance stack for manufacturing/design is an acceptable method in an pragmatic environment, but still might stuff you up.

Tolerance and statistical analysis are different beasts.

We also have the discussion on the theory of trailing zeros.

If you subscribe to the practice of No Trailing Zeros then 5mm is exactly the same as 5.00000mm.

If you don’t then 5mm is 100,000 times less precise than 5.00000mm.

Personally I am a No Trailing Zeros type of guy. If you need to define a dimension then you apply tolerance directly to it.

But yes, we are seriously starting to mix our disciplines here me thinks.

Jim Gorman
Reply to  Clyde Spencer
December 14, 2022 6:52 am

If I remember right, if the first decimal digit of the uncertainty is an”1″ then you state the second digit, so ±0.14.

davezawadi
Reply to  Nick Stokes
December 10, 2022 1:16 am

You are making a mistake immediately Nick, you are assuming the errors have a Gaussian distribution! Now why do you think that is true for temperature measurements made with many instruments in many various places? You are simply saying that is a starting assumption, but it cannot be as there are several uncorrelated sources of the errors!

Nick Stokes
Reply to  davezawadi
December 10, 2022 1:19 am

No, in fact the additivity of variances is not dependent on Gaussian distribution. Nor of course is the existence of a standard deviation.

bnice2000
Reply to  Nick Stokes
December 10, 2022 2:43 am

WRONG !

ThinkingScientist
Reply to  bnice2000
December 10, 2022 10:08 am

Nope, Nick Stokes is actually correct.

Tim Gorman
Reply to  Nick Stokes
December 10, 2022 7:19 am

The additivity of variances is *NOT* the standard deviation of the mean.

The standard deviation of the mean is *only* useful if you have a Gaussian distribution of error around a mean. In this case the average becomes the true value and the standard deviation of the mean applies.

When you have measurements of different things you do *NOT* have a true value, only a calculated value, and the uncertainties (variances) add by root-sum-square.

Why, in climate science, everyone seems to assume that mid-range values represent some kind of true value is beyond me, let alone the idea that the individual measurements form some kind of a Gaussian distribution where measurement error cancels. Daily temperatures follow a sine wave growth and an exponential decay, neither of which are Gaussian nor is the mid-range value an actual average.

michael hart
Reply to  Tim Gorman
December 10, 2022 10:24 pm

“Daily temperatures follow a sine wave growth and an exponential decay, neither of which are Gaussian nor is the mid-range value an actual average.”

Yup. A key point.

ThinkingScientist
Reply to  Tim Gorman
December 11, 2022 12:56 pm

That is not what is actually measured historically.only min and max daily temps are generally recorded. So your answer is irrelevant.

bdgwx
Reply to  davezawadi
December 10, 2022 6:26 pm

davezawadi said: “Nick, you are assuming the errors have a Gaussian distribution!”

No, he isn’t. The errors do not have to be gaussian. You can prove this out for yourself using the NIST uncertainty machine which allows you to enter any distribution you want.

bdgwx
Reply to  Kip Hansen
December 10, 2022 7:37 pm

There’s no need to wait. Using the NIST uncertainty machine select two input quantities (x0 and x1) and assign them rectangular (or any non-gaussian) distribution. Then for the output enter (x0 + x1) / 2 and see what you get.

Last edited 1 month ago by bdgwx
Tim Gorman
Reply to  bdgwx
December 11, 2022 9:22 am

The NIST machine is ONLY applicable for multiple measurements of the same thing. Uncertainty of a single measurement simply can’t be assigned a distribution since you don’t have multiple values with which to define a probability distribution.

Clyde Spencer
Reply to  bdgwx
December 10, 2022 8:50 pm

Explain how the errors cancel without a residual if the distribution isn’t at least symmetrical.

bdgwx
Reply to  Clyde Spencer
December 11, 2022 5:07 am

Not only do the distributions not have to be gaussian, but they don’t even have to be symmetrical. The only requirement is that be random. The reason is the CLT.

You can prove this out using the NIST uncertainty machine. Select 3 input quantities each being a gamma distribution (non-symmetrical). Then enter (x0+x1+x2) / 3 as the output quantity. Notice that despite the input quantities having non-symmetrical distribution the output quantity forms into a normal distribution with a standard deviation scaled per 1/sqrt(N).

And what happens if they aren’t random? That is the “component of uncertainty arising from a system effect” that NIST TN 1297 talks about which can be removed by exploiting the additive identity of algebra when doing anomaly analysis.

karlomonte
Reply to  bdgwx
December 11, 2022 7:11 am

Here a little homework problem for you:

1) stuff X+Y into your NIST machine
2) write down the result as A
3) stuff 2 into machine
4) write down result as B
5) stuff A / B into NIST machine
6) write result as C

Come back and tell everyone C.

Tim Gorman
Reply to  karlomonte
December 11, 2022 9:33 am

Are you willing to expect an answer?

karlomonte
Reply to  Tim Gorman
December 11, 2022 12:40 pm

He did try, sort of (below), with a million Monte Carlo steps. He got back “u(y) = 0.707” which is supposed to be one of the inputs!

bdgwx
Reply to  karlomonte
December 11, 2022 11:40 am

You didn’t specify the distribution of x or y so I made them gaussian with a mean of 0 and std of 1.

Here is the configuration.

version=1.5
seed=97
nbVar=2
nbReal=1000000
variable0=x;7;0;1
variable1=y;7;0;1
expression=a=x+y;b=2;c=a/b;c
symmetrical=false
correlation=false

And here is the result.

===== RESULTS ==============================

Monte Carlo Method

Summary statistics for sample of size 1000000 

ave     = 3e-04
sd      = 0.708
median  = -3e-04
mad     = 0.71 

Coverage intervals

99% (    -1.8,      1.8)	k =      2.5 
95% (    -1.4,      1.4)	k =        2 
90% (    -1.2,      1.2)	k =      1.7 
68% (   -0.71,     0.71)	k =        1 

ANOVA (% Contributions)

            w/out Residual w/ Residual
x                      100       50.03
Residual                NA       49.97

--------------------------------------------

Gauss's Formula (GUM's Linear Approximation) 

        y  = 0
      u(y) = 0.707 

             SensitivityCoeffs Percent.u2
x                          0.5         50
y                          0.5         50
Correlations                NA          0
============================================
karlomonte
Reply to  bdgwx
December 11, 2022 12:36 pm

You didn’t specify the distribution of x or y so I made them gaussian with a mean of 0 and std of 1.

Distribution!?!??? From where did you pull this nonsense?

Duh #3

X has combined uncertainty u(X).
Y has combined uncertainty u(Y).

Tim Gorman
Reply to  bdgwx
December 11, 2022 9:32 am

The CLT only applies when determining how close the sample means are to the population mean. It tells you nothing about the accuracy of that mean.

If there is *any* systematic bias in the measurements the population mean can be very inaccurate.

How does your meme that all individual uncertainties cancel allow for that?

Clyde Spencer
Reply to  bdgwx
December 11, 2022 1:36 pm

Are these other functions you mention representative of natural phenomena such as temperatures? If not, then it is mathematically interesting, but not germane to the discussion.

Can you explain how it is that if random events are expected to cancel each other that it will happen if there are more measurements above the mean than below?

bdgwx
Reply to  Clyde Spencer
December 11, 2022 3:37 pm

It is true for all distribution sources whether natural or otherwise.

If there the errors do not average to 0 then there is a systematic effect or bias. That will bias will drop out when working with anomalies via the additive identity of algebra.

Last edited 1 month ago by bdgwx
karlomonte
Reply to  bdgwx
December 11, 2022 6:22 pm

Bullshit, total BS.

Tim Gorman
Reply to  karlomonte
December 12, 2022 6:40 am

It’s right back to assuming that all measurement uncertainty cancels in all situations!

karlomonte
Reply to  Tim Gorman
December 12, 2022 7:05 am

Its all they have to hang their hats on.

Tim Gorman
Reply to  bdgwx
December 12, 2022 6:40 am

Really?

You get 10 crankshaft journals from a jobber to use in your engine. They are from different manufacturers with different lot numbers. I.e. different things! The boxes are all marked 2.46″ (i.e. .01 over standard).

Why would you assume the distribution of measurement errors would average to zero?

Why would you assume a systematic bias in their measurement?

Why would you assume that the average value of the journals (2.46″) would be 100% accurate?

What if one of the journal bearings was actually 2.47″? Is that a systematic measurement error or did the wrong bearing get put in the box?

That will bias will drop out when working with anomalies via the additive identity of algebra.”

Huh? How does uncertainty drop out for anomalies? Uncertainty ADDS whether you are adding stated values or subtracting them. They either add directly or by root-sum-square.

The absolute average will have propagated uncertainty and the absolute temperature will have uncertainty. When you subtract the two stated values to form an anomaly their uncertainties ADD, they don’t subtract.

As usual you are just assuming that all measurement uncertainty cancels and the stated values are 100% accurate.

You would *never* be able to make a living as a machinist.

Clyde Spencer
Reply to  Tim Gorman
December 12, 2022 5:22 pm

When you subtract the two stated values to form an anomaly their uncertainties ADD, they don’t subtract.

And, because historical measurements are likely to have greater uncertainty than recent measurements, the act of subtraction may increase the absolute uncertainty. Because the anomaly is smaller than the original measurements, the relative uncertainty will be greater for the anomaly than the original.

Tim Gorman
Reply to  Clyde Spencer
December 13, 2022 5:46 am

Thanks, Clyde. I hadn’t considered the relative uncertainty impacts.

Matt Kiro
Reply to  Nick Stokes
December 10, 2022 2:29 am

You have it backwards, the anomaly is the red herring. How can we even have a measurable anomaly if don’t have a base point to work from? And if this anomaly is within the area of uncertainty, then is it even an anomaly? Which I believe is ultimately the purpose of the essay, that we really can’t say how much the world has warmed in the last 150 years.
But you got lost in the technical words and definitions of the post again. And you were in such a hurry to point out mistakes or misinterpretations that you had to correct yourself multiple times.

Tim Gorman
Reply to  Matt Kiro
December 10, 2022 7:25 am

You hit the nail on the head. The assumption in the climate science world is that the long term average is 100% accurate with no uncertainty and that current temps are 100% accurate with no uncertainty so any anomaly is 100% accurate.

The accepted uncertainty of most measuring device, even today is between +/- 0.5C and +/- 0.3C. No amount of averaging can reduce that uncertainty. It simply doesn’t allow the resolution down to the hundredths digit. In fact, since the temperatures in the data base are separate, individual, single measurements of different things, their uncertainty should add, either directly or in root-sum-square. Just like Nick quoted in GUM 2.3.4.

karlomonte
Reply to  Nick Stokes
December 10, 2022 7:37 am

It actually doesn’t mean that in statistical usage. It means there is a distribution of values about 2.5 with standard deviation 0.5.

Wrong. In nearly all cases of formal uncertainty analysis the actual distribution is unknown.

So the essay drifts into nonsense about absolute uncertainty.

The expert on absolutely everything hath spake, so mote it be.

Nick Stokes
Reply to  Kip Hansen
December 10, 2022 10:32 am

Kip,
values with stated “absolute uncertainty” (which is also clearly defined)”

You’ve misunderstood the definition. It isn’t talking about a different kind of uncertainty. It’s really just about the units. Here is the definition they give of the underlying uncertainty, which is standard:

Uncertainty. Synonym: error. A measure of the the inherent variability of repeated measurements of a quantity. A prediction of the probable variability of a result, based on the inherent uncertainties in the data, found from a mathematical calculation of how the data uncertainties would, in combination, lead to uncertainty in the result. This calculation or process by which one predicts the size of the uncertainty in results from the uncertainties in data and procedure is called error analysis.

See: absolute uncertainty and relative uncertainty. Uncertainties are always present; the experimenter’s job is to keep them as small as required for a useful result. We recognize two kinds of uncertainties: indeterminate and determinate. Indeterminate uncertainties are those whose size and sign are unknown, and are sometimes (misleadingly) called random. Determinate uncertainties are those of definite sign, often referring to uncertainties due to instrument miscalibration, bias in reading scales, or some unknown influence on the measurement.”

bdgwx
Reply to  Kip Hansen
December 10, 2022 6:34 pm

Kip, I think you have misinterpreted that definition. The difference between absolute and relative uncertainty is that absolute retains the units whereas relative is unitless. The formula for converting between the two is R = A/X where R is the relative or fractional uncertainty, A is the absolute or measured uncertainty, and X is the quantity itself. Absolute and relative uncertainties are not different types of uncertainties. They are just different ways of expressing it. See Taylor section 2 for details.

I think the source of confusion arises from Gavin Schmidt’s statement. When used the word “absolute” he was talking about the actual temperature (~288 K) and not anomalies. Do not conflate the discussion of absolute temperature with absolute uncertainty.

Last edited 1 month ago by bdgwx
bdgwx
Reply to  Kip Hansen
December 10, 2022 7:35 pm

The point I’m making is this.

The uncertainty in an absolute temperature has one set of contributing factors. The uncertainty in an anomaly temperature has another, albeit overlapping, set of contributing factors. Those sets are different and thus the two metrics have different uncertainties. It is important not to conflate the uncertainty of the absolute temperature with “absolute uncertainty”. Those are two different concepts.

I think the terminology you are actually wanting to use in the context of absolute temperature which anomaly temperature does not have is “component of uncertainty arising from a systematic effect” as defined in NIST TN 1297. (see caveat below).

We can model the uncertainty of absolute temperature as Ta = Tm + Urm + Us where Ta is the actual temperature, Tm is the measured temperature, Urm is the component of uncertainty arising from a random effect for the measurement, and Us is the component of uncertainty arising from a systematic effect. Then if we want to convert absolute temperature measurements (Tm) into anomaly measurements (Am) we do so using Am = Tm – Tb where Tb is the baseline temperature. But Tm and Tb have uncertainty so Aa = (Tm + Urm + Us) – (Tb + Urb + Us). Notice that the Us terms cancel here leaving us with Aa = (Tm – Tb) + (Urm – Urb). Note that because Us is caused by a systematic effect it is included in both Tm and Tb.

This is the power of anomalies and why GISTEMP anomaly temperatures have a much lower uncertainty than the absolute temperatures. The reason…the algebraic identity x – x = 0. The component of uncertainty arising from systematic effect cancels out. That is what Gavin Schimdt is discussing.

caveat: We can actually further subdivide the component of uncertainty arising from a systematic effect into a time variant portion and a time invariant portion. It is only the time invariant portion that will cancel when doing anomalies.

Last edited 1 month ago by bdgwx
karlomonte
Reply to  bdgwx
December 10, 2022 8:45 pm

caveat: We can actually further subdivide the component of uncertainty arising from a systematic effect into a time variant portion and a time invariant portion. It is only the time invariant portion that will cancel when doing anomalies.

Total word salad.

Clyde Spencer
Reply to  bdgwx
December 10, 2022 9:04 pm

Note that because Us is caused by a systematic effect it is included in both Tm and Tb.

Note also that the uncertainty for historical data is commonly larger than current measurements. Thus, they do not completely cancel.

old cocky
Reply to  bdgwx
December 10, 2022 10:53 pm

Oh, dear.

There are a lot of assumptions in that equation.

Tim Gorman
Reply to  bdgwx
December 11, 2022 9:41 am

Notice that the Us terms cancel here leaving us with Aa = (Tm – Tb) + (Urm – Urb).”

You keep making this same mistake over and over and over again.

The uncertainties only cancel if they are equal. The systematic bias uncertainties will only cancel totally if you are using the same instrument to make all measurements. The random uncertainties will only cancel if you are measuring the same thing for all measurements.

You simply can *NOT* just assume that the uncertainties will cancel unless both restrictions are met – multiple measurements of the same thing using the same measurement device.

Neither restriction is met for temperature measurements.

And the uncertainty of the absolute temps carry over to the anomalies.

If T_avg has the propagated uncertainties associated with all the individual, single temperature measurements and the single, individual daily temperatures are subtracted then the total uncertainty of the anomaly will be at lest the root-sum-square uncertainty of T_avg and Tm.

bdgwx
Reply to  Kip Hansen
December 11, 2022 3:34 pm

That’s actually not the primary purpose of using anomalies. But it is a convenient consequence that is obviously exploited. I do take issue with the statement “seems smaller than reality”. On the contrary, it is the reality. That is the reality is that anomalies have lower uncertainty than the absolutes because a large portion of the systematic effect cancels out.

karlomonte
Reply to  bdgwx
December 11, 2022 6:23 pm

Hey! You forgot to yammer on and on about the uncertainty machine in this post.

HTH

sherro01
Reply to  bdgwx
December 11, 2022 5:14 pm

bdgwx,
Sometimes when I am about to spend some money, I am uncertain if my account has enough credit money. So, I would be happy to reduce this uncertainty by use of a proven statistical formula. I lack confidence that useful formulae exist.
Many formulae, out of no more than laziness and ignorance, express uncertainty symmetrically as in +/-5 or whatever. If I have but $1 in my account, I might have +5/-O unless I have loan arrangements and if the money counting is inaccurate. Likewise, water temperature near freezing phase change cannot be shown as 1 +/-2 degrees. Better is 1 +2/-1.
Look, I could go on for pages about the clash of reality and theory in measurement, but I did that on WUWT earlier this year.
I still miss a theoretical statistical method to minimise my bank account uncertainty.
Geoff S

Clyde Spencer
Reply to  bdgwx
December 10, 2022 8:59 pm

… whereas relative is unitless.

Commonly expressed as a percentage.

bdgwx
Reply to  Clyde Spencer
December 11, 2022 5:35 am

CS said: “Commonly expressed as a percentage.”

Sure, though you could use the Taylor 2.21 approach and leave it as fractional without multiplying by 100. This has the advantage of being used in propagation formulas.

Last edited 1 month ago by bdgwx
karlomonte
Reply to  bdgwx
December 11, 2022 7:11 am

How many formal UAs have you actually performed?

Tim Gorman
Reply to  bdgwx
December 11, 2022 9:42 am

Relative uncertainties are typically applicable when the uncertainties depend on the size of the measurand.

karlomonte
Reply to  Tim Gorman
December 11, 2022 12:44 pm

And they are totally useless for temperature unless working only in Kelvin.

Clyde Spencer
Reply to  Nick Stokes
December 10, 2022 8:57 pm

Note that if you calculate the variance of a time-series with a trend, one will get a larger value than if the variance were calculated from the same time-series that has been de-trended. The implication of that is one can expect variance to grow with the number of samples, rather than decrease.

bigoilbob
Reply to  Clyde Spencer
December 11, 2022 6:28 am

“Note that if you calculate the variance of a time-series with a trend, one will get a larger value than if the variance were calculated from the same time-series that has been de-trended”

Not true. The standard error – and hence the variance – of the zero expected value of detrended data is the same as for the original set. Since this is your claim, please supply the data set and it’s detrended equivalent that shows this. Maybe use wft.

I’ve done this, detrending the data myself. I did it again just now to be sure. I’m open to the rebut that my detrending method is faulty, but I don’t think so.

Last edited 1 month ago by bigoilbob
karlomonte
Reply to  bigoilbob
December 11, 2022 7:13 am

The main goal of trendology—make the numbers as small as possible, then make a lot of noise if anyone dares to point out they are nonphysical nonsense.

bigoilbob
Reply to  karlomonte
December 11, 2022 7:19 am

I was responding to a specifically false claim. I actually provided the rebut data in another post. As opposed to fact free whining – are your eyes burning?

karlomonte
Reply to  bigoilbob
December 11, 2022 8:20 am

Are you watching CNN this morning, blob?

And no, you didn’t “rebut”. You don’t comprehend the implications of a time-series measurement.

Tim Gorman
Reply to  bigoilbob
December 11, 2022 9:44 am

The standard error – and hence the variance “

The term “standard error” is typically used to describe the uncertainty of the mean calculated from a set of samples. That is *NOT* the variance of the population or even of the samples themselves.

karlomonte
Reply to  Tim Gorman
December 11, 2022 12:46 pm

Around and around the hamster wheel spins, there is no getting off…how many times has this one been told?

ThinkingScientist
Reply to  bigoilbob
December 11, 2022 1:01 pm

Bigoilbob I disagree with you and Stokes on almost everything here, but on statistics I agree with most of what has been Neen said by the both of you here.

Still a sceptic, but sticking to the science I know.

bigoilbob
Reply to  ThinkingScientist
December 12, 2022 6:26 am

It’s not fair to stick Stokes and bdgwx with my posts. I have neither the breadth nor depths of their training and experience.

All petroleum engineers end up about half statistician. We take a couple of extra courses in our undergrad and grad lessons, and then use what we’ve learned to apply statistical software commercially. Workover costs, stochastic inputs for reservoir simulations (which, in spite of posters here claiming that it is impossible, use many different sources for ranged geologic and rheologic inputs), economic evaluations of development campaigns – like that.

But I don’t have nearly the ability to use the fundamentals in my posts that Stokes and bdgwx do. Which does give my the humility to avoid wanting to disregard centuries of accumulated knowledge with prejudged intuition, per Pat Frank, the Gorman’s and others.

karlomonte
Reply to  bigoilbob
December 12, 2022 7:08 am

Statistics über alles!

Go right ahead and believe nonphysical nonsense if it floats your boat, blob. You are free to do so.

KB
Reply to  Kip Hansen
December 10, 2022 1:09 pm

I’m saying your arithmetic is indeed incorrect.

Tim Gorman
Reply to  KB
December 10, 2022 2:32 pm

Show your work!

KB
Reply to  Kip Hansen
December 11, 2022 8:55 am

Your method of propagating uncertainty is clearly incorrect. They add in quadrature, not simple addition like you have done.

KB
Reply to  Kip Hansen
December 11, 2022 2:40 pm

(7 +/- 1) + (3 +/- 1) =

10 +/- SQRT(1^2 + 1^2) = 10.0 +/-1.4

Assuming both uncertainties are normally distributed (for example they are random measurement uncertainties).

If the uncertainties are actually tolerances, they will have a rectangular distribution. In which case you would divide each by SQRT(3) to obtain their standard uncertainties first.

TimTheToolMan
Reply to  KB
December 12, 2022 11:13 am

“Assuming both uncertainties are normally distributed (for example they are random measurement uncertainties).”

And if they’re not? Or at least there is no evidence they should be?

TimTheToolMan
Reply to  KB
December 13, 2022 10:28 am

So let’s apply this to a hypothetical progressive calculation such as output from a GCM.

We have a starting temp of 10 +/- 1 “measured”
Uncertainty is then 10 +/- 1.4 when we apply quadrature rule.
That gets fed into the next calculation which returns say 10.1 +/- 1.6
Uncertainty was larger to begin with.

That gets fed into the next calculation which returns 10.1 +/- 1.6
That gets fed into the next calculation which returns 10.1 +/- 1.6

And the uncertainty is slowly increasing unbounded.
After millions of iterations it’s huge.

Tim Gorman
Reply to  TimTheToolMan
December 13, 2022 10:36 am

Yup!

bdgwx
Reply to  Kip Hansen
December 11, 2022 11:32 am

It’s not easy, but I will show you where. You start with the law of propagation of uncertainty given as equation 10 in the JCGM 100:2008 (the GUM).

Let Y = f(x_1, x_2) = x_1 + x_2.

It follows that

∂f/∂x_1 = 1
∂f/∂x_2 = 1

Then using GUM equation 10

u(Y)^2 = Σ[(∂f/∂x_n)^2 * u(x_n)^2, 1, N]

u(Y)^2 = (∂f/∂x_1)^2 * u(x_1)^2 + (∂f/∂x_2)^2 * u(x_2)^2

u(Y)^2 = (1)^2 * u(x_1)^2 + (1)^2 * u(x_2)^2

u(Y)^2 = u(x_1)^2 + u(x_2)^2

u(Y) = sqrt[u(x_1)^2 + u(x_2)^2]

This is the well known root sum square or summation in quadrature rule.

karlomonte
Reply to  bdgwx
December 11, 2022 12:47 pm

Did you even read Kip’s article? Obviously not.

KB
Reply to  bdgwx
December 11, 2022 2:35 pm

I agree. Why can’t they see this?

Jim Gorman
Reply to  bdgwx
December 14, 2022 7:58 am

Tell everyone the assumptions behind both straight addition and using quadrature. They have different assumptions for their use!

bdgwx
Reply to  Jim Gorman
December 14, 2022 8:21 am

For u(x+y) = sqrt[u(x)^2 + u(y)^2] the correlation must be r(x, y) = 0.

For u(x+y) = u(x) + u(y) the correlation must be r(x, y) = 1.

See JCGM 100:2008 equation 15 and 16 for details.

Captain Climate
Reply to  Nick Stokes
December 11, 2022 5:09 am

You are truly an ignorant person Nick. When I tell you a ruler can measure to +/-1 inch, nothing about the distribution is assured at all. You learned a little statistics and have embarrassed yourself by parading your overconfidence and ignorance daily.

May Contain Traces of Seafood
Reply to  Captain Climate
December 11, 2022 10:45 pm

Actually Nick MAY be correct. It depends on what he is trying to do.

Using RMS as an analysis method for discussing tolerance stack is completely valid. If you add an item with a tolerance to a second (or third… fourth…) item (say a pile of discs you cut from a cylinder) then you need to be able to calculate the possible variations in the final stack size.

If I ask you to cut each disc 5mm thk and assign a tolerance limit of 6mm/4mm then any single disc can be cut within that range and still be accepted.

The theory is that while a single disc could be 6mm thick, the odds of getting two cut at that thickness is rather low. Hence using RMS as a method is valid under most circumstances in an engineering world.

Tolerance cost money. If you can get away with it you design the need to have tolerance on parts out of the design.

It is probably not valid for discussion individual measurements. Probably. Not my skill set.

Tim Gorman
Reply to  May Contain Traces of Seafood
December 12, 2022 7:54 am

Except Nick doesn’t use root-sum-square of the uncertainties with temperatures. He assumes all uncertainties of the individual measurement cancel and the standard deviation of the sample means is the accuracy of the average.

It’s like putting 5 shots into the 4-ring of a pistol target and they all hit the same hole. Nick would say that implies that the standard deviation of the sample mean of those 5 shots is very small which makes them very accurate – when in actuality all the shots missed the 10 ring (the bullseye) by a large margin implying that his standard deviation of the sample means is *NOT* the actual accuracy of his shots. His accuracy is very low.

Luke B
Reply to  Nick Stokes
December 11, 2022 8:55 pm

And you know the statistical distribution how?

strativarius
December 10, 2022 12:50 am

Plus or minus 57 Varieties

“”I’m asking my people to give me a better download on exactly what the emissions implications are going to be””

– J Kerry on a Cumbrian coal mine…

https://www.theguardian.com/environment/2022/dec/10/john-kerry-examining-likely-impact-of-new-uk-coalmine

Download???

Last edited 1 month ago by strativarius
karlomonte
Reply to  strativarius
December 10, 2022 7:46 am

Spoken like a true, clueless politician!

The Dark Lord
December 10, 2022 12:55 am

oh good lord … they mix water temperatures with air temperatures … pure apples and oranges … there is no global temperature data set that is fit for purpose … this haggling about “uncertainty” is worrying about the napkins in the Titanic dining room …

Matt Kiro
Reply to  The Dark Lord
December 10, 2022 2:15 am

While I agree with your overall point, the climate activists need to be called out on every point. There’s no way we can pinpoint the average temperature of the earth in 1880 to +/- half a degree.

Pat Frank
Reply to  Matt Kiro
December 10, 2022 12:13 pm

Kip’s Gavin Schmidt source cites P. D. Jones, et al., *1999) Surface Air Temperature and its Changes Over the Past 150 Years. Rev Geophys. 37(2), 173-199, who quote global air temperature anomalies with an uncertainty of ±0.1 C in 1880 declining to ±0.05 C in 1995.

These uncertanties are smaller than the lower limit of resolution of the meteorological instruments. That should inform one of the level of competence we’re dealing with here.

Workers in the field also don’t seem to understand that the uncertainty in an air temperature anomaly must be larger than the uncertainty in the monthly measurement mean (minimally ±0.5 C) and the normal (also minimally ±0.5 C) used to calculate it.

The minimal uncertainty in any given 20th century GMST anomaly is then sqrt[(0.5)²+(0.5)²] = ±0.7 C.

The whole global warming thing is mortally subject to the scientific rigor learned at the level of the sophomore science or engineering major.

Tim Gorman
Reply to  Kip Hansen
December 11, 2022 9:50 am

Because they learned their statistics in STAT 101 which never addresses uncertainty and how to handle that. It’s like my youngest son when he was taking a microbiology major – his advisor told him not to bother with any statistics or engineering classes,,just find a math major to analyze his collections of data in the lab.

It just boils down to the blind leading the blind. Most math majors and climate science majors have never done any real world work with metrology where either customer relations or monetary/criminal liability can ensue from ignoring proper propagation of uncertainty. So they’ve never learned.

old cocky
Reply to  Kip Hansen
December 11, 2022 2:13 pm

Kip,

seriously trained (brainwashed?) 

the problem that many stats and numbers people have — they can only see things a certain way

That seems to be the crux of it. Everybody’s education trains them to see the world and approach problems in a particular way.
The deeper the knowledge, the more this applies.

JCM
Reply to  old cocky
December 11, 2022 2:29 pm

The deeper the knowledge, the more this applies.

I tend to argue that it is actually the uneducated who are more preoccupied with rules and procedures, and keeping up appearances. You may notice that children are totally obsessed with rule structures, whereas the elderly often couldn’t care less. A new graduate or wannabe will stop at nothing to share new things which they have learned, often lacking the proper context to understand what it means.

It takes wisdom to understand the array of tools, when it’s appropriate to apply them, and what they are telling you. These are tools to aid understanding for a research scientist, quite different than the way the rules are applied to lab standards organizations and technicians. The scientist’s job is to take in information and to make an informed judgement, not to just regurgitate computer output and proclaim it as meaningful. The first thought for any good scientist should be, “does this seem reasonable?”.

JCM
Reply to  JCM
December 11, 2022 2:45 pm

I should add that the job of the scientist is to inform – not to deceive. There seems to be a race to the bottom when it comes to finding data manipulation tricks to indicate certainty today. This has come about since the advent to desktop number crunching software. p-hacking routines and such. It may be a cultural thing, with the influx of hordes of students and handing out advanced degrees like candy. It is very competitive, and everyone wants to be seen. What is lost in the fold is the duty to inform. It should be understood that often the most useful insights are drawn from an admission of uncertainty, which allows new hypotheses to evolve. Hiding this uncertainty is actually a hindrance to advancement. It is the hardest part of any research to determine how to report the uncertainty, and to understand the subsequent consequences of these active cognitive decisions. What is for certain is that often the most useful information comes about from an appreciation of what is not known. This concept should be embraced.

Last edited 1 month ago by JCM
old cocky
Reply to  JCM
December 11, 2022 3:19 pm

In my more cynical moments, I think that programs such as SPSS did a great disservice to the world by allowing statistical analysis by those with insufficient understanding of the concepts, and knowing where particular approaches are and are not applicable.

old cocky
Reply to  JCM
December 11, 2022 2:55 pm

Those are all good points, but I wasn’t thinking of rules and procedures or showing off newly acquired knowledge.

There will always unquestionable axioms in any field – those things which you know so well that you don’t even know you know them.
We’re seeing that here with the slightly incongruent approaches of the mathematicians and metrologists.

JCM
Reply to  old cocky
December 11, 2022 3:03 pm

I see. I would recommend to start thinking like a scientist then. There is a risk of getting lost in the recesses of our datasheets and conceptual axioms, and to lose sight of reality.

old cocky
Reply to  JCM
December 11, 2022 3:15 pm

Recognising and questioning those axioms seems to be extremely difficult, even for scientists.
Those who can do so often seem to achieve great things. Marshall & Warren and their work on H. pylori should be an inspiration to all.

JCM
Reply to  old cocky
December 11, 2022 3:34 pm

agreed. we can put lipstick on a pig but at the end of the day it’s still a pig. If we forget this we all miss out on the good eatn.

ThinkingScientist
Reply to  Pat Frank
December 11, 2022 1:02 pm

Pat, do you mean the accuracy of the instruments or the precision of the instruments?

Jim Gorman
Reply to  Pat Frank
December 14, 2022 8:12 am

Pat,
Nice comment. Short, sweet, and to the point.

Somehow these folks have missed the fact that resolution is a method of displaying available information. Quoting calculations with a higher resolution than was measured is adding information from nowhere and that is fantasy.

It is the reason for Significant Digit rules. Too many here simply ignore
SigFigs as a childish endeavor not worthy of real scientific consideration.

The two issues combined is tragic.

Randy Stubbings
Reply to  The Dark Lord
December 10, 2022 1:41 pm

No kidding!

If the air in a 1 m^3 box from Churchill, Manitoba has a temperature of exactly -25 C (measurement error is zero) and the air in a 1 m^3 box from Honolulu has a temperature of +29 C (again assuming no measurement error), the (mathematical) average temperature is precisely 2 C. So what? If I join the boxes together while holding constant the physical properties of the two air masses and then remove the boundary between them to create a 2 m^3 box, the temperature of the combined air mass is unlikely to be 2 C after a new equilibrium is reached. Without additional information, such as pressure and humidity, about the initial state of the air in the two boxes, we don’t know what the equilibrium temperature of the joined system will be. Average energy or enthalpy would be more meaningful than average temperature, though I expect the uncertainties would be vastly larger because of yet more uncertainties in yet more measurements. Of course, no global average is of much use on a local level because there is no global “climate.”

Tim Gorman
Reply to  Randy Stubbings
December 11, 2022 9:52 am

Bravo!

Jim Gorman
Reply to  The Dark Lord
December 14, 2022 8:02 am

Other than these uncertainties also blow many of the predictions out of the water. Do you ever wonder why you never ever see either a standard deviation or error bars on GMT?

ThinkingScientist
December 10, 2022 1:02 am

So for length the uncertainty is additive. As Nick Stokes points out, if the uncertainty is expressed as standard deviation then they add in quadrature because variances are additive.

However, the uncertainty goes down from a daily measure to a monthly measure because the variation in monthly values is clearly less than daily values. In this case there is a change of scale (known as support in geostatistics) and the central limit theorem applies. Again, the variance is the linear quantity.

Note also that in change of scale there is an equality related to variance. For example for a set of a year of daily temperature measures the total variance (of the daily measures) equals the within monthly variance + the between monthly variance. So when changing scale the variance is redistributed but always preserved.

The change of support is why it is wrong to splice modern temperature series at, eg monthly resolution onto paleo-temperature series where the resolution might be several hundred years. The modern temps should shown as a single value with the reduced variance.

Last edited 1 month ago by ThinkingScientist
Nick Stokes
Reply to  Kip Hansen
December 10, 2022 10:46 am

“Stokes should explain to Gavin Schmidt”

No, you are misreading your Gavin quote. In fact he sets out the requirement to add in quadrature explicitly, and exactly as I did:

comment image

All he is doing in the part you quote is saying that you should add in quadrature (hence 0.502) and then round the uncertainty to 0.5 (because of uncertainty about the uncertainty).

Last edited 1 month ago by Nick Stokes
Nick Stokes
Reply to  Nick Stokes
December 10, 2022 11:05 am

Kip,
In fact, if Gavin had followed your rules, the uncertainty would have been 0.5+0.05=0.55°C.
But he used quadrature
sqrt(0.25+0.0025)=0.50249
which he then rounds to 0.5 (his rule 2)

Nick Stokes
Reply to  Kip Hansen
December 10, 2022 4:14 pm

Kip,
Where did he get the 0.5°? From the original measurement uncertainty”
There was no “original measurement”. Jones formed an average of many global temperatures over all times. The 0.5 is a statistical estimate. I think his global absolute mean temperature is based on data too inhomogeneous to average usefully, and should be paid little attention. Oddly, the GISS item that Gavin is modifying was saying just that, but somehow he is getting sucked in.

The fact is, no useful information is gained by adding that 14 to the anomaly average.

Clyde Spencer
Reply to  Kip Hansen
December 10, 2022 9:21 pm

I suspect that the anomalies will be correlated with the raw data, thus disqualifying the use of addition in quadrature.

Clyde Spencer
Reply to  Nick Stokes
December 10, 2022 9:18 pm

Adding in quadrature is justified if the measurements are uncorrelated. If there is autocorrelation, as in a time series with a trend, or it is not known whether or not there is correlation, then the more conservative simple addition is preferable.

Jim Gorman
Reply to  Nick Stokes
December 14, 2022 10:57 am

Dr. Schmidt has not proven that there is cancelation in the combined uncertainties of different temperature measurements. He also uses 0.5 which disagrees with the documentation from the NWS and NOAA. They show ±1°F for both ASOS and MMTS. I suspect older LIG readings are even larger.

In any case, one can not dismiss the direct addition of uncertainty intervals of different temperatures. This provides an upper bound on the uncertainty interval.

From Dr. Taylor’s book, An Introduction to Error Analysis. Page 59.

“That is, our old expression (3.14) for δq is actually an upper bound that holds in all cases. If we have any reason to suspect the errors in x and y are not independent and random (as in the example of the steel tape measure), we are not justified in using the quadratic sum (3.13) for δq. On the other hand , the bound (3.15) guarantees that δq is certainly no worse than δx + δy, and our SAFEST course is to use the old rule.”

3.13. δq = √[(δx)^2 + (δy)^2]
3.14. δq ~ δx + δy
3.15. δq ≤ δx + δy

The ±0.5 is basically caused by rounding error. It is not random and one does not know the exact temperatures that were rounded. One would have to assume that not only the signs cancel but every value between +0.5 and -0.5 of each temperature would cancel in the two daily values, the 30/31 monthly values, and 12 annual values.

Jim Gorman
Reply to  Kip Hansen
December 14, 2022 9:18 am

Kip,, you are exactly correct. The NWS and NOAA describe what is basically the combined standard uncertainty for ASOS and MMTS as ±1°F. This number applies to each and every reading taken by these system. I read somewhere that prior to 1900, the uncertainty was assumed to be ±4°F.

One only has to do simple math to confirm this. Find the mean of two numbers with ±1. The mean will also vary by ±1. Now try with more numbers.

81/82->81.5
80/81->80.5. 80.5 ±1
79/80->79.5

79/80/81/82/83/84 -> 81.5
80/81/82/83/84/85 -> 82.5. 82.5 ± 1
81/82/93/84/85/86 -> 83.5

You can use all the statistics you want but this doesn’t lie. With single temperature readings averaged together with all having the same combined uncertainty, the mean will carry that uncertainty also.

Singular readings have no distribution to cancel error in a given reading nor any other single readings. This is where Tim Gorman has tried to make folks aware that calculating a mean using exact numbers is what is being done rather than considering that each and every data point has an uncertainty that moves through each statistical calculation.

In essence much of the argument about uncertainty is really moot. There are published documents from the NWS and NOAA that tell what the intervals should be. Climate scientists should be using these intervals and not replacing them with some statistical calculations where averages and dividing by “n” reduces the intervals.

Tim Gorman