UAH v6.1 Global Temperature Update for December, 2024: +0.62 deg. C

Do you really not see that saying the relative uncertainty of the mean equals the relative uncertainty of the sum, implies that the absolute uncertainty of the mean is equal to the absolute uncertainty of the sum divided by N?

“u_cr(X_bar) * X_bar = u_r[ sum (X_i) ] * sum (X_i)

u_c(X_bar) = u[ sum (X_i) ]”

Wrong.

“u_cr(X_bar) * X_bar = u_r[ sum (X_i) ] * sum (X_i)”

Is wrong. You’ve multiplied both sides by different amounts. X_bar on the left, sum (X_i) on the right.

Multiply both sides by the same value X_bar, and remember that X_bar = sum (X_i) / N.

u_cr(X_bar) * X_bar = u_r[ sum (X_i) ] * X_bar

=> u_cr(X_bar) * X_bar = u_r[ sum (X_i) ] * sum (X_i) / N

=> u_c(X_bar) = u[ sum (X_i) ] / N

But we could trade equations all day. If you can’t see that 1% of 20 is smaller than 1% of 2000, I doubt you are going to be persuaded by algebra.

“No root(N) in here.”

No. It’s N, not root(N). As I’ve said from the start, if the uncertainty of the sum of 100 thermometers is ±5°C, the uncertainty of the average is 5 / 100 = 0.05°C.

“If the relative uncertainty of the sum is 1%, the relative uncertainty of the mean is also 1%.”

Good, so we are all agreed that the relative uncertainty of the average is the same as the relative uncertainty of the sum. Now what do you think that does to the absolute uncertainty of an average. Does it get larger as N increases or does it get smaller?

“And do you not see that the average uncertainty is a meaningless number.”

The uncertainty of the average. Not the average uncertainty. And, no, I don’t think it’s a meaningless number, given that it’s what we’ve been arguing about for the last 4 years. If you think it’s meaningless, why do you worry about how large it is?

“Look at it from a measurement standpoint not some math possibility.”

Measurement is math.

“A measurement of the same thing has a probability distribution made up of the observations you take of that given measurand.”

You keep saying that, and I’m not sure if it’s even worth correcting you at this point. The observations do not make up a probability distribution. They are random values taken from a probability distribution.

So anyway, you describe finding the experimental standard deviation of the mean of four different values. And say

“The combined uncertainty is determined by the correct addition of the uncertainties of each input variable, not an average of all four uncertainties.”

And the correct propagation, assuming you are taking an average of the four values is given by using the law of propagation, with the function (a + b + c + d) / N. Which is karlo correctly points out leads you to the uncertainty of the sum of the 4 values divided by 4,

Uc(Avg) = Uc(Sum) / 4

or dividing through by the average gives

Uc(Avg) / Avg = Uc(Sum) / Sum

“You and bdgwx are wanting to skip the process of analyzing the probability distribution of a series of measurements which results in the average and variance.”

It doesn’t matter what the process is. You can use your 4 mean values, as a Type A uncertainty, or you can use a Type B uncertainty. The point we are trying to make is regardless of how you came by the uncertainties, you need to understand how to use the law of propagation. Everything you keep saying is just trying to distract from the obvious point, that the law does not result in uncertainties increasing the more things you average.

“Therefore, you get what the GUM 4.2 says.y = f(a, b, c, d).”

Yes, that’s what a function looks like. I’ve no idea why you keep pointing out these things as if they were some sort of magic.

“Each “a, b, c, d” has its own unique and independent uncertainty which are added to find a combined uncertainty.”

They might have different uncertainties, we are assuming they are independent, that’s a basic assumption of equation 10. But you are not just “adding” the uncertainties. You are multiplying each by the partial derivative and adding in quadrature.

“what you are trying to do is scale everything by a value, in your case by “n”.”

Yes, that’s what happens. If you would try to understand why the equation works, you might realize that scaling any value is the most basic application of the general law. If the function is Cx, where C is a constant, then the derivative is C, and the uncertainty is multiplied by C. If that wasn’t true then none of it would work.

“However, “n” can be any value, not just the number of terms you “average”.”

Yes, that’s the point. But in the case of an average 1/n is the scaling factor.

“If they don’t make up a distribution then how do you get any statistical descriptors?”

The statistical descriptors are for the sample. They are estimates of probability distribution the sample came from.

Of course, the sheets must be known to be equally thick.

Could you explain why you think Taylor makes that assumption, and how the calculation would change if it didn’t hold?

“Dr. Taylor doesn’t state it, but one must also assume that the uncertainty of each sheet must be identical.”

That makes no sense. There is no direct measurement uncertainty of the individual sheets. You are making just one measurement of the stack with an associated uncertainty. Then estimating the width of a single from that measurement, with the uncertainty of that estimate calculated from the uncertainty of the stack. How could different sheets have different uncertainties?

Nobody should suggest the paper example is a good practice. It’s impossible to claim that all sheets are identical without measuring each one first. And it ignores all the problems with the physics of stacking paper.

“You immediately look for mathematical methods to make the uncertainty as small as possible.”

You keep saying that as if it’s a bad thing. Why would you not want to design an experiment to reduce uncertainty. If I were doing any sort of survey I think it would be common sense to try to reduce uncertainty. That can be by choosing a sensible sample size, trying to reduce the risk of systematic bias, or any other good practice.

I should say that in the real world there is often a balance between uncertainty and cost or ethics.

“That is why you never consider using an expansion factor to calculate total uncertainty.”

Pardon? The whole idea of expanded uncertainty comes from statistics. That’s what significance testing does. That’s what a confidence interval is.

“I want to know what range of measurements to expect at a 68% or even better at 95% interval.”

And as I keep saying,you can do that, and on many cases it’s the most important statistic you want. But it is not the uncertainty of the mean.

Could you explain why you think Taylor makes that assumption, and how the calculation would change if it didn’t hold?

And once again, rather than answer that question a Gorman just cuts and pastes the entire section from Taylor. I know what Taylor says. I’m asking you, why you think he said it.

“This can result in different sheets having different thicknesses. If their thicknesses are different then their uncertainties will be different as well.”

OK, so you are not talking about their measurement uncertainties, rather the uncertainty in the specification. But as the assumption is that they are all have identical thickness, why claim there’s an addition assumption that they all have the same “uncertainty”?

“It’s not even obvious that you have ever run multiple sheets through the auto sheet feeder on a home ink jet printer to copy them.”

You wouldn’t believe how much I’ve had to do this.

“It’s not unusual to have a jam because an individual sheet is too thick or too thin for the feed rollers to handle it.”

Uncertainty in paper size shouldn’t do that. Misaligned paper, dirty rollers, or paper that is stuck together is much more likely. That and the fact that ink jet printers are the spawn of Satan.

If you ever paid attention to the discussion you would realise I’ve been going about it since the start.

You did claim to have taken a course in logic didn’t you?

There’s really no excuse for not getting it by this point. When you want a SUM of multiple measurements you ADD the uncertainties. When you want the AVERAGE you have to also divide the uncertainty of the sum by N.

This follows either from the special rule you quote from Taylor, where x is the sum, q is the average and B is 1/200.

Or you can do it directly from the law of propagation (eq 10), and using the function f(x1, x2, …, xN) = (x1 + x2 + … + XN) / N.

“There is a reason Dr. Taylor shows the equation 3.9 using the |B|. Can you guess why?’

If you are asking why he uses the absolute value of B, it’s because uncertainty cannot be negative.

I don’t want to guess your motives, but is it possible you think absolute means it can’t be less than 1?

Comments will be closed soon, so I’ll have to bookmark this comment as a useful digest if every mistake the Gormans and co keep making, no matter how many times it is explained.

You manage to go through just about every way you could address the question of how to estimate the uncertainty of an average

Using equation 10 the input quantities are the measurements. The assumption here is we want an exact average of the values, and want to see what effect the measurement uncertainty has on the uncertainty of that average.

If you want to include the variation in the actual measurements, then that’s what you get with the SD/√N equation. This is more appropriate as the different measurements can be considered a sample from the population, and it’s the population mean we are really intrested in.

In most cases the sampling uncertainty will be much larger than the measurement uncertainty, which is why uncertainty from individual measurements is not a major concern.

But whatever uncertainty you are interested in, it does not grow with sample size, and usually gets smaller.

Scale as in, if x is a measurement with uncertainty u(x) and q is a quantity derived from X by multiplying by B, where B is an exact number with no uncertainty, then

u(q) = |B|u(x)

“There is one other interesting thing about the stack of paper in Taylor.
Taylor Eq. 3.9 δq = |B|δx”

Yes, he’s using the thing we’ve been saying since year 0. The uncertainty scales with the measurement. The stack of paper example is just one example of how to use this. Other examples would be dividing the circumference of circle by 2π to get the radius, or multiplying the radius by 2π to get the circumference. It’s this simple point that you keep denying, as your whole claim that you never reduce the uncertainty in order to claim that the uncertainty of the sum is the same as the uncertainty of the average.

“His example used δq/|B| = δx”

You can say that, or do what Taylor does and use |B|δx with B = 1/200.

“Let’s assume we can measure one single sheet whose thickness is 0.01 mm ±0.005 mm.”

That’s very thin paper. No wonder you get so many jams.

“Funny how those uncertainties are ADDITIVE, and directly additive at that!”

It’s directly additive because you adding things with complete dependence in the uncertainties. If for some reason you wanted to measure every single sheet, the uncertainty would be (assuming you had the capabilities to measure to that much precision).

√200 x ±0.005 = ±0.07 mm

I’ve no idea why you get so exited by the word “additive”. As a guess, are you trying to imply uncertainty can only increase never decrease?

“Uncertainty is an intimate function of the end-to-end system used to measure a quantity, this statement is meaningless.”

Good grief. I am not doing an “end-to-end” analysis. Simply pointing out how you keep misunderstanding the part about propagating the uncertainty when you scale measurement. All you distractions involving additional aspects of uncertainty is irrelevant if you can’t get that aspect correct.

I don;t care how many aspects there might be in designing a building, if you keep telling me that the hypotenuse is equal to the sum of other two sides, I know your result will be wrong.

“Uncertainty is not an abstract number”

All numbers are abstract.

“that you try to minimize to make your results look good.”

Nobody is doing that. What we are trying to do is explain how to correctly calculate those uncertainties.

“It is supposed to be an honest assessment of the quality of a measurement result.”

Which is what I hope we all want. But claiming the measurement uncertainty of an average can be much bigger than any individual measurement does not strike me as honest.

Spell what out? It’s your claim that the uncertainties are huge and it’s impossible to reduce uncertainty by averaging.

Tim says the average of 100 thermometers with an uncertainty of 0.5°C will be 5°C. That’s all that has been spelled out. Yet when I challenge that basic mistake, you insist that I’m the one who has to create a defined measurement procedure for these 100 thermometers.

And now we are back to the ad homs.

Why does that person we are constantly insulting and saying he doesn’t understand basic algebra, constantly defending himself. Must be because he’s paid to do it.

The same karlomonte who objects to me answering questions is also the first to bring out the old “you couldn’t answer” insult, if I don’t respond within ten minutes.

Right on cue.