UAH v6.1 Global Temperature Update for September, 2026: +0.61 deg. C

You keep asking that and ignore everyone who points out that you don’t need to subtract to make a number smaller. The uncertainty of an average gets smaller not because you are subtracting but because you are dividing by √N.

“The uncertainty of the mean gets smaller by 1/√n. The measurement uncertainty of an input quantity is the standard deviation. The standard deviation is not decreased by 1/√n.”

Correct, except an input quantity may be a mean of multiple measurements in which case it’s uncertainty is divided by √n.

“The only average shown in the GUM Section 4 is when there are multiple observations of a measurand that form a random variable “q” that contains the qₖ observations.”

This is unrelated to equation 10 – but yes, section 4 explains the “experimental standard deviation of the mean”, the same as SDOM in Taylor’s book. Both cases are using the same concept as the SEM, but applied to measuring a single value.

“The mean of qₖ observations is the best estimate of the value and the measurement uncertainty is the standard deviation σ. Neither of these are divided by √n. ”

No. You keep ignoring section 4.2.3. The section that says the experimental standard deviation of the mean may be used as measure of the uncertainty of the mean of the qₖ observations. And that’s when you are dividing by √n.

“The reliability of the mean, the interval where the mean may lay, is the standard deviation of the mean (SDOM). It can be estimated by σ/√n. It is not the measurement uncertainty.”

They tell you in 4.2.3 that it is the measurement uncertainty of the mean.

“I know you base your assertions by making a functional relationship describing the measurand to be an average.”

Yes, that’s where equation 10 comes in. This is different from measuring the same thing multiple times as in 4.2.3. It’s saying if you want to find the combined measurement uncertainty of any functional relationship, you can use that equation to propagate the individual measurement uncertainties.

“You cannot have both a functional relationship average using observations AND a Type A evaluation using a random variable that contains the observations also.”

Why not? You were the ones who insisted that you had to use equation 10 to determine the uncertainty if the mean, then when it turned out it still gave you a division by √n, you say you can’t use it.

“That results in two averages of the same observations.”

No it doesn’t. The assumption of using equation 10 for the average is that you are averaging a number of different things, each with a different value. You have a measurement for each thing, which may come from a single measurement with a Type B uncertainty, or the result of measuring one thing multiple times with a Type A uncertainty. It doesn’t matter in order to use equation 10. It’s just propagating the individual uncertainties into a combined uncertainty.

“If you want to impress people take the temperatures from NIST TN 1900 Example 2”

Please no, not that again.

They are not using equation 10. It’s just applying a sampling uncertainty to the mean, i.e. calculating the SEM of the different daily values. They justify this as measurement uncertainty terms by considering the daily values as measurements of the same thing under conditions of repeatability, and so use 4.2.3 and calculate the experimental standard deviation of the mean.

“show the math about how you would perform a Type A evaluation using GUM Section 4.2,”

Take the daily values and average them. Take the standard deviation and divide that by √n. Calculate an interval using a student-t distribution with n – 1 degrees of freedom.

As I keep saying, I think it’s questionable to call this the measurement uncertainty of the monthly average, as it has little to do with any uncertainty in the individual measurements.

“then create a combined uncertainty using a functional relationship reusing the temperatures again to calculate a monthly average measurand.

For that you would need to know the Type B uncertainty of the station, which they don’t state. As the resolution of the stated values appears to be 0.25°C, let’s assume that this is the standard uncertainty. So now it’s just a question of putting in these uncertainties into equation 10 with partial derivatives of 1/22 for each input. Which will come to

√[22 * (0.25/22)²] = 0.25 / √22 = 0.053°C.

That would be the measurement uncertainty of the exact average of those 22 daily values.

However, you then have the big problem that there are 9 missing days, so if you want to know the uncertainty of the actual monthly average.

A simple way of handling that would be to treat them as coming from the same distribution as the other 22 days, then each of the missing days has an uncertainty of 4.1°C

Reworking equation 10, but now with 22 days with an uncertainty of 0.25 and 9 with an uncertainty of 4.1 and we get (hopefully)

√[22 * (0.25/31)² + 9 * (4.1/31)²] = √[22 * 0.25² + 9 * 4.1²] / 31 = 0.40°C.

This is smaller than the TN1900 version because it’s assuming we want the uncertainty of the actual monthly average rather than treating all the values as a random sample.

“You are STILL CONFLATING SAMPLING UNCERTAINTY WITH MEASUREMENT UNCERTIANTY!”

Where? Equation 10 is about measurement uncertainty.