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[Phys-L] uncertainty on the uncertainty



Hi Folks --

Over the years, I have spent an unreasonable amount of time looking
into the issue of uncertainty on the uncertainty, especially for
small sample sizes.

Suppose there is a sample consisting of N readings, drawn from
some parent population, and the task is to compute the mean and
standard deviation.

In particular, consider the factor of (N-1) that appears in the
denominator in the formula for the standard deviation. That shows
up in lots of places, including e.g. Baird equation 2.9, where it
is touted as the "best estimate". It is baked into the stdev()
function in every spreadsheet I've ever seen.

The question arises, where did that come from????!?!?!!!!!?

Contrary to what is nearly-universally claimed, (N-1) does not
produce an unbiased or optimal estimator for the standard
deviation of the parent population. It turns out that (N-1.5)
is vastly better across the whole range, and even that's not
perfect for small N.

On top of all that, there's the fact that even if it were unbiased,
it would still be an insanely /noisy/ estimator (for small N).
There's a huge uncertainty on the uncertainty.

There shouldn't be any big mystery about this. The properties
of the chi distribution have been understood for quite a while
now.

I take this as yet more evidence that most of the people who
insist that it's super-important to calculate a number for
the uncertainty never actually use it for any authentic
purpose, and never check it against experimental reality
(or even mathematical reality).

I am reminded of the proverb of which the contrapositive
says: If it's not worth doing right, it's not worth doing.

Why do people pretend this is important, in introductory lab
situations where it clearly isn't? Why not just skip it most
of the time, and save it for the rare occasions where you
have a big-enough N to make it worthwhile?