Joe wrote in message ...
A professor once said that you need to list all uncertainties in your
lab, for all measurements, all numbers. Whenever you quote a number,
it must have some error bars. In fact, he said, it is meaningless to
quote a number without error bars. But using that logic, why don't we
put error bars on the error bars?

Using the Student's t distribution takes into account the uncertainty
of the uncertainty. So if your measurements are good you CAN be 95%
certain that the true number is in your confidence interval. That's
what the error bars stand for, a confidence interval.

Once the sample size gets above about 30 then Student's t is
practically identical to the Gaussian/Normal distribution, because the
uncertainty of the uncertainty is almost zero.