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? Error bars, and all uncertainties,
are not made up, they coonstitute measurements too. They are numbers
too. And like all such things, they have their own uncertainties.
Musn't we find them and quote them??? Why give a value for the
electron charge, and quote the uncertainty in the measurement, without
quoting the uncertainty in the uncertainty! In fact, to quote the
professor, it is meaningless to quote uncertainties, without values
for their uncertainties. I am taking a lab now. Like a robot I want
to quote the uncertainties in my uncertainties, but I have been
explictly told not to do this. They will take points off to
discourage me. And yet, if I quote measurements without any
uncertainties, they'll give me a zero. The hippocrites!! I propose
finding the uncertainties in our uncertainties, and the uncertainties
of those uncertainties, and the uncertainties of those uncertainties,
and so on add infinitum! I wonder if in fact a measurement is ever
complete until this is done.

You are in principle correct. For example, a standard deviation, which
expresses the width of a distribution of a measured quantity in a
sample of measurements, also has an uncertainty associated with it.
And scientific papers that quote a measured number and an error bar
have to defend their methods both for arriving at the number and for
arriving at the error bar. But there comes a point of diminishing
returns. After all, the point of the original uncertainty is to
estimate how far you should trust that measured number. So a little
tweak in the value of the uncertainty (because that's uncertain in
itself) will deflate your trust a little more, but so what? The series
rapidly converges in most cases.

In practice, the main purpose of the error bar is so that, when you
have two independent experiments that have measured the same number
and come up with slightly different values, you can look at the
respective error bars and deduce the odds of that being a significant
disagreement.

PD