Joe wrote:
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.

I haven't had much lab experience, but here goes:

If I measure a ball to weigh 5kg, and I have measured the uncertainty
of that measurement to be 2 grams, and the uncertainty of that
calculation to be 0.1 grams, I will just say I measured the ball
to weigh 5kg, give or take 2.1 grams.

I think that in most if not all cases, the margin of error
gradually decreases: if you have some fairly effective way of
measuring things, your uncertanties will be 1 or more orders of
magnitude lower than the measurement.

By selecting a worst case scenario early in the process, you might
miss out on realizing that the error of measurement on the balls
weight was actually 2.101 grams, but isn't 2.1 or even 2 grams
accurate enough? Perhaps someone could invent an "order of error", so in
my case I would state that the ball is measured to weigh 5kg, with an
uncertainty of 2.101g, order of error 3, or 2.1g, OOE 3.

I suspect that even if you set out to calculate the errors on the errors
on the errors on the.... of your measurements, you would hit on the
limits of your equipment/methods pretty soon.

In the end, isn't this a case of pushing well-meant statements beyond
their limits? :-)

Bart