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Re: [Phys-L] uncertainty



On 01/12/2015 12:40 PM, Paul Lulai wrote:
Question 2: I know uncertainty can also be done in quadrature, and
through the use of partial derivatives. I never learned how these
are known to be valid, or how they can be justified. If anyone has a
link that demonstrates this, I would appreciate some help.

The conditions for validity of the calculus approach are
highly technical. The approach is better for getting a
qualitative feel for what might happen. That's because
actually verifying the validity is so much work that almost
nobody ever does it. And if you don't verify it, the results
are quite likely to be invalid.

Adding uncertainties in quadrature is a corollary of the
calculus approach. You need the calculus in order to tell
you /what/ you are adding in quadrature (relative uncertainty,
absolute uncertainty, or whatever). Also note that if you
are trying to estimate the uncertainty of tan(89° ± 2°) you
aren't "adding" any uncertainties.

Question 3: Do folks have a preference btn 3 crank, quadrature, and
partial derivatives?

Crank three times is appropriate at every level from
grade school on up. It is a stepping stone to Monte
Carlo, which is extreeeeemely powerful.

The calculus approach is wildly inappropriate at the
high school level.

The special case of plain old sums looks simple but really
is not. If you want the errors to add in quadrature,
you need to assume things are Gaussian distributed (which
often they are not), and assume they are IID (independent
and identically distributed) (which often they are not),
et cetera. This works often enough that it is worth
mentioning, but it is not a good starting point and not
a good ending point, because there are waaaay too many
things that can go wrong.

Let me add one thing to the list: A proverb:
When all else fails, look at the data.

Find a way to plot the data, and use that to estimate
the uncertainty.

The following is probably fancier than what you need in
high school, but it is super-useful, and you probably won't
see it discussed anywhere else:
https://www.av8n.com/physics/probability-intro.htm#sec-scatter-diaspogram
especially
https://www.av8n.com/physics/probability-intro.htm#sec-diaspogram