Question regarding errors & uncertianty

[SimpliciusH]

I have a simple task of measuring the changes in volume (delta V) of water due to rising temperatures. I have trouble combining certain errors.


How would one get the error for (V0-V)/V if one already knows the error for V0(volume at start of measurement) and V(measurement of volume).


I know how to combine relative and absolute errors when dealing with +,-,*, / I'm assuming I can't just use these rules here since the errors of V-V0 and V are... well related.


I would also really appreciate if anyone could also give a link to a good online or downloadable resource which I can use to study and perhaps solve examples of how to calculate and combine errors.


Thanks for all your help. :) Sorry for bothering you with such easy problems.

 

[ideasrule]

(V0-V)/V = V0/V-1. Now you can use the usual rules for combining errors when one number is divided by another.

 

[diazona]

I have a simple task of measuring the changes in volume (delta V) of water due to rising temperatures. I have trouble combining certain errors.How would one get the error for (V0-V)/V if one already knows the error for V0(volume at start of measurement) and V(measurement of volume).I know how to combine relative and absolute errors when dealing with +,-,*, / I'm assuming I can't just use these rules here since the errors of V-V0 and V are... well related.

I don't think there's much to say about that without specifying just how the errors are related.

Usually (as you may know) whatever measurement error you're dealing with can be split into systematic errors, which are the same (or at least are strongly correlated) for all measurements, and random errors, which are, well, random. Any systematic errors you know about, you might as well correct for before plugging the numbers into your calculations, and then all you can do in general is assume that any remaining errors are random, specifically with a Gaussian distribution (because that tends to be the case in practice). So I guess in so many words I've just justified doing what ideasrule told you to

There is a general formula for error propagation, which looks like this:
$$\delta f = \sqrt{\biggl(\frac{\partial f}{\partial x}\biggr)^2\delta x^2 + \biggl(\frac{\partial f}{\partial y}\biggr)^2\delta y^2 + \cdots}$$
where f is the function you're trying to calculate (in your case $(V_0 - V)/V$, and x, y, etc. are the measurements you're using to calculate it. So in your case,
$$\delta f = \sqrt{\biggl(\frac{\partial f}{\partial V}\biggr)^2\delta V^2 + \biggl(\frac{\partial f}{\partial V_0}\biggr)^2\delta V_0^2}$$
and then you can do the math from there. (Or if you can't, I guess that formula doesn't do you a whole lot of good anyway).

I don't know of a good online resource but the book "Introduction to Error Analysis" by John Taylor is usually considered to be pretty good, if you don't already have it.

 

[SimpliciusH]

(V0-V)/V = V0/V-1. Now you can use the usual rules for combining errors when one number is divided by another.

:facepalm:

I feel so stupid. Thank you both for the answer. :)