# Tricky error calculation

1. Jan 22, 2005

### JamesJames

How do I calculate the uncertainty of

q = [exp(x/2)][t^(1/2)]

where both x and t have known uncertainties.

I could have done the thing if there was no exp(x/2) term. But that term is causing me a lot of stress.

I am really confused.
James

2. Jan 22, 2005

### cepheid

Staff Emeritus
In one of my lab courses, we were given a short write-up on error analysis. It gave the formula that I'm going to put below. So you are trying to calculate a quantity q that depends on two other quantities you have measured (and therefore whose experimental uncertainties you know): x and t. Well, the uncertainty in both x and in t will affect the error in q, because q depends on both. Assuming that x and t are independent variables, the formula we were given for "Errors propagating through a functional relationship:"

$$\delta q^2 = \left(\frac{\partial q}{\partial t}\right)^2 \delta t^2 + \left(\frac{\partial q}{\partial x}\right)^2 \delta x^2$$

where delta q represents the uncertainty in q, for example. This formula makes sense to me sort of, because the uncertainty in q depends on the individual uncertainties in x and t, as well as the rate at which q changes with each one. For instance, if dq/dt (<--meant to be partial) is large, the even a small delta t will affect q significantly. However, a friend of mine with a math BSc was telling me that these formulas aren't strictly correct, and that you're not actually supposed to be doing calculus per se. Error analysis seems to me to be a very complicated, convoluted subject. Hopefully somebody here will be able to comment on whether using this formula is indeed the best method.

3. Jan 23, 2005

### Dr Transport

At this time the formula that cephid quoted is correct. You have KNOWN errors, the problem arises when you do not know the errors. I have been reading alot about error calculation lately because my employer needs to know the no (&()&) error for some of the apparatuses we use and bought commercially. The calculation of errors is essentially a mathematical problem, a very complex mathematical problem. If you go to the NIST website (www.nist.gov) or the NPL (www.npl.co.uk) and look for uncertainty analysis you'll find a massive amount of material out there. A good place to start is with John Mandel, Statistical Analysis of Experimental Data, I have learned enough to be somewhat dangerous.