# Dealing with experimental data

I have some experimental data about a value $n$, now, I am supposed to give, in the ending, a single value with an error: $n=a\pm b$. I have originally 6 values of $n$, each one comes as an indirect measurement from direct measurement, each one with it's systematic errors, so in the ending I have those 6 values, each one with an error.

So what I guess I have to do is to mix the systematic error with the random error the way I've been taught $(E_{sys}^2+E_{rand}^2)^{1/2}$. The systematic is already calculated, what do I use for the error? the mean of systematic errors?

Another question is that those values, which are by the way moles of a quantity of a gas, have been got from different ways (basically from calculating different isothermic curves and getting the $n$ value that best fits each of them), so they're actually not from the same kind of measurement, but from different ones. This makes me doubt about how this would affect the calculation of the final error, if it does, or if I can just do it the way I said above.

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I think that you may have mis-transcribed the formula for combining errors above: it has the wrong units. The usual prescription is to combine them "in quadrature" meaning $\left( \sum E_i^2 \right)^{1/2}$. –  dmckee Mar 5 '13 at 22:04
@dmckee Yes it's wrong, I'm fixing it, thanks, that was a typo. –  MyUserIsThis Mar 5 '13 at 22:05

The answers depend on a number of details, and without knowing more about the actual situation you face I can give only a very general prescription.

Assuming that you have uncorrelated errors, you would form the error-weighted mean

$$\bar{n} = \sigma^2 \sum_i \frac{n_i}{E_i^2} \, ,$$

where the variance of the mean is

$$\sigma^2 = \sum_i \frac{1}{E_i^2} \, .$$

With more information it might be possible to do better, but this is the way to punt in the absence of a better scheme.

That said, you describe the errors as "systematic", which introduces a very real possibility that they are not uncorrelated and this analysis will under-estimate the real uncertainty that you face. There is quite a bit of detail in the linked wikipedia article, though it is somewhat terse.

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Thanks for your answer. Systematic is probably a bad translation, I mean the error you make because of the instrument of measurement. –  MyUserIsThis Mar 5 '13 at 22:04
I understood about the errors, the problem is that if your instrument or method gets a, say low, value then it might get a low value on all measurements. Those errors are correlated and invalidate the above analysis. –  dmckee Mar 5 '13 at 22:07
Ok, I see, what else would you need to know? The experiment is measuring values of pressure-volume to calculate isothermic curves. We then fit those curves to some equation of state, tipically Van der Waals, as we're dealing with phase transition, and we get the van der Waals coefficients, a, b, and the number of moles, n, I have to get these done for all three numbers. So what I have is 6 values (from six fits to 6 isothermic curves) for a,b,n. And now thinking about it, the error for each of them is the error of the fitting, not a systematic one. How would one approach this? –  MyUserIsThis Mar 5 '13 at 22:11
The biggest thing here is to examine the tools and methods to try to identify what correlation might exist in your data (in my field we would also assign a "model dependent" uncertainty by examining how the number vary under different assumptions about the equation of state (from among 2 or three of the best available models)). Repeating a single measurement six or more times would allow you to work out what part of you error is random (uncorrelated), but evaluating true systematics is hard work. At some point you have to ask your self if it is worth it. –  dmckee Mar 5 '13 at 22:19
Ok, that's good enough, I will accept this answer and follow your advice (it's totally not worth it). I will simplify things. –  MyUserIsThis Mar 5 '13 at 22:37