Friday, May 6, 2016

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.



Answer



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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