This is my first post, so I would like to say hi to everybody.
This is not exactly homework, but I thought that this was the most appropriate sub-forum... I have problems with measurement uncertainty propagation, so I was wondering how other people cope with it. This is one problem that I couldn't solve so far.
We perform radon measurements, and each lasts 3 days. Let's denote the result of radon concentration with xi, and associated uncertainty with uxi. Now we need to calculate average monthly concentration, which is trivial, but also associated uncertainty, which proves to be almost impossible, for me at least. Obviously, standard deviation and standard error are inadequate, because we do not measure the same thing 10 times, but 10 different things. For example, if we can get results 100±1, 200±1, 300±2, the uncertainty of average value would be very low, because our measurements are very precise (less than 1%). However, standard deviation would be very large and is totally inadequate for this problem. The same is true if you have results like 30±8, 29±7, 31±6. Uncertainty of average is obviously much higher than standard deviation would indicate.
So, another solution is to use the formula for propagation of uncertainty, u2f = Ʃ(±δf/δxi)2uxi2. In our case, f is function for average, so f = 1/nƩxi, so we get u2f = 1/n2Ʃuxi2, which looks nice. However, imagine that all uxi are the same values, or nearly the same. In that case, u2f =1/n2 * nuxi2, or uf=uxi/√n. Sooo, if we find the average of infinite number of radon concentration measurements, we have zero uncertainty. But it is not true, because there are some "systematic" errors associated with the same measurements and some of them are not random, i.e. they influence each measurement the same way. So we need also to take into account the correlation between measurement uncertainties, which seems very complicated to me. Each uxi has at least 7 inputs and all would need to be investigated.
I asked around, but it seems that nobody in my surroundings has given it any thought. How do you deal with similar situation? Is there an easier way to calculate this, or do you use some approximation, or do you actually estimate all the covariances? Or do you somehow split uxi2 in two parts, one "systematic" and other random and then apply formula u2f = 1/n2Ʃuxi2 for random part and add systematic effects to that?
