I have a set of $N$ bodies, which is a random sample from a population whose mean and variance I want to estimate. A property of each body is being measured $m_i$ times ($m_i>1$) and different for each body index $i$ identifies which body it is; the property is expected to be distributed around zero). I would like to describe the resulting measurement. Particularly I'm interested in average property value and in the variance.
The average value is simple. First calculate the mean values for each body and then calculate the mean of means.
The variance is more tricky. There are two variances: the variance of measurement and the variance of property values. In order to have an idea on the confidence we have in any single measurement, we need to account for both the sources. Unfortunately, I can't think of a good method. It is obvious that putting all the numbers in a single pool and calculating the stdev of this pool isn't a good idea.
EDIT Colin Gillespie suggests applying Random Effects Model. This model seems to be the right solution for my case, except for the fact that it is described (in Wikipedia) for the cases where each group (body in my case) is sampled equally ($m_i$ is constant for all the bodies), which is not correct in my case