I have a set of *N* bodies. The size of each body is being measured $m_i$ times ($m_i>1$ and different for each body). I would like to describe the resulting measurement. Particularly I'm interested in average body size and in the variance.

The average body size is simple. First calculate the mean sizes 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 sizes. In order to have an idea on the confidence we have in any single measurement, we need to account for both the sources. Can anyone help me with this part?

Thank you

**Updates and clarifications **

- The size of the i-th body is measured mi times, so that the index i identifies which body it is
- The set of N bodies supposed to be a random sample from a population whose mean and variance I want to estimate