Suppose we have a PDF p(x) from which we generate a series N of averages \mathbb{E}[x_i] . Each mean value \mathbb{E}[x_i] is viewed as the average of a specific measurement, e.g., throwing dice 100 times and then taking the average value, or producing a certain amount of random numbers. For notational ease, we set \mathbb{E}[x_i]=x_i in the discussion which follows. We do the same for \mathbb{E}[z]=z .
If we compute the mean z of m such mean values x_i
z=\frac{x_1+x_2+\dots+x_m}{m},the question we pose is which is the PDF of the new variable z .