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 \( z \).