The probability of obtaining an average value \( z \) is the product of the probabilities of obtaining arbitrary individual mean values \( x_i \), but with the constraint that the average is \( z \). We can express this through the following expression
$$ \tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m) \delta(z-\frac{x_1+x_2+\dots+x_m}{m}), $$where the \( \delta \)-function enbodies the constraint that the mean is \( z \). All measurements that lead to each individual \( x_i \) are expected to be independent, which in turn means that we can express \( \tilde{p} \) as the product of individual \( p(x_i) \). The independence assumption is important in the derivation of the central limit theorem.