  Given a permutation statistic $s : S_n \to \mathbb{R}$, define the mean
statistic $\bar{s}$ as the statistic which computes the mean of $s$ over
conjugacy classes. We describe a way to calculate the expected value of $s$ on
a product of $t$ independently chosen elements from the uniform distribution on
a union of conjugacy classes $\Gamma \subseteq S_n$. In order to apply the
formula, one needs to express the class function $\bar{s}$ as a linear
combination of irreducible $S_n$-characters. We provide such expressions for
several commonly studied permutation statistics, including the excedance
number, inversion number, descent number, major index and $k$-cycle number. In
particular, this leads to formulae for the expected values of said statistics.
