Population genetics and the fitness probability distribution. Why is the arithmetic mean all we need?
It's simplest to think about a haploid population with size $N$. Say there are two alleles, $A$ and $a$, with $A$ having frequency $p$ in this generation. Each $A$ individual will have some realized fitness (i.e., number of offspring) in this generation; let's call this number $w_A^{(i)}$ for the $i^\text{th}$ individual. The total number of $A$ individuals in the next generation is then $\sum_{i=1}^{Np}w_A^{(i)}=Npw_A$, where $w_A$ is the arithmetic mean of the fitness distribution, $w_A\equiv\sum_{i=1}^{Np}w_A^{(i)}/(Np)$. So regardless of how weird the distribution of the $w_A^{(i)}$ was, all that matters (in the large-$N$ limit) is the arithmetic mean.