Dimension statistics of representations of finite groups

The first part of this paper deals with unipotent and reductive groups over finite fields with $q$ elements in which either $q$ goes to infinity or $G=GL_n(q)$ and $n$ goes to infinity. The second part of the paper deals with the symmetric group $S_n$. The main conclusion that we want to bring out in the case of reductive groups $G(q)$, $q$ varying, is that the dimension data, resp. the size of conjugacy classes, is in a statistical sense, ``roughly'' constant and the same (up to taking the squares). We introduce the notion of {\it asympototically constant}, and {\it asympototically log constant} to make precise these notions, which we apply to various groups discussed in this paper including the symmetric groups $S_n$.