A histogram of the dimensions of the irreducible representations of the symmetric group $S_n$ becomes sharply peaked for large $n$. For example, here is a $\log_{10}$ histogram for $S_{50}$:
(This is showing that most of the dimensions of $S_{50}$ irreps have 26-29 decimal digits.)
Based on looking at histograms for $S_{30}$, $S_{40}$, $S_{50}$, etc, it seems likely that as $n\to\infty$, the variance of the distribution goes to zero as the maximum shifts toward infinity.
Is the asymptotic form of this distribution for large $n$ known?