What is the distribution of the standard deviation of $n$ normal variates? That is, if $X_1,...,X_n$ are i.i.d. normal random variables with mean $\mu$ and s.d. $\sigma$ and $M=\sum X_i/n$, then what is the distribution of $\sqrt{\sum(X_i-M)^2/(n-1)}$?

If I could pretend that $D_i=X_i-M$ are independent then the s.d. is a scaled Chi (not squared) variable with $n$ degrees of freedom, but $D_i$ are not independent.

If it makes it simpler, feel free to assume that $\mu=0$, and if necessary then even $\sigma=1$.