  Let $X_1, X_2,\dots, X_n$ denote i.i.d.~centered standard normal random
variables, then the law of the sample variance
$Q_n=\sum_{i=1}^n(X_i-\bar{X})^2$ is the $\chi^2$-distribution with $n-1$
degrees of freedom. It is an open problem in classical probability to
characterize all distributions with this property and in particular, whether it
characterizes the normal law. In this paper we present a solution of the free
analog of this question and show that the only distributions, whose free sample
variance is distributed according to a free $\chi^2$-distribution, are the
semicircle law and more generally so-called \emph{odd} laws, by which we mean
laws with vanishing higher order even cumulants. In the way of proof we derive
an explicit formula for the free cumulants of $Q_n$ which shows that indeed the
odd cumulants do not contribute and which exhibits an interesting connection to
the concept of $R$-cyclicity.
