  We show that the distribution of self-normalized sums of free self-adjoint
random variables converges weakly to Wigner's semicircle law under appropriate
conditions and estimate the rate of convergence in terms of the Kolmogorov
distance. In the case of free identically distributed self-adjoint bounded
random variables, we retrieve the standard rate of order $n^{-1/2}$ up to a
logarithmic factor, whereas we obtain a rate of order $n^{-1/4}$ in the
corresponding unbounded setting. These results provide free versions of certain
self-normalized limit theorems in classical probability theory.
