  The localization measures $A$ (based on the information entropy) of localized
chaotic eigenstates in the Poincar\'e-Husimi representation have a distribution
on a compact interval $[0,A_0]$, which is well approximated by the {\em beta
distribution}, based on our extensive numerical calculations. The system under
study is the Bunimovich' stadium billiard, which is a classically ergodic
system, also fully chaotic (positive Lyapunov exponent), but in the regime of a
slightly distorted circle billiard (small shape parameter $\varepsilon$) the
diffusion in the momentum space is very slow. The parameter $\alpha=t_H/t_T$,
where $t_H$ and $t_T$ are the Heisenberg time and the classical transport time
(diffusion time), respectively, is the important control parameter of the
system, as in all quantum systems with the discrete energy spectrum. The
measures $A$ and their distributions have been calculated for a large number of
$\varepsilon$ and eigenenergies. The dependence of the standard deviation
$\sigma$ on $\alpha$ is analyzed, as well as on the spectral parameter $\beta$
(level repulsion exponent of the relevant Brody level spacing distribution).
The paper is a continuation of our recent paper (B. Batisti\'c, \v{C}. Lozej
and M. Robnik, Nonlinear Phenomena in Complex Systems {\bf 21}, 225 (2018)),
where the spectral statistics and validity of the Brody level spacing
distribution has been studied for the same system, namely the dependence of
$\beta$ and of the mean value $<A>$ on $\alpha$.
