  In a disordered system, a quantity is self-averaging when the ratio between
its variance for disorder realizations and the square of its mean decreases as
the system size increases. Here, we consider a chaotic disordered many-body
quantum system and search for a relationship between self-averaging behavior
and the properties of the distributions over disorder realizations of various
quantities and at different timescales. An exponential distribution, as found
for the survival probability at long times, explains its lack of
self-averaging, since the mean and the dispersion are equal. Gaussian
distributions, however, are obtained for both self-averaging and
non-self-averaging quantities. Our studies show also that one can make
conclusions about the self-averaging behavior of one quantity based on the
distribution of another related quantity. This strategy allows for
semianalytical results, and thus circumvents the limitations of numerical
scaling analysis, which are restricted to few system sizes.
