  The study of quenched random systems is facilitated by the idea that the
ensemble averages describe the thermal averages for any specific realization of
the couplings, provided the system is large enough. Careful examination
suggests that this idea might have a follow, when the correlation length
becomes of the order of the size of the system. We find certain bound
quantities are not self-averaging when the correlation length becomes of the
order of the size of the system. This suggests that the strength of
self-averaging, expressed in terms of properly chosen signal to noise ratios,
may serve to identify phase boundaries. This is demonstrated by using such
signal to noise ratios to identify the boundary of the ferromagnetic phase and
compare the findings with more traditional measures.
