The renormalization group approach is used to study the effects of a “canonical” constraint (e.g., a fixed number of occupied bonds) on critical quenched disordered systems. The constraint is found to be always irrelevant, even near the “random” fixed point. This proves that α<0, or that ν>2/d. “Canonical” and “grand canonical” averages thus belong to the same universality class. Related predictions concerning the universality of non-self-averaging distributions are tested by Monte Carlo simulations of the site-diluted Ising model on the cubic lattice. In this case, the approach to the asymptotic distribution for “canonical” averaging is slow, resulting in effectively smaller fluctuations.