The distributions P(X) of singular thermodynamic quantities, on an ensemble of d-dimensional quenched random samples of linear size L near a critical point, are analyzed using the renormalization group. For L much larger than the correlation length ξ, we recover strong self-averaging (SA): P(X) approaches a Gaussian with relative squared width RX~(L/ξ)−d. For L≪ξ we show weak SA (RX decays with a small power of L) or no SA [P(X) approaches a non-Gaussian, with universal L-independent relative cumulants], when the randomness is irrelevant or relevant, respectively.