In a previous paper (Parisi G. and Sourlas N., Phys. Rev. Lett., 89 (2002) 257204) we found that in the random field Ising model at zero temperature in three dimensions the correlation length is not self-averaging near the critical point and that the violation of self-averaging is maximal-This is due to the formation of bound states in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensions near the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperature but the violation of self-averaging is weaker than in the random field case. In the random Ising model we find still weaker violations of self-averaging and we cannot rule out the possibility of the restoration of self-averaging in the infinite volume limit.