A quantity is self-averaging if its sample to sample fluctuations vanish in the thermodynamic limit.

Non-self-averaging quantities are characteristic of Disordered systems

Statistical physics - How can a statistical description be right? (thermodynamic limit)

Selfaveraging explanation in the context of Spin glasses (from book Theory of Spin Glasses and Neural Networks)

Traditional way of speculations, why the selfaveraging phenomenon should be expected to take place, is in the folbwing. The free energy of the system is known to be proportional to the volume of the system, which in our case is N. Therefore, in the thermodynamic limit $N\to \infty$ the main contribution to the free energy in such macroscopic system must come from the volume, and not from the boundary, which usually produces the effects of the next orders in the small parameter 1/N.

Any macroscopic system could be divided into, again, macroscopic number of macroscopic subsystems. Then the total free energy of the system would consist of the sum of the free energies of the subsystems, plus the contribution which comes from the interactions of the subsystems, on their boundaries. If all the interactions in the system are short range (which takes place in any normal system), then the contributions from the mutual interactions of the subsystems are just the boundary effects which are vanishing in the thermo- dynamic limit. Therefore, the total free energy could be represented as a sum of the macroscopic number of terms. Each of these terms would be a random quenched quantity since it contains as the parameters the elements of the ran- random spin-spin interaction matrix. Next, in accordance with the law of large numbers, the sum of many random quantities can be represented as their aver- average value, obtained from their statistical distribution, times their number (all this is true, of course, only under certain requirements on the characteristics of the statistical distribution in the limit $N\to \infty$). Therefore, the conclusion which comes out from such speculations is that the free energy of a macroscopic system must be selfaveraging over the realizations of the random interactions in accordance with their statistical distribution.