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We argue that when the number of spins N in the SK model is finite, the Parisi scheme can be terminated after K replica-symmetry breaking steps, where K(N ) ∝ N 1/6 . We have checked this idea by Monte Carlo simulations: we expect the typical number of peaks and features R in the (non-bond averaged) Parisi overlap function P J (q) to be of order 2K(N ), and our counting (for samples of size N up to 4096 spins) gives results which are consistent with our arguments. We can estimate the leading finite size correction for any thermodynamic quantity by finding its K dependence in the Parisi scheme and then replacing K by K(N ). Our predictions of how the Edwards-Anderson order parameter and the internal energy of the system approach their thermodynamic limit compare well with the results of our Monte Carlo simulations. The N -dependence of the sample-to-sample fluctuations of thermodynamic quantities can also be obtained; the total internal energy should have sample-to-sample fluctuations of order N 1/6 , which is again consistent with the results of our numerical simulations.