We investigate the large deviation behaviour of the overlap probability density in the Sherrington–Kirkpatrick (SK) model using the coupled replica scheme, and we compare with the results of a large-scale numerical simulation. In the spin glass phase we show that, generically, for any model with continuous replica symmetry breaking (RSB), 1/N log P_N(q)≈ −A(|q| − q_EA)³, and we compute the first correction to the expansion of A in powers of T_c − T for the SK model. We also study the paramagnetic phase, where results are obtained in the replica symmetric scheme that do not involve an expansion in powers of q − q_EA or T_c − T. Finally we give precise semi-analytical estimates of P(|q| 1). The overall agreement between the various points of view is very satisfactory.