Wave group statistics predicted by linear theories are compared to numerical simulations, thus determining ranges of spectral shapes for which the theories are valid. It is found that these theories are not generally valid for ocean data because of many assumptions and simplifications beyond linearity and random phase or because their range of applicability does not include the vast majority of ocean conditions. The simulations also provide quantitative information about the variability of linear wave group statistics which is useful when examining ocean field data The simulation technique is used to show that important ocean gravity wave group statistics are not inconsistent with an underlying wave field composed of linearly superposed random wave. The majority of the field data examined were collected in 10 m depth; significant wave heights varied from about 20 to 200 cm, and the spectral shapes ranged from fairly narrow to broad (1<Qp<6). For the 10-m depth data, the observed mean run length, variance of run length, and probabilities of runs of a given number of waves were statistically consistent with the simulations. In contrast to the apparently linear groups observed in 10 m depth, waves in 2--3 depth showed marked departures from the linear simulations.