The short?term statistics of 10 million individual waves observed with buoys in deep water have been investigated, corrected for a sample?rate bias, and normalized with the standard deviation of the surface elevation (the range of normalized wave heights is 0 < H < 10). The observed normalized trough depths are found to be Rayleigh distributed with near?perfect scaling. The normalized crest heights are also Rayleigh distributed but 3% higher than given by the conventional Rayleigh distribution. The observed normalized wave heights are not well predicted by the conventional Rayleigh distribution (overprediction by 9.5% on average), but they are very well predicted by Rayleigh?like distributions obtained from linear theories and by an empirical Weibull distribution (errors