The existence of non-Rayleigh statistics in phantom experiments and computer simulations is demonstrated. The histogram of speckle magnitudes in low scatterer concentration is compared with the Rayleigh distribution probability density function. The difference is not significant by direct observation. Higher order moments of the speckle are then computed; they are clearly different from the Rayleigh moments. In Rayleigh statistics the phase of the speckle is assumed uniformly distributed. It is shown that this assumption is not always true. When regularly spaced scatterers exist, the phase distribution may be biased. It is noted that these non-Rayleigh properties can be used to estimate scatterer concentration and size for the purpose of ultrasound tissue characterization.