Abstract. The median window operation is being increasingly used to process images. Although the deterministic properties of the median are fairly well known, its statistical properties are not. Consider a median window of width N scanning a noisy background image with white power spectrum. We present here the probability law for the median outputs, its mean, variance, and signal-to-noise ratio, and the probability that two successive median outputs are equal. Specialization is made to speckle imagery. Key results are as follows: the probability law is of a Bernoulli multinomial form; the mean is asymptotic with N to the average background times In 2, and hence is about 30% less than the background value; the variance is asymptotic with N to a 1/N dependence; signal-to-noise ratio is asymptotic with N to Ni171 In 2. Finally, the probability that two successive median outputs are equal is 2-1(N-1)/N, or slightly less than 0.5 for N ⪆ 7. This is independent of the type of image data at hand, i.e., whether speckle, Poisson, or normal, provided that it has a white power spectrum.