The relative power of the Mann-Whitney statistic, the t-statistic, the median test, a test based on exceedances (A,B), and two special cases of (A,B) the Tukey quick test and the revised Tukey quick test, was investigated via a Monte Carlo experiment. These procedures were compared across four population probability models: uniform, beta, normal, and double exponential. Sample sizes of (5.5), (10,10), (20,20), (5,10), and (5,20) were among those used. Results indicate the median test should be considered for distributions which certain outliers. The exceedances tests can be powerful alternatives to more standard procedures if the underlying distributions are platykurtic. (Author)