A class of statistical tests based on molecular polymorphism data is studied to determine size and power properties. The class includes Tajima's D statistic as well as the D* and F* tests proposed by Fu and Li. A new method of constructing critical values for these tests is described. Simulations indicate that Tajima's test is generally most powerful against the alternative hypotheses of selective sweep, population bottleneck, and population subdivision, among tests within this class. However, even Tajima's test can detect a selective sweep or bottleneck only if it has occurred within a specific interval of time in the recent past or population subdivision only when it has persisted for a very long time. For greatest power against the particular alternatives studied here, it is better to sequence more alleles than more sites.