Rarefaction is a technique that corrects for unbalanced sample sizes, which are often a major problem in comparisons of diversity. The rarefaction curve is the expected number of higher taxonomic groups, such as families or genera, represented in a random selection of lower taxonomic units, such as species or individuals. The shapes of these curves are analyzed by finding the best possible uniform upper and lower bounds for a fixed number of units and groups. The position of a rarefaction curve between these limits provides a natural measure of evenness of diversity. Asymptotic formulae are also given. The results are applied to the distribution of species within families of recent echinoids and bivalves.