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Abstract It has long been recognized that diversity has many measurable aspects, such as richness, evenness, and similarity among species. However, given a diversity index, it is unclear whether it necessarily can be decomposed into components that reflect these different aspects. Here, we present a scheme to decompose the Leinster and Cobbold diversity index, which subsumes and generalizes many other indices, into the components of richness, evenness and taxonomic similarity. Our approach addresses the problem that in general a vector of equal relative abundances does not maximize diversity. Furthermore, our approach uses all available information to give unbiased estimates of both evenness and similarity.