What is Diversity?

For a community of $S$ species with relative abundances $p_{1},\dots, p_{S}$ ($\sum p_{i} = 1$): the richness is $S$; the Shannon index $H = -\sum_{i} p_{i}\ln p_{i}$ measures the uncertainty about the species of a randomly drawn individual, and $e^{H}$ is the number of equally abundant species that would give the same uncertainty (the effective number of species); the Simpson index $D = \sum_{i} p_{i}^{2}$ is the probability that two individuals drawn at random are of the same species, and $1/D$ is another effective number, weighted toward the common species. Richness depends on how many individuals one has looked at: a rarefaction curve plots the species found against the number of individuals sampled, and flattens when most species have been seen.