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Simpson's diversity index

Suppose $R(\pi_k) = {1 - \pi_k}$, the probability that the next species you encounter is different from the one you have just seen. Then

\begin{eqnarray*}
\Delta_1(C) &=& \sum_{k=1}^s\pi_k(1-\pi_k) \cr
&=& 1 - \sum_{k=1}^s\pi_k^2
\end{eqnarray*}

For those of you who've had population genetics, you can think of the $\pi_k$ as allele frequencies and Simpson's diversity as the panmictic heterozygosity. It's the probability that any two individuals chosen at random from the community belong to different species.



Kent Holsinger 2013-10-02