Simpson index

The Simpson index was introduced in 1949 by Edward H. Simpson to measure the degree of concentration when individuals are classified into types.

The measure equals the probability that two entities taken at random from the dataset of interest represent the same type.

The same index was rediscovered by Orris C. Herfindahl in 1950. The square root of the index had already been introduced in 1945 by the economist Albert O. Hirschman. As a result, the same measure is usually known as the Simpson index in ecology, and as the Herfindahl index or the Herfindahl–Hirschman index (HHI) in economics.

Since mean proportional abundance of the types increases with decreasing number of types and increasing abundance of the most abundant type, λ obtains small values in datasets of high diversity and large values in datasets of low diversity.

This is counterintuitive behavior for a diversity index, so often such transformations of λ that increase with increasing diversity have been used instead.

The most popular of such indices have been the inverse Simpson index and the Gini–Simpson index. Both of these have also been called the Simpson index in the ecological literature, so care is needed to avoid accidentally comparing the different indices as if they were the same.

The index is also as a measure of the effective number of parties.

The original Simpson index λ equals the probability that two entities taken at random from the dataset of interest (with replacement) represent the same type.

Its transformation therefore equals the probability that the two entities represent different types. This measure is also known in ecology as the probability of interspecific encounter (''PIE''). It can be expressed as a transformation of true diversity of order 2:

The Gibbs–Martin index of sociology, psychology and management studies, which is also known as the Blau index, is the same measure as the Gini–Simpson index.