The Statistical Entropy of a Sample from a Community of Species

Abstract

Shannon's measure of entropy,-E Pi log Pi, has been widely used to describe the diversity of ecological communities. There are, however, several different sets of proportions (or probabilities) that arise in the specification of a community and the study of samples obtained from it. The purpose of this note is to examine the relationships between the corresponding entropies. We consider a community comprising a finite or countable set of species of similar type. The abundance of Species i, denoted by Ai, is defined to be the expected number of individuals of that species caught by the expenditure of unit trapping effort. In the case of an infinite set of species it will be assumed that 2 Ai =A<°°, and in either case the proportional abundance of Species i will be denoted by Tri = Ai( Ai)-l. If the trapping method acts indiscriminately on all species, as for example with a light trap for moths or a core extraction method for soil insects, the operational definition of abundance coincides with the usual one. The quantity H=-fi ri log ri will be called the diversity entropy of the community. Alternative measures used to describe diversity are the number of species, the variance of their proportional abundances, and the coefficient HS = 1S2 = 1 T2, where S2 iS the probability that two individuals chosen randomly and independently (with replacement) from the community, will belong to the same species. The meaning of these and other coefficients of diversity, and the contrasts between them, have been discussed by Pielou (1975), Patil and Taillie (1976, 1977), Engen (1978) and Routledge (1979).