On the Shannon-Weaver index of diversity, in relation to the distribution of species in bird censuses

A correlation found by Tramer (1969) between the diversity of bird censuses and the number S of species is shown to agree fairly well with the distribution of species described by a Dirichlet series, and remarkably well with MacArthur's “broken-stick” distribution.

Abstract

The "diversity" of a statistical sample is studied in relation to some simple ideal distributions of species within the sample. A correlation found by Tramer (1969) between the diversity of bird censuses and the number S of species is shown to agree (a) fairly well with the distribution of species described by a Dirichlet series, and (b) remarkably well with MacArthur's "broken-stick" distribution. It is shown that for large values of S the "broken-stick" distribution tends to a continuous gamma-distribution, from which some simple relations can be derived theoretically; for instance the number of species in the modal octave is to the total number of species in the sample in the ratio (log 2)e = 0.255. This is in agreement with some observations by Preston (1962a). The gammadistribution can be approximated by a special log-normal distribution—one having standard deviation σ = 1 on the natural scale. This again agrees with Preston's (1962a) data. The general log-normal distribution is also studied and it is shown that if the "tails" of the distribution are not truncated, then the diversity H is given by the simple relation H = log S − 12σ2. More generally, when the curve is truncated at a veil-point on the left, then the difference between H and log S is thereby reduced. The difference ¦ H − log S ¦ is tabulated, and some applications are discussed.