Oecologia (1989) 80:140-141 Oecologia 9 Springer-Verlag 1989 Short communications An undesirable property of Hill's diversity index N2 Raghavendra Gadagkar Centre for Ecological Sciences and Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012, India Summary. A desirable property of a diversity index is that it be a convex function of the proportions of different species that constitute a community. N2, a diversity index belonging to the Hill's series of diversity indices and which is equiva- lent to the reciprocal of Simpson's diversity index violates this requirement when the proportion of one of the species is close to 1. Recommendations of N2 as the best possible index of diversity therefore need to be examined judiciously. Key words: Species diversity - Diversity indices - Commun- ity ecology - Hill's diversity indices In spite of considerable controversy, ecologists find it useful to describe biological communities using a single number which incorporates the number of species as well as their relative abundances. One such commonly used diversity in- dex is the Simpson's Index given by S ,~=1- E p{ i=1 where S is the number of species and Pl, the proportion [ of the ith species in the community (Simpson, 1949; actually S \ p{ in his paper). The meaning of 2 is intuitively clear; i=l ! it is the probability that two randomly picked individuals belong to different species. Another popular index of diver- sity is the Shannon Wiener Index (Margalef, 1958) given by S H'=- ~ piOnpi) i=1 Hill (1973) has derived a family of diversity indices Ni with a unified notation where N1 = exp(H') and N2= p. i Debate on the advantages and disadvantages of various diversity indices and indeed, on the usefullness of the con- cept of diversity itself has continued (see for eg. Hurlbert 1971) but several authors recommend 5/1 and N2 of the Hill's series over H' and 2 primarily because the former are in units of species numbers and therefore more easily interpretable (eg. Peet, 1974; Alatalo and Alatalo, 1977). In fact Routledge (1979) goes a step further and recom- mends that "N2 is the best, single measure of diversity, and that the only other index worth considering is Nx". Such recommendations have now begun to trickle down to text- books and primers (eg. Ludwig and Reynolds 1988, P. 103). As a result the original Simpson's and Shannon Wiener indices and N1 too (if the recommendation of Routledge, 1979 is taken seriously) face the prospect of being discarded in favour of N2. In an attempt to estimate diversity using 2, H', N~ and N2 I repeatedly encountered what might be called an unde- sirable property which was shown by Ne but not by the remaining three indices. It is desirable that when communi- ties are pooled the value of the diversity index for the resul- tant community is greater than or equal to, but not less than, the mean of the diversity indices for the original com- munities. Although not always explicitly stated (but see Lewontin 1972), the reason for this is intuitively obvious. When two communities are pooled the unevenness of spe- cies distribution decreases. Any index of diversity should therefore reflect this property. A simple example illustrated in Table 1 shows this condition to be satisfied by 2, H', and N1, but violated by N2. To estimate how often such a condition arises, consider the simplest case of two communities with two species each. Let X and 1-X denote the proportion of the two species in the first community and Yand 1 - Ythose in the second. Table 1. An example of 2 communitiesthatillustratetheanomalous behaviour of Nz Community Community Pooled 1 2 community X or Y 0.95 0.75 0.85 1-X or 1-Y 0.05 0.25 0.15 2 0.0950 0.3750 0.255 Mean=0.235 H' 0.1985 0.5623 0.4227 Mean=0.3804 N1 1.2196 1.7547 1.5261 Mean= 1.4872 5/2 1.1050 1.6000 1.3423 Mean= 1.3525