## Introduction

The numbers of individuals of each of several species in a sample, assemblage or community produces a species–abundance distribution (SAD). There have been many suggestions as to which statistical distribution fits a particular species–abundance distribution or a set thereof (for reviews see May 1975; Magurran 1988, 2004; Tokeshi 1993; Marquet, Keymer & Cofré 2003). Most suggested distributions have little theoretical justification.

Hubbell (2001) has developed a well-known neutral theory that leads to a new distribution, the zero sum multinomial (ZSM). This is based on continuum vegetation (Williamson 2003), where every point is occupied and occupied by precisely one individual. In its basic form the Hubbell theory applies, even roughly, to rather few animal communities (such as some coral reefs). Nevertheless, it is a SAD derived from theory rather than from an empirical fit and so there is interest in seeing how well it performs compared with empirical SADs. So far, the ZSM has been compared only with the lognormal distribution (in which the logarithms of the abundances of the different species follow a normal (Gaussian) distribution), which has been put forward as an appropriate null model for a SAD. McGill (2003a) talks of ‘reasonable null models’ and ‘the null lognormal hypothesis’, Nee & Stone (2003) of ‘[the lognormal] an older, simpler null model’ and justify that by the central limit theorem. Harte (2003), too, says ‘A hand-waving justification for the lognormal distribution is that it could arise from the central limit theorem’ and, going back a little, Taylor (1978) said ‘Common logic suggests that the frequency distribution for N individuals (which vary logarithmically) in S species (which is a Poisson variate) … is a Log-normal’.

Our object in this paper is to explore a number of reasons why the lognormal is not an appropriate null model, or indeed an appropriate model of any sort, for a SAD. Others before us have objected to the lognormal (Lambshead & Platt 1985; Hughes 1986; Dewdney 2003) primarily on features of empirical distributions. We also start with examples which show why the lognormal is popular, and we then consider three issues. Under fitting, we discuss not only the results of statistical tests but also a new point, that there is a closely related distribution, the logit-normal, which often fits better. Under causes, we look at the central limit theorem and other derivations and provide a new argument against the use of the former for SADs. Under mathematical consequences, we discuss the canonical hypothesis and a new suggestion, the log-binomial, and argue that the lognormal cannot, on theoretical grounds, be a SAD. Our examples will often, for convenience, be drawn from our own publications; we are not attempting a comprehensive review.