## Introduction

Species-abundance distributions (SADs), also called relative-abundance distributions (RADs), record the relative or absolute number of a set of species in a sample. As a description of an ecological community, they sit conveniently between a simple listing of the species present and multidimensional analysis (McGill *et al.* 2007) and so have been much studied. Generally, they are unimodal on a logarithmic scale of abundance and that has lead ecologists to seek a simple mathematical form for SADs. Despite this simplicity, McGill *et al.* (2007) list 27 models that have been suggested and note an absence of agreement about which models are best.

Chiarucci *et al.* (1999) calculated SADs for 35 individual plots, fitting broken-stick, geometric, lognormal and Zipf–Mandelbrot functions (models 20, 19, 6 and 11, respectively, in McGill *et al.* 2007) on point quadrat cover and biomass data. The most frequent (22 plots) best fitting model for the point quadrat cover data was the Zipf–Mandelbrot, and in contrast, the most frequent best fitting model for the biomass data (16 plots) was the lognormal. The Zipf–Mandelbrot model was the best fitting model for biomass data in 10 plots, whilst the lognormal was the best fitting model for the point quadrat cover data in only two plots.

There are two simple reasons for this lack of agreement. The first is that none of the models is additive (Williamson & Gaston 2005; Šizling *et al.* 2009) over either taxa or areas; indeed Šizling *et al.* (2009) show that such an additive model involves an indefinitely large number of parameters. This means that a model that fits at one scale or one set of species cannot also fit at another scale or over an enlarged set of species. The second, and the one we are concerned with here, is that the models are not invariant under different abundance measures; SADs change shape in relation to the abundance measure used. This has been recognised to some extent by reference to Preston’s veil line (Williamson 2010) or by classing data sets as fully censused vs. incompletely sampled (Ulrich, Ollik & Ugland 2010). But the veil line is an unsatisfactory approximation (Chisholm 2007; Williamson 2010) and all data sets are in some sense and to some degree samples, so sampling is a continuous (and universal) variable, not a discrete one. Chisholm (2007) has a thorough discussion of previous arguments, particularly Dewdney (1998). Williamson (2010) showed that the investigator choice of using individuals as opposed to biomass for sampling has a mathematical consequence; describing the phenomenon of differential veiling using marginal plots of the same community measured as individuals and as biomass. Much SAD work has been with taxa in which individuals are readily distinguished (e.g. Morlon *et al.* 2009; who studied trees, fishes, birds and mammals). For applied plant ecology, both biomass and density have serious drawbacks as abundance measures (Kershaw 1973). Collecting biomass data is destructive and time-consuming and plant density has the inherent problem that individuals are often difficult to distinguish, if it is possible at all, and where there may be great variation in size within a species (Jonasson 1988). Here, we consider the effect of sampling plant communities using different species-abundance measures. Objective abundance measures commonly used with plant communities often attempt to estimate cover, for example, point quadrat cover or local frequency (Greig-Smith 1964). Neither of these measure individuals *per se* but in common with density they are discrete variables, that is, the count of numbers of pin hits or subsquares occupied. In contrast, both biomass and basal area are continuous variables. Williamson (2010) linked the differential veiling to individuals. In contrast, we show that whilst differential veiling does occur with individuals this is not a property of individuals *per se*, rather differential veiling is a consequence of using a discrete rather than continuous abundance measure. Counting individuals is, along with a multitude of other sampling measures discrete, and it is the *discreteness*, not the *individualness* which produces the differential veil line. This is an important distinction, especially relevant to fields, such as plant sampling, where discrete abundance measures other than counting individuals are common.