J. theor. Biol. (1992) 158, 109-128 Detecting and Displaying Size Bimodality: Kurtosis, Skewness and Bimodalizable Distributions TOMASZ WYSZOMIRSKI Department of Phytosociology and Plant Ecology, Institute of Botany, Warsaw University, Al. Ujazdowskie 4, PL-00-478 Warsaw, Poland (Received on 25 July 1991, Accepted on 5 March 1992) Bimodality of size distributions is often found both in plant and animal populations, but any widely accepted measure of it does not exist. Kurtosis coefficient g2 can be, with some reservations, considered to be a measure of bimodality in symmetric distributions. Bimodality-generating mechanisms usually make distributions platy- kurtic. Strongly asymmetric distributions are ieptokurtic and also may remain as such when an additional mode emerges. This disqualifies kurtosis as a bimodality measure. The logarithmic transformation, which is often used to make distributions less asymmetric, may create bimodality. This leads to the concept of bimodalizable and platykurtizable distributions, i.e. distributions becoming bimodal or platykurtic, respectively, after transformation. The log-transformation is appropriate only in some cases. In this paper, the Box-Cox (BC) transformation to symmetry is proposed as a basis for assessing bimodalizability. Bimodalizable and platykurtizable distribu- tions are defined as distributions becoming bimodai or platykurtic, respectively, after the symmetrizing BC-transformation. Further transformation by the normal cumulative distribution function allows us to present platykurtizability graphically. Kurtosis of BC-transformed data indicates the operation of bimodality-promoting mechanisms much better than the common kurtosis coefficient. Properties of the proposed procedure are illustrated by its application to distribution mixtures. Examples of its behaviour are also presented for data drawn from simulation of growth and competition in even-aged plant populations. 1. Introduction Size frequency distributions are usually far from normality both in natural and experimental populations. Mass distributions are often positively skew (see e.g. Uchmafiski, 1985). Apart from asymmetry, bimodality of size distributions is found in many cases in plant (Ford, 1975; Mohler et al., 1978; Rabinowitz, 1979) as well as in animal populations (see Huston & DeAngelis, 1987 for a review). The study of bimodality is strongly restricted by the lack of its measure. Usually, analyses were based mainly on the visual inspection of histograms. This is a rather unreliable method, as the appearance of a histogram depends markedly on the number of class intervals used. Histograms can only reveal gross distribution changes (Hutchings, 1975). Existing statistical procedures designed for the analysis of bimod- ality are oriented mainly at the resolution of empirical distribution into two or more components (Everitt & Hand, 1981). Such methods are dependent upon the assumptions about the form of component distributions. This imposes serious 109 0022-5193/92/170109 + 20 $08.00/0 © 1992 Academic Press Limited