Impact of the Lognormal Distribution on Studies of Phenotypic Variation and Evolutionary Rates

Abstract

Kerfoot, W. Charles and Arnold G. Kluge (Museum of Zoology and Department of Zoology, University of Michigan, Ann Arbor, Michigan 48104). 1971. Impact of the Lognormal Distribution on Studies of Phenotypic Variation and Evolutionary Rates. Syst. Zool., 20:459-464.-Both normal and lognormal statistics describe the distribution of vertebral numbers in local samples of lower vertebrates. When samples from populations possessing different means are compared, however, standard deviations of vertebral counts are correlated with mean counts, a relationship predicted by the lognormal, but not the normal distribution. Fit to the lognormal requires recognition in significance testing, neoessitates log transformation of variables in studies of evolutionary rates, and predicts exponential rates of character divergence under constant selective pressure. Numerous papers discuss the variability of vertebral number in the context of systematics, embryology, and genetics. Studies show that vertebral number is heritable and polygenic both in natural populations and in laboratory strains (Green and Russell, 1951; McLaren and Michie, 1954), and that vertebral number is susceptible to temperature shock during embryonic development (Fox, Gordon and Fox, 1961; Kubota, 1967; Lindsey, 1962; Taning, 1944). Other studies suggest the selective advantage of the modal number found in natural populations (Ford and Bull, 1926). Most of these works implicitly assume that vertebral numbers are normally distributed. The distributions appear bell-shaped, and slight departures from normality have little effect on the results of the above mentioned studies. Comparison of vertebral distributions between organisms indicates a higher order relationship between the variance and the mean, which suggests that vertebral distributions more closely fit a lognormal distribution, and thus may provide additional insights into the development and evolution of vertebral number.