Arithmetic or geometric normality of biological variation: an empirical test of theory

Variation of biological populations is required for evolution by natural selection, and variance is a fundamental component in quantitative characterization of evolutionary differences and rates of change. Biological variation is widely understood to be normally distributed because of a general theoretical law of error. The law of error has two forms, and resulting normality may be arithmetic-where equivalent positive and negative deviations from expectation differ by equal amounts, or normality may be geometric-where equivalent deviations differ by equal proportions. Which law of error applies in biology can only be determined empirically, and this is surprisingly difficult. A new likelihood approach is developed here using data from anthropometric surveys of humans in two states in India: Maharashtra and Uttar Pradesh. Each state sample is large, but more importantly, each includes a large number of smaller subsamples. Likelihood support is additive, and subsamples are advantageous because (1) they are more homogeneous, (2) they yield probabilities and support scores in every case, and (3) significance can be evaluated first by tracing signs of the subsample support scores and then by comparing subsample support sums. Sign traces that fluctuate randomly show arithmetic and geometric normality to be indistinguishable. Two of 14 measurement variables studied here have subsample support sign traces differing from random, and one is significant in having a subsample support sum falling outside a 95% prediction interval for the 12 fluctuating traces: geometric normality is favored by a factor of ca. 10(60). Six of 14 index variables have support sign traces differing from random, and all are significant in having subsample support sums falling outside a 95 % prediction interval for the 8 fluctuating traces: geometric normality is favored by factors of 10(8) or more. Arithmetic and geometric normality cannot be distinguished for 21 of 28 variables studied here, but whenever alternatives are distinguishable geometric normality is consistently and strongly favored. This means that the applicable law of errors is proportional. In practical terms, arithmetic measurements must be transformed using logarithms to represent both the geometric normality of biological variation and the relative functional significance of measurements appropriately.