Defensive spines: Inverse relationship between coefficients of variation and size 1

Coefficients of variation for spine lengths are not constant over a range of mean values, but show an inverse relationship with sample means. The nearly hyperbolic relationship is evident in selection experiments, in seasonal samples from a single lake, and in widely dispersed geographic samples. The observed inverse relationship could indicate an interesting developmental property (e.g. that smaller, rudimentary spines are more variable than larger functional ones) or could reflect one of several possible statistical artifacts or properties. Detailed examination of Bosmina suggests that the strong inverse trend arises from a combination of biological and statistical properties; it is not an artifact of measurement error, although measurement error contributes to inflation of small values. Small spines are inherently more variable than longer spines, and the latter have more uniform component parts. Moreover, there are several alternative arrangements of components that sum to the same length and hence that achieve nearly equivalent fitness value. These alternative arrangements lower part‐whole correlations within the population. The two statistical properties that contribute to observed trends include a version of the central limit theorem and measurement error. The percentage variation of a character which is the sum of a number of randomly varying parts will be considerably less (roughly as the inverse square root of the number of parts) than the mean percentage variation of the parts themselves. Thus a statistical property of large numbers contributes to the observed inverse trend. In addition, coefficients of variation as fractions are especially sensitive to measurement error, so that precise measurements are recommended for small defensive spines.