The use and misuse of the coefficient of variation in analysing geographical variation in birds

Abstract

The coefficient of variation (CV) provides a method of measuring intrinsic variation in a population because increases in variance caused by increases in means are appropriately adjusted in a common percentage scale. Lewontin (1966) described a statistical test for the difference between two CVs that approximates the result obtained by transforming each original datum to its natural logarithm, then testing for equality of variances in the usual manner. Two ornithologists, at least, have erred in using this test in their work consistently in the form of a one-tailed rather than a two-tailed hypothesis. Here I consider using a measure of the standard error of the CV in Student's t-test as a possible alternative to the F-test of Lewontin. The relative performance of the two tests was compared empirically in 1,770 comparisons generated from published data. The total number of significant results did not differ if only the five per cent level of probability was used. But at finer levels when samples were both small (5–22) or both moderately large (23–66) the F-test yielded a higher number of significant results than the t-test. With samples of mixed size, the t-test produced more significant results. It is obvious that the sizes of sample commonly published in studies of relative variation in morphological features of birds are much too small to be expected to show any but the most gross differences. The most powerful and flexible technique appears to be that of transforming data to their natural logarithms, then using the methods of ordinary analysis of variance.