Human cranial variability: a methodological comment.

Measurement of morphological variation is essential to our understanding of human evolution. At least two classes of statistics can be used to describe variation: (1) statistics such as the variance that measure average dissimilarity (distance), and (2) statistics such as the range of variation that measure maximal dissimilarity. Either kind of measure may be meaningfully employed depending on one’s interest, but each has radically different sampling properties that must be taken into account. Sampling properties are particularly important because we generally want to infer, from the sample we happen to have, aspects of the underlying distribution from which the sample was drawn. In a recent letter, Wolpoff (1992) compares cranial variation in a sample of 13 Paleolithic humans from the Levant to that in a sample of 388 modern humans from London. By basing his conclusions on the sample range and failing to consider sampling bias associated with this statistic, Wolpoff seriously misrepresents the difference in variability between Paleolithic and modern humans. It has long been known that, in addition to being very sensitive to outliers, the range depends strongly on sample size (Pearson, 1926). This obvious dependence partly underlies the preferred use of sample statistics such as the variance (which reflects average difference), and related statistics, as measures of dispersion (Van Valen, 1974; Uytterschaut and Wilmink, 1983; Franciscus and Long, 1991; van Vark and Bilsborough, 1991). The sample variance provides an unbiased estimate of the parametric variance (Fig. 1A) (Sokal and Rohlf, 1981). However, the sample range is generally biased, increasing monotonically with sample size. Because a biological population is of finite size, it does have a true range that is knowable in principle, but one must sample the entire population to determine this range. The smaller a sample one draws from a population, the smaller the sample range will be. This point is particularly important if we compare large modern samples to small fossil samples. Dependence between sample size and range of variation can be conveniently illustrated with a normal distribution (Fig. 1B). Although the nature of this dependence is determined here by simulation, it can also be determined analytically (Pearson, 1926). Given a univariate normal distribution, regardless of its variance, expected sample range increases roughly as the logarithm of sample size (Rohlf and Sokal, 1981; see also Foote, 1992). In a comparison between N = 13 Paleolithic humans and N = 388 modern humans, we would expect the range of the modern sample to be about 1.8 times that of the Paleolithic sample if the two samples were drawn from normal distributions with the same variance (Fig. 1C). This expected difference is remarkably similar to the twofold difference in range that Wolpoff finds for some cranial measures! The apparent difference in variation between the two samples probably reflects sampling bias much more than it shows true differences in variation between the distributions that these samples represent. Wolfpoff s claim that “the amount of variation in measurements from the Middle Paleolithic. . . appears to be less than that in a modern population” (p. 142; emphasis in original), is unjustified, unless (1) his emphasis is on the appearance rather than the existence of a true difference in variation, or (2) his emphasis is only on the samples, rather than the underlying populations that these sam-