Is Human Height Bimodal?

Abstract

The combined distribution of heights of men and women has become the canonical illustration of bimodality when teaching introductory statistics. But is this example appropriate? This article investigates the conditions under which a mixture of two normal distributions is bimodal. A simple justification is presented that a mixture of equally weighted normal distributions with common standard deviation σ is bimodal if and only if the difference between the means of the distributions is greater than 2σ. More generally, a mixture of two normal distributions with similar variability cannot be bimodal unless their means differ by more than approximately the sum oftheirstandard deviations. Examination of national survey data on young adults shows that the separation between the distributions of men's and women's heights is not wide enough to produce bimodality. We suggest reasons why histograms of height nevertheless often appear bimodal.