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Bloomberg School BIO 751 - HEIGHT

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Teacher’s CornerIs Human Height Bimodal?Mark F. SCHILLING, Ann E. WATKINS, and William WATKINSThe combi ned distribution of heights of men and women hasbecom e the canonical illu stration of bimodality when teachingintrod uctory stati stics. But is this exampl e appropriate? This ar-ticle investiga tes the conditions unde r which a mixture of twonormal distributions is bimodal. A simp le justi cation is pre-sented that a mi xture of equally weighted normal distributionswith common standard deviation ¼ is bimodal if and only if thedifference between the means of the distributio nsis greater than2¼ . Mo re generally, a mixture of two normal distributions withsimilar variability cannot be bimodal unless their means differby more than approximatelythe sum of their standard deviations.Examin ation of national survey data on you ng adult s shows thatthe separation between the distributions of men’s and women’sheigh ts is not wide enough to produce bimodality. We sugg estreasons why histograms of height nevertheless often app ear bi-modal .KEY WORDS: Bimod al distribution;Living histogram; Nor-mal distribution.1. THE DISTRIBUTION OF HUMAN HEIGHTBrian Joiner’s (1975 ) living histogram (Figure 1) of his stu-dents at Penn State grouped by height inspired the standardclassroo m examp le of bimodality resulting from a mixture oftwo populations.Figure 1. Joiner’s living histogram of student height.Mark F. Schilling is Professor, Ann E. Watkins is Professor, and WilliamWatkins is Professor, Department of Mathematics, California State University,Northridge Northridge, CA 91330-8313 (E-mail: [email protected]). Theauthors thank Rebecca Walker and Deborah Nolan for helpful comments.Altho ugh the separate distributions of his male and femalestude nts are approximately normal, the histogram of men andwomen together is clearly bimodal. Joiner wrote, “Note that thishisto gram has a bi-modal shape due to the mixing of two sepa-rate groups, males and females.” This appealing idea ap pears inseveral introductory textbooks:“A histogramof the heights of students in a statistics class would be bi modal,for example, when the class contains a mix of men and women.” (Iversen andGergen 1997, p. 132)“Bimodality often occurs when data consists of observations made on twodifferent kinds of individuals or objects. For example, a histogram of heights ofcollege students would show one peak at a typical male height of roughly 700 0and another at a typical female height of about 650 0.” (Devore and Peck 1997,p. 43)“If you look at the heights of people without separating out males and fe-males, you get a bimodal distribution : :: ” (Wild and Seber 2000, p. 5 9)Still other textbooks include a problem that asks students topredic t the shape of the height distribution of a group of stu-dents when there are an equal number of males and females.The expect ed answer is “bimodal.” The distribution of heightoffers a plausibleexample of bimodalityand is easy for studentsto visualize.Rath er than sending our students out onto the football  eld³alaJoine r to demon strate bimodality, we decided to get somegovernme nt data and construct the appro ximate theoretical den-sity function for the mixture of the male and female popula-tions . The most recent Nation al Health an d Nut rition Exami-natio n Survey (NHANES III), conducted in 1988–1994 by theUnited States National Center for Health Statistics, rep orts thecumul ative distribution of height in inches for males and for fe-c®2002 American Statistical Association DOI: 10.1198/0003130 0265 The American Statistician, August 2002, Vol. 56, No. 3 223males in the 20–29 age bracket (U. S. Census Bureau 1999).The data for each sex have the means and standard deviations inTable 1 and ea ch follow a normal distribution rea sonably we ll.Table 1. Summary Statistics for NHANES Height DataMean SD*Males 69.3 2.92Females 64.1 2.75*Standard deviations were computed from the cumulative distributions, as they were notsupplied by NHANES.Using the NHANES means and standard deviations as param-eters for two normal densities and assuming equal numbers ofeach sex, we get the graph in Figure 2 for the theoretical mixturedistribu tion of height for persons aged 20–29.Figure 2. Theoretical distribution of U.S. young adult human height ininches as a mixtureof two normal distributions using means and standarddeviations from NHANES data.This obviously is not a bimodal distri bution! Yet Joiner’s liv-ing histogram shows two clear peaks. We start our investigationof thi s paradox by studying the modality of a mixture of twounimo dal den sities, with particular focus on normal densities.2. AN INVESTIGATION OF BIMODALITYMost students are willing to believe that the mixture of twounimo dal den sities with differing mod es will necessarily be bi-modal , as each of the component modes will gen erate a peak inthe mixture distribution. It is quite easy to show that this is notthe case. Consider a mixture of two triangular distributions, asshown in Figure 3. If neither distribution’s support overlaps thea bFigure 3. Dashed lines represent an equal mixture of the componentdistributions (solid lines).other’s mode (Figure 3a), then the mixture distribution is indeedbimod al. However, if each m ode is contained within the sup portof the other distribution (Figure 3b), then outside the compo-nent modes the mixture is monotone, while between modes t hemixtu re is the sum of two li near functions, hence is itself linear.Thus the mix ture is not bimodal.Now consider mixture densit ies of the form f(x) = pf1( x) +(1¡p)f2(x), where f1and f2are each normal densities withmeans ·1< ·2and variances ¼21and ¼22, respect ively, and 0 <p < 1. Consider  rst the simplest situationwhere ¼21= ¼22= ¼2and p = 0:5. If ·1and ·2are far apart, then f clearly will bebimod al, resembling two bell curves side by side. This certainlyhapp ens wh en ·2¡·1is greater than about 6¼ . Furthermore,it is easy to see that the modes of f occur not at ·1and ·2, butbetween them: Both f1and f2are strictly increasing below ·1,so f has positive derivative th ere. At ·1, f1has a derivative ofzero and f2is increasing, so f has posi tive derivative at ·1. Thusany mode must be larger than ·1. Simil arly, any mode must besmaller than ·2.But what happens when ·1and ·2are much closer together?The result for this case is generally credited to Cohen’s (1956)probl em in theAmerican


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