Short communicationThe source of differences in population distributions☆
Highlights
► If the distribution of a variable in a population is affected by many factors, then differences in population distributions may be due exclusively to differences in the distribution of one factor.
Introduction
Many important variables display different distributions in different populations. Examples range from differences in the income distributions of individuals of different races (US Census, 2002, p. 41) to academic test scores of white and black children (Fryer and Levitt, 2004), to different levels of prowess of different ethnic groups in various sports, to the different success rates of men and women in different professions and vocations,1 to the debate on nature or nurture (Pinker, 2002). Explanations offered for these differences typically link genetic and environmental factors,2 suggesting, explicitly or implicitly, that the differences in the shape of the distribution in observable traits between different populations are affected by differences in the distributions of both genetic and environmental factors in the two populations, the relevant question being which has more weight.3 Below, I sketch a simple example which does in fact suggest that the link between the distribution of the underlying factors and the distribution of the relevant observed outcome affected by these factors is in general not straightforward. In view of this, it seems useful to show in this note that, in some specific but relevant cases, the relative distribution of the observed outcome in two different populations is fully determined by the relative distribution of only one of the underlying factors in the populations. Specifically, under some conditions, given two populations with identical distributions for one underlying characteristic, say the genetic features, the first-order and second-order stochastic dominance and hazard rate ordering are determined exclusively by the distribution of other characteristics, for example environmental influences.
While the results are immediate consequences of well-known results in statistical theory (see for example Shaked and Shanthikumar, 2007), their practical importance in social science applications suggests that it may be useful to state them formally and explicitly.
Section snippets
The setting and the result
There are two populations of individuals, each with measure normalised to 1. Each individual is characterised by a parameter , which for definiteness measures the potential for “succeeding”. For example, the expected labour market income, the expected SATs score (the standardised test used for college admissions in the United States), and the probability of reaching the top ranks in a sport are increasing functions of . With large populations, the population distributions of this variable
Conclusion
It stands to reason and is generally accepted that many indicators of “success” depend on both environmental and genetic factors: success in sport depends on training and on innate skills (hand to eye coordination, body build, and so on), labour market income depends on education and social background and on innate ability (intelligence, determination, ambition and so on). Thus, differences in the success of different individuals are caused both by differences in the genetic input, and
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Cited by (0)
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I would like to thank Subir Bose for his encouragement while writing this note and for his comments on the earlier draft, and the editor and a referee of this journal for helpful suggestions which led to improvement of the earlier version.