A Note on Over- and Under-Representation Among Populations with Normally-Distributed Traits

TL;DR

This paper investigates the unique tail-overrepresentation property of normal distributions within finite mixtures, highlighting its absence in other symmetric unimodal distributions like Laplace and Cauchy.

Contribution

It identifies and characterizes a distinctive tail-overrepresentation property exclusive to normal distributions among symmetric unimodal distributions.

Findings

Normal mixtures have a uniquely dominant subpopulation in the right tail.

Other distributions like Laplace and Cauchy do not exhibit this property.

The property is not shared by non-normal symmetric distributions.

Abstract

In every finite mixture of different normal distributions, there will always be exactly one of those distributions that not only is over-represented in the right tail of the mixture, but even completely overwhelms all other subpopulations in the rightmost tails. This property, although not unique to normal distributions, is not shared by other common continuous centrally-symmetric unimodal distributions such as Laplace, nor even by other bell-shaped distributions such as Cauchy (Lorentz) distributions.