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Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions

Author

Listed:
  • Theodore P. Hill

    (School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
    These authors contributed equally to this work.)

  • Ronald F. Fox

    (School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
    These authors contributed equally to this work.)

Abstract

Given several different populations, the relative proportions of each in the high (or low) end of the distribution of a given characteristic are often more important than the overall average values or standard deviations. In the case of two different normally-distributed random variables, as is shown here, one of the (right) tail ratios will not only eventually be greater than 1 from some point on, but will even become infinitely large. More generally, in every finite mixture of different normal distributions, there will always be exactly one of those distributions that is not only overrepresented in the right tail of the mixture but even completely overwhelms all other subpopulations in the rightmost tails. This property (and the analogous result for the left tails), 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.

Suggested Citation

  • Theodore P. Hill & Ronald F. Fox, 2022. "Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions," Stats, MDPI, vol. 5(4), pages 1-8, October.
    • Handle: RePEc:gam:jstats:v:5:y:2022:i:4:p:57-984:d:948909
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    Cited by:

    1. Hill, Theodore P. & Arden, Rosalind, 2023. "Recurring errors in studies of gender differences in variability," LSE Research Online Documents on Economics 119788, London School of Economics and Political Science, LSE Library.
    2. Theodore P. Hill & Rosalind Arden, 2023. "Recurring Errors in Studies of Gender Differences in Variability," Stats, MDPI, vol. 6(2), pages 1-7, April.

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