Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions
1
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
2
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Academic Editor: Wei Zhu
Stats 2022, 5(4), 977-984; https://doi.org/10.3390/stats5040057
Received: 3 October 2022
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Revised: 17 October 2022
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Accepted: 18 October 2022
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Published: 20 October 2022
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.
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MDPI and ACS Style
Hill, T.P.; Fox, R.F. Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions. Stats 2022, 5, 977-984. https://doi.org/10.3390/stats5040057
AMA Style
Hill TP, Fox RF. Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions. Stats. 2022; 5(4):977-984. https://doi.org/10.3390/stats5040057
Chicago/Turabian StyleHill, Theodore P., and Ronald F. Fox. 2022. "Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions" Stats 5, no. 4: 977-984. https://doi.org/10.3390/stats5040057
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