# Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions

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## Abstract

**:**

## 1. Introduction

## 2. Tail Ratios and Right-Tail Dominance

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

**Proof.**

**Example**

**1.**

- (i)
- Let ${P}_{1}$ and ${P}_{2}$ be Cauchy distributions with medians ${m}_{1}=0$ and ${m}_{2}=0.5$ and scale parameters ${s}_{1}=1$ and ${s}_{2}=0.5$, i.e., with density functions ${{f}_{P}}_{i}\left(x\right)=$${\left(\pi {s}_{i}[1+{\left((x-{m}_{i})/{s}_{i}\right)}^{2}]\right)}^{-1},$ respectively. Then by Lemma 1,$$\underset{c\to \infty}{lim}\frac{{\overline{F}}_{{P}_{1}}\left(c\right)}{{\overline{F}}_{{P}_{2}}\left(c\right)}=\underset{x\to \infty}{lim}\frac{{f}_{{P}_{1}}\left(x\right)}{{f}_{{P}_{2}}\left(x\right)}=\underset{x\to \infty}{lim}\frac{\pi (1+2{x}^{2}-2x)}{\pi (1+{x}^{2})}=2,$$
- (ii)
- Let ${P}_{1}$ and ${P}_{2}$ be Laplace distributions with medians ${m}_{1}=1$ and ${m}_{2}=0$ and scale parameters ${s}_{1}={s}_{2}=1$, i.e., with density functions ${{f}_{P}}_{i}\left(x\right)=$ ${\left(2{s}_{i}exp\left(-\left(\right|x-{m}_{i}\left|\right)/{s}_{i}\right)\right)}^{-1}$, respectively. Then,$$\underset{c\to \infty}{lim}\frac{{\overline{F}}_{{P}_{1}}\left(c\right)}{{\overline{F}}_{{P}_{2}}\left(c\right)}=\underset{c\to \infty}{lim}\frac{{e}^{1-c}}{{e}^{-c}}=e;$$
- (iii)
- Let ${P}_{1}$ and ${P}_{2}$ be Laplace distributions with medians ${m}_{1}={m}_{2}=0$ and scale parameters ${s}_{1}=1$ and ${s}_{2}=0.5$, respectively. Then, ${P}_{1}$ strongly dominates ${P}_{2}$ in the right tail since$$\underset{c\to \infty}{lim}\frac{{\overline{F}}_{{P}_{1}}\left(c\right)}{{\overline{F}}_{{P}_{2}}\left(c\right)}=\underset{c\to \infty}{lim}\frac{{e}^{-c}}{{e}^{-2c}}=\infty .$$
- (iv)
- Let ${P}_{1}$ and ${P}_{2}$ be normal distributions with identical variances $+1$ and with means 1 and 0, respectively. Then, the density functions ${f}_{{P}_{1}}\left(x\right)=(1/\sqrt{2\pi}){e}^{-(1/2){(x-1)}^{2}}$ and ${f}_{{P}_{2}}\left(x\right)=(1/\sqrt{2\pi}){e}^{-(1/2){x}^{2}}$ for ${P}_{1}$ and ${P}_{2}$, respectively, satisfy $\left({f}_{{P}_{1}}\left(x\right)\right)/\left({f}_{{P}_{2}}\left(x\right)\right)={e}^{x-(1/2)}\to \infty $ as $x\to \infty $; so, by Lemma 1, ${P}_{1}$ strongly dominates ${P}_{2}$ in the right tail.

## 3. Tail Ratios in Normal Distributions

**Example**

**2.**

- (i)
- Let ${P}_{1}\sim N(100,{10}^{2})$ and ${P}_{2}\sim N(110,{10}^{2})$. It is clear that the unique crossing point of the density functions of ${P}_{1}$ and ${P}_{2}$ is at $x=105$, which implies that the proportion of ${P}_{2}$ that is above any cutoff $c>105$ is greater than the proportion of ${P}_{1}$ above c, i.e., the tail ratio of ${P}_{2}$ to ${P}_{1}$ is greater than 1 for all cutoff values c strictly greater than 105. Conversely, the proportion of ${P}_{1}$ below any $c<105$ is greater than the proportion of ${P}_{2}$ below c.
- (ii)
- Let ${P}_{1}\sim N(100,{10}^{2})$ and ${P}_{2}\sim N(101,{11}^{2})$. By basic algebra, the two crossing points of the density functions of ${P}_{1}$ and ${P}_{2}$ are seen to be at ${x}_{1}\cong 83.52$ and ${x}_{2}\cong 106.95$, which implies that the tail ratio of ${P}_{2}$ to ${P}_{1}$ is greater than 1 for all cutoffs $c>{x}_{2}$. Similarly, in this case ${P}_{2}$ also dominates ${P}_{1}$ in the lower tail in that the proportion of ${P}_{2}$ that is below any cutoff $c<{x}_{1}$ is also greater than the proportion of ${P}_{1}$ below c.

**Theorem**

**1.**

- (i)
- either${P}_{1}$strongly dominates${P}_{2}$ in the right tail or ${P}_{2}$ strongly dominates ${P}_{1}$ in the right tail;
- (ii)
- if ${P}_{1}$ strongly dominates ${P}_{2}$ in the right tail, then either ${P}_{1}$ has greater mean (average value) than ${P}_{2}$ or ${P}_{1}$ has greater variance than ${P}_{2}$ or both;
- (iii)
- if ${P}_{1}$ has greater variance than ${P}_{2}$, then ${P}_{1}$ strongly dominates ${P}_{2}$ in both right and left tails, independent of the means.

**Proof.**

**Corollary**

**1.**

**Proof.**

**Example**

**3.**

## 4. Overrepresentation in the Right Tail

**Definition**

**3.**

**Theorem**

**2.**

**Proof.**

**Example**

**4.**

**Figure 2.**Figure from [13], p. 7, titled “The numbers College Board didn’t publish” showing statistics for nearly two million students for the 2016 Edition of the Scholastic Aptitude Test, with breakdown by gender and score ranges. Note that the proportions of males in various score ranges, i.e., the tail ratios, increase as the score range increases and the left tail ratios also increase as the score range decreases.(About 10% more females participated than males, which is reflected in the Adjusted Male/Female Ratios).

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Universal empirical rules for all normal (

**left**) and Cauchy (

**right**) distributions. Here m represents the median, and s represents the distance (called standard deviation for normal distributions and scale parameter for Cauchy) from m to the inflection point. Although both families, normal and Cauchy, have similar bell-shaped density functions, normal distributions satisfy the main strong-domination and overrepresentation properties presented in this note, but Cauchy distributions do not.

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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 Style**

Hill, 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