On the Order Between the Standard Deviation and Gini Mean Difference
Nawaf Mohammed
TL;DR
This paper investigates when the standard deviation ${\mathrm{SD}}[X]$ dominates or is dominated by the Gini mean difference ${\mathrm{GMD}}[X]$ by expressing both in terms of pairwise differences via $Y=|X-X'|$ and linking their comparison to the mean excess function $m_Y(t)$. It shows sharp sufficient conditions: heavy-tailed regimes (decreasing hazard rate $h_X$ or increasing reverse hazard rate $r_X$) lead to ${\mathrm{SD}}[X] \ge {\mathrm{GMD}}[X]$, while light-tailed regimes (increasing $h_X$ and decreasing $r_X$, or equivalently log-concave density $f_X$) yield ${\mathrm{SD}}[X] \le {\mathrm{GMD}}[X]$. The results are preserved under affine transformations, mixtures, convolutions, and tail truncation, with extensions to discrete distributions showing how tail behavior and distributional regularity govern dispersion ordering. The framework unifies dispersion ordering principles and provides guidance for selecting robust variability measures in risk-sensitive contexts. Key tools include representations of $m_Y(t)$, Chebyshev’s inequality, and log-concavity properties of tails and densities.
Abstract
In this paper, we study the order between the standard deviation (SD) and the Gini mean difference (GMD) and derive sharp, interpretable sufficient conditions under which one exceeds the other. By expressing both the SD and the GMD in terms of pairwise differences and linking their comparison to the mean excess function of the absolute difference of two i.i.d.\ copies, we reduce the problem to structural properties of the underlying distribution. Using tools from reliability and survival analysis, we show that SD dominance arises under heavy-tailed regimes, characterized by decreasing hazard rates or increasing reverse hazard rates. Conversely, when both tails are light -- equivalently, when the hazard rate is increasing and the reverse hazard rate is decreasing -- the GMD dominates the SD. We further demonstrate that these dominance relations are preserved under affine transformations, mixtures, convolutions, and tail truncation, and we extend the analysis to discrete distributions. Numerous examples illustrate the sharpness of the results and highlight the distinct roles played by tail behavior and distributional regularity. Our findings provide a unified framework for understanding dispersion ordering and offer clear guidance for the choice of variability measures in risk-sensitive applications.