On the Relationship Between the Gini Coefficient and Skewness

Abstract

Skewness, a measure of the asymmetry of a distribution, is frequently employed to reflect a biologically important property. Another statistic, the Gini coefficient (GC), originally used to measure economic inequality, has been validated in measuring the inequality of biological size distributions. Given that the GC and skewness control overlapping domains and interact with each other, researchers are perplexed by their relationship (varying with the biological [organ, tissue or cell] size distributions) and use both of them together to provide a more complete picture of the data. This study provides analytical forms of the GC for biological size distributions, including two-parameter Weibull, uniform, normal, two-parameter lognormal, gamma, three-parameter Weibull, three-parameter lognormal, and three-parameter gamma distributions. Two empirical data sets and simulation data sets were used to clarify the GC–skewness relationships under different distributions. For the aforementioned distributions, the GC–skewness relationships can be divided into three types: (i) for a symmetrical distribution, the skewness is 0, and the GC ranges from 0.56 to 0.58 multiplied by the standard deviation divided by the mean irrespective of its relationship with the skewness; (ii) for an asymmetric distribution with a zero threshold, the GC is a monotonously increasing function of the skewness, and the two measures are equivalent; (iii) for an asymmetric distribution with a non-zero threshold, the GC is determined by the skewness and an additional correction factor. We showed the differences in improving the accuracy of GC calculations based on small-sample adjustments among various calculation methods, including the polygon (trapezoidal set) area method and the rotated Lorenz curve method. The present study turns the GC into a property of the distribution and offers a clear understanding for the GC–skewness relationship. This work provides insights into selecting and using the GC to measure inequality in ecological data, facilitating more accurate and meaningful analyses.