MAD (about median) vs. quantile-based alternatives for classical standard deviation, skewness, and kurtosis

Frontiers in Applied Mathematics and Statistics

Applied Mathematics and Statistics

Eugene Pinsky¹  Sidney Klawansky² 

[1]Department of Computer Science, Metropolitan College, Boston University, Boston, MA, United States

[2]Department of Health Policy and Management, Harvard School of Public Health, Boston, MA, United States

关键词: computational statistics;    mean absolute deviation;    kurtosis;    quantiles;    distributions;    data analysis;    skewness;   

DOI  :  10.3389/fams.2023.1206537

 received in 2023-04-15, accepted in 2023-05-15,  发布年份 2023

来源: Frontiers

【 摘 要 】

In classical probability and statistics, one computes many measures of interest from mean and standard deviation. However, mean, and especially standard deviation, are overly sensitive to outliers. One way to address this sensitivity is by considering alternative metrics for deviation, skewness, and kurtosis using mean absolute deviations from the median (MAD). We show that the proposed measures can be computed in terms of the sub-means of the appropriate left and right sub-ranges. They can be interpreted in terms of average distances of values of these sub-ranges from their respective medians. We emphasize that these measures utilize only the first-order moment within each sub-range and, in addition, are invariant to translation or scaling. The obtained formulas are similar to the quantile measures of deviation, skewness, and kurtosis but involve computing sub-means as opposed to quantiles. While the classical skewness can be unbounded, both the MAD-based and quantile skewness always lies in the range [−1, 1]. In addition, while both the classical kurtosis and quantile-based kurtosis can be unbounded, the proposed MAD-based alternative for kurtosis lies in the range [0, 1]. We present a detailed comparison of MAD-based, quantile-based, and classical metrics for the six well-known theoretical distributions considered. We illustrate the practical utility of MAD-based metrics by considering the theoretical properties of the Pareto distribution with high concentrations of density in the upper tail, as might apply to the analysis of wealth and income. In summary, the proposed MAD-based alternatives provide a universal scale to compare deviation, skewness, and kurtosis across different distributions.

【 授权许可】

Unknown   
Copyright © 2023 Pinsky and Klawansky.