MAD (about median) vs. quantile-based alternatives for classical standard deviation, skewness, and kurtosis - Pinsky, Eugene; Klawansky, Sidney (Frontiers in Applied Mathematics and Statistics, 2023)
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 $[0, 1]$. For the six well-known theoretical distributions considered, we present a detailed comparison of MAD-based, quantile-based, and classical metrics. 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.