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

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