Computation and interpretation of mean absolute deviations by cumulative distribution functions - Pinsky, Eugene (Frontiers in Applied Mathematics and Statistics, 2025)
In recent years, there has been an increased interest in using the mean absolute deviation (MAD) around the mean and median (the L1 norm) as an alternative to standard deviation σ (the L2 norm). Till now, the MAD has been computed for some distributions. For other distributions, expressions for mean absolute deviations (MADs) are not available nor reported. Typically, MADs are derived using the probability density functions (PDFs). By contrast, we derive simple expressions in terms of the integrals of the cumulative distribution functions (CDFs). We show that MADs have simple geometric interpretations as areas under the appropriately folded CDF. As a result, MADs can be computed directly from CDFs by computing appropriate integrals or sums for both continuous and discrete distributions, respectively. For many distributions, these CDFs have a simpler form than PDFs. Moreover, the CDFs are often expressed in terms of special functions, and indefinite integrals and sums for these functions are well known. We compute MADs for many well-known continuous and discrete distributions. For some of these distributions, the expressions for MADs have not been reported. We hope this study will be useful for researchers and practitioners interested in MADs.