Computation and Interpretation of Mean Absolute Deviations by Cumulative Distribution Functions

Eugene Pinsky ^1,2*

• ¹ College of Metropolitan, Boston University, Boston, United States
• ² Department of Computer Science, Metropolitan College, Boston University, Boston, United States

In recent years, there has been an increased interest in using the mean absolute deviation around the mean and median (the L 1 norm) as an alternative to standard deviation σ (the L 2 norm). To date, the mean absolute deviation has been computed for some distributions. For other distributions, expressions for mean absolute deviations have not been available or reported.Typically, mean absolute deviations are derived using the probability density functions (PDFs).By contrast, we derive simple expressions in terms of the integrals of the cumulative distribution functions. We show that mean absolute deviations have simple geometric interpretations as areas under the appropriately folded CDF. As a result, mean absolute deviations 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 mean absolute deviations for many well-known continuous and discrete distributions. For some of these distributions, the expressions for mean absolute deviations have not been reported. We hope this work will be useful for researchers and practitioners interested in mean absolute deviations.