A measure of asymmetry based on a new necessary and sufficient condition for symmetry

Abstract

It is common practice to make assertions about the symmetry or asymmetry of a probability density function based on coefficients of skewness. Since most coefficients of skewness are designed to be zero for a symmetric density, they do, overall, provide an indication of symmetry. However, skewness, as opposed to asymmetry, is primarily influenced by the tail behavior of a density function. Therefore, coefficients of skewness do not reliably calibrate asymmetry in the density curve. Our goal in this paper is to present a new measure of asymmetry. With this purpose, we first provide a new necessary and sufficient condition for a continuous probability density function to be symmetric. This condition is then used to produce an asymmetry measure - a coefficient of asymmetry- for a continuous probability density function on the scale of −1 to 1. We show through examples that the proposed measure does an admirable job of capturing the visual impression of asymmetry of a continuous density function. Further, we discuss its implementation in practice and conclude by demonstrating its usefulness via an application to a real world dataset.