A comparative evaluation of some measures of skewness

Abstract

In this paper some rigorous definitions of left (right) skewness, of regular left (right) skewness and of comparison between "simply" skew distributions (strong comparison), and between regular distributions (strog comparison) are reviewed with reference to Frosini (1984). This to judge the coherence of the measures of skewness to be discussed. Six main indices of skewness are examined. The coherence— in general—of an index proposed by Bonferroni (A1) and of an index proposed by Gini (A2) is shown; moreover the validity of the index A3 introduced by Frosini, in the set of regular skew random variables, is recalled. The other indices studied are: the difference between the mean and the median related to the standard deviation (A4); the difference between the mean and the mode related to the standard deviation (A5); the ratio between the third central moment and the cube of the standard deviation. The behaviour of these indices is considered with reference to six groups of comparable —and not—skew distributions; thus we have been able to verify the coherence and to point out some unexpected features of the indices; the results prove the invalidity of A4 and A5. Besides, the comparison between the most important graphs of the indices is confirmed by the correlation coefficients computed for couples of indices, for each group of distributions.