Kelley’s coefficients of skewness using R

Abstract

Kelley developed a robust measure of asymmetry based on quantiles. His proposal was an absolute index which, when divided by the median, results in its relative expression. If the additive complement is standardized with the semi-percentile range, the percentile coefficient of skewness (PCS) is obtained. Additionally, Kelley provided its standard error in case of normal distribution. However, no statistical software currently computes these measures. The aim of this methodological article is to determine their sampling distribution and facilitate their use. Three random samples of 10,000 data points were generated from three symmetric distributions: semicircular (platykurtic), normal (mesokurtic), and logistic (leptokurtic). By bootstrapping, the sampling distribution was obtained for absolute and relative indices, as well as the PCS. The sampling distributions of the absolute index and the PCS conformed to normality, while that of the relative index was leptokurtic with an excessive bootstrap standard error. Furthermore, a script was developed for the R program, adjusted based on these findings, to obtain point and interval estimates of these indices. The script was applied to a random sample as an example. It is concluded that dividing the absolute index by the semi-percentage range is a better standardization option than dividing by the median.