# Measuring Skewness with Respect to the Mode

Measuring Skewness with Respect to the Mode Abstract There are several measures employed to quantify the degree of skewness of a distribution. These have been based on the expectations or medians of the distributions considered. In 1964, van Zwet showed that all the standardized odd central moments of order 3 or higher maintained the convex or c-ordering of distributions that he introduced. This ordering has been widely accepted as appropriate for ordering two distributions in relation to skewness. More recently, measures based on the medians have been shown to honor the convex ordering. The measure of skewness (μ – M) / [sgrave] where μ, [sgrave], and M are, respectively, the expectation, standard deviation, and mode of the distribution was initially proposed by Karl Pearson. It unfortunately does not maintain the convex ordering. Here we introduce a measure based on the mode of a distribution that maintains the c-ordering. For many classes of right-skewed distributions, it is easily computed as a function of the shape parameter of the family and the distribution function of the distribution. The measure γ M satisfies −1 ≤ γ M ≤ 1, with 1(−1) indicating extreme right (left) skewness. As γ M can be found explicitly in the gamma, log-logistic, lognormal, and Weibull cases, and its influence function suggests appropriate properties as a skewness measure, it may be considered as an attractive competitor to other measures based on the mean or median. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The American Statistician Taylor & Francis

# Measuring Skewness with Respect to the Mode

, Volume 49 (1): 5 – Feb 1, 1995

## Measuring Skewness with Respect to the Mode

, Volume 49 (1): 5 – Feb 1, 1995

### Abstract

Abstract There are several measures employed to quantify the degree of skewness of a distribution. These have been based on the expectations or medians of the distributions considered. In 1964, van Zwet showed that all the standardized odd central moments of order 3 or higher maintained the convex or c-ordering of distributions that he introduced. This ordering has been widely accepted as appropriate for ordering two distributions in relation to skewness. More recently, measures based on the medians have been shown to honor the convex ordering. The measure of skewness (μ – M) / [sgrave] where μ, [sgrave], and M are, respectively, the expectation, standard deviation, and mode of the distribution was initially proposed by Karl Pearson. It unfortunately does not maintain the convex ordering. Here we introduce a measure based on the mode of a distribution that maintains the c-ordering. For many classes of right-skewed distributions, it is easily computed as a function of the shape parameter of the family and the distribution function of the distribution. The measure γ M satisfies −1 ≤ γ M ≤ 1, with 1(−1) indicating extreme right (left) skewness. As γ M can be found explicitly in the gamma, log-logistic, lognormal, and Weibull cases, and its influence function suggests appropriate properties as a skewness measure, it may be considered as an attractive competitor to other measures based on the mean or median.

/lp/taylor-francis/measuring-skewness-with-respect-to-the-mode-d3SmCS2uBi

# References (23)

Publisher
Taylor & Francis
Copyright Taylor & Francis Group, LLC
ISSN
1537-2731
eISSN
0003-1305
DOI
10.1080/00031305.1995.10476109
Publisher site
See Article on Publisher Site

### Abstract

Abstract There are several measures employed to quantify the degree of skewness of a distribution. These have been based on the expectations or medians of the distributions considered. In 1964, van Zwet showed that all the standardized odd central moments of order 3 or higher maintained the convex or c-ordering of distributions that he introduced. This ordering has been widely accepted as appropriate for ordering two distributions in relation to skewness. More recently, measures based on the medians have been shown to honor the convex ordering. The measure of skewness (μ – M) / [sgrave] where μ, [sgrave], and M are, respectively, the expectation, standard deviation, and mode of the distribution was initially proposed by Karl Pearson. It unfortunately does not maintain the convex ordering. Here we introduce a measure based on the mode of a distribution that maintains the c-ordering. For many classes of right-skewed distributions, it is easily computed as a function of the shape parameter of the family and the distribution function of the distribution. The measure γ M satisfies −1 ≤ γ M ≤ 1, with 1(−1) indicating extreme right (left) skewness. As γ M can be found explicitly in the gamma, log-logistic, lognormal, and Weibull cases, and its influence function suggests appropriate properties as a skewness measure, it may be considered as an attractive competitor to other measures based on the mean or median.

### Journal

The American StatisticianTaylor & Francis

Published: Feb 1, 1995

Keywords: Asymmetry; Convex ordering; Log-logistic distribution; Lognormal distribution