Skewness is a measure of the degree of asymmetry of a distribution. If the left tail (tail at small end of the distribution) is more pronounced than the right tail (tail at the large end of the distribution), the function is said to have negative skewness. If the reverse is true, it has positive skewness. If the two are equal, it has zero skewness.

Several types of skewness are defined, the terminology and notation of which are unfortunately rather confusing. "The" skewness of a distribution is defined to be


where mu_i is the ith central moment. The notation gamma_1 is due to Karl Pearson, but the notations alpha_3 (Kenney and Keeping 1951, p. 27; Kenney and Keeping 1962, p. 99) and sqrt(beta_1) (due to R. A. Fisher) are also encountered (Kenney and Keeping 1951, p. 27; Kenney and Keeping 1962, p. 99; Abramowitz and Stegun 1972, p. 928). Abramowitz and Stegun (1972, p. 928) also confusingly refer to both gamma_1 and beta=gamma_1^2 as "skewness." Skewness is implemented in the Wolfram Language as Skewness[dist].

An estimator g_1=<gamma_1> for the skewness gamma_1 is


where the ks are k-statistics (Kenney and Keeping 1962, p. 101). For a normal population with a sample size of N, the variance of g_1 is

 var(g_1) approx 6/N

(Kendall et al. 1998).

The following table gives the skewness for a number of common distributions.

Several other forms of skewness are also defined. The momental skewness is defined by


The Pearson mode skewness is defined by


Pearson's skewness coefficients are defined by




The Bowley skewness (also known as quartile skewness coefficient) is defined by


where the Qs denote the interquartile ranges. The momental skewness is


See also

Bowley Skewness, Gamma Statistic, Graph Skewness, h-Statistic, Kurtosis, Mean, Momental Skewness, Standard Deviation

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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972.Kendall, W. S.; Barndorff-Nielson, O.; and van Lieshout, M. C. Current Trends in Stochastic Geometry: Likelihood and Computation. Boca Raton, FL: CRC Press, 1998.Kenney, J. F. and Keeping, E. S. "Skewness." §7.10 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 100-101, 1962.Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Moments of a Distribution: Mean, Variance, Skewness, and So Forth." §14.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 604-609, 1992.Stuart, A.; and Ord, J. K. Kendall's Advanced Theory of Statistics, Vol. 1: Distribution Theory, 6th ed. New York: Oxford University Press, 1998.

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Cite this as:

Weisstein, Eric W. "Skewness." From MathWorld--A Wolfram Web Resource.

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