Skewness

The degree of asymmetry of a distribution. If the distribution has a longer tail less than the maximum, the function has Negative skewness. Otherwise, it has Positive skewness. Several types of skewness are defined. The Fisher Skewness is defined by

$$\gamma_1 = {\mu_3\over {\mu_2}^{3/2}} = {\mu_3\over\sigma^3},$$ (1)

where $\mu_3$ is the third Moment, and ${\mu_2}^{1/2}\equiv \sigma$ is the Standard Deviation. The Pearson Skewness is defined by

$$\beta_1 = \left({\mu_3\over\sigma^3}\right)^2 = {\gamma_1}^2.$$ (2)

The Momental Skewness is defined by

$$\alpha^{(m)}\equiv {\textstyle{1\over 2}}\gamma_1.$$ (3)

The Pearson Mode Skewness is defined by

$${[{\rm mean}]-[{\rm mode}]\over \sigma}.$$ (4)

Pearson's Skewness Coefficients are defined by

$${3[{\rm mean}]-[{\rm mode}]\over s}$$ (5)

and

$${3[{\rm mean}]-[{\rm median}]\over s}.$$ (6)

The Bowley Skewness (also known as Quartile Skewness Coefficient) is defined by

$${(Q_3-Q_2)-(Q_2-Q_1)\over Q_3-Q_1} = {Q_1-2Q_2+Q_3\over Q_3-Q_1},$$ (7)

where the $Q$s denote the Interquartile Ranges. The Momental Skewness is

$$\alpha^{(m)} \equiv {\textstyle{1\over 2}}\gamma = {\mu_3\over 2\sigma^3}.$$ (8)

An Estimator for the Fisher Skewness $\gamma_1$ is

$$g_1 = {k_3\over {k_2}^{3/2}},$$ (9)

where the $k$s are k-Statistic. The Standard Deviation of $g_1$ is

$${\sigma_{g_1}}^2\approx {6\over N}.$$ (10)

See also Bowley Skewness, Fisher Skewness, Gamma Statistic, Kurtosis, Mean, Momental Skewness, Pearson Skewness, Standard Deviation

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972.

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.