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Kurtosis


Kurtosis is defined as a normalized form of the fourth central moment mu_4 of a distribution. There are several flavors of kurtosis, the most commonly encountered variety of which is normally termed simply "the" kurtosis and is denoted beta_2 (Pearson's notation; Abramowitz and Stegun 1972, p. 928) or alpha_4 (Kenney and Keeping 1951, p. 27; Kenney and Keeping 1961, pp. 99-102). The kurtosis of a theoretical distribution is defined by

 beta_2=(mu_4)/(mu_2^2),
(1)

where mu_i denotes the ith central moment (and in particular, mu_2 is the variance). This form is implemented in the Wolfram Language as Kurtosis[dist].

The "kurtosis excess" (Kenney and Keeping 1951, p. 27) is defined by

gamma_2=beta_2-3
(2)
=(mu_4)/(mu_2^2)-3
(3)

and is commonly denoted gamma_2 (Abramowitz and Stegun 1972, p. 928) or b_2. Kurtosis excess is commonly used because gamma_2 of a normal distribution is equal to 0, while the kurtosis proper is equal to 3. Unfortunately, Abramowitz and Stegun (1972) confusingly refer to beta_2 as the "excess or kurtosis."

For many distributions encountered in practice, a positive gamma_2 corresponds to a sharper peak with higher tails than if the distribution were normal (Kenney and Keeping 1951, p. 54). This observation is likely the reason kurtosis excess was historically (but incorrectly) regarded as a measure of the "peakedness" of a distribution. However, the correspondence between kurtosis and peakedness is not true in general; in fact, a distribution with a perfectly flat top may have infinite kurtosis, while one with infinite peakedness may have negative kurtosis excess. As a result, kurtosis excess provides a measure of outliers (i.e., the presence of "heavy tails") in a distribution, not its degree of peakedness (Kaplansky 1945; Kenney and Keeping 1951, p. 27; Westfall 2014).


See also

Central Moment, Excess, k-Statistic, Kurtosis Excess, Mean, Skewness, Standard Deviation

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References

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.Darlington, R. B. "Is Kurtosis Really Peakedness?" Amer. Statist. 24, 19-22, 1970.Dodge, Y. and Rousson, V. "The Complications of the Fourth Central Moment." Amer. Statist. 53, 267-269, 1999.Kaplansky, I. "A Common Error Concerning Kurtosis." J. Amer. Stat. Assoc. 40, 259, 1945.Kenney, J. F. and Keeping, E. S. "Kurtosis." §7.12 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 102-103, 1962.Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.Moors, J. J. A. "The Meaning of Kurtosis: Darlington Reexamined." Amer. Statist. 40, 283-284, 1986.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.Ruppert, D. "What is Kurtosis? An Influence Function Approach." Amer. Statist. 41, 1-5, 1987.Westfall, P. H. "Kurtosis as Peakedness, 1905-2014. R.I.P." Amer. Statist. 68, 191-195, 2014.

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Kurtosis

Cite this as:

Weisstein, Eric W. "Kurtosis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Kurtosis.html

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