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Kurtosis Excess


The "kurtosis excess" (Kenney and Keeping 1951, p. 27) is defined in terms of the usual kurtosis by

gamma_2=beta_2-3
(1)
=(mu_4)/(mu_2^2)-3.
(2)

It 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 excess, 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).

The following table gives terms sometimes applied to different regimes of gamma_2.

An estimator g_2=<gamma_2> for the kurtosis excess gamma_2 is given by

 g_2=(k_4)/(k_2^2),
(3)

where the ks are k-statistics (Kenney and Keeping 1961, p. 103). For a normal distribution, the variance of this estimator is

 var(g^^_2) approx (24)/N.
(4)

The following table lists the kurtosis excess for a number of common distributions.


See also

Central Moment, Excess, k-Statistic, Kurtosis, 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.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.

Cite this as:

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

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