Kurtosis is defined as a normalized form of the fourth central moment 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 (Pearson's notation; Abramowitz and Stegun 1972, p. 928) or (Kenney and Keeping 1951, p. 27; Kenney and Keeping 1961, pp. 99-102). The kurtosis of a theoretical distribution is defined by
(1)
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where denotes the th central moment (and in particular, 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
(2)
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(3)
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and is commonly denoted (Abramowitz and Stegun 1972, p. 928) or . Kurtosis excess is commonly used because of a normal distribution is equal to 0, while the kurtosis proper is equal to 3. Unfortunately, Abramowitz and Stegun (1972) confusingly refer to as the "excess or kurtosis."
For many distributions encountered in practice, a positive 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).