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)
|
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)
|
 |  |  |
(3)
|
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).
