The "kurtosis excess" (Kenney and Keeping 1951, p. 27) is defined in terms of the usual kurtosis by
 |  |  |
(1)
|
 |  |  |
(2)
|
It 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 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 
.
| regime | term |
 | platykurtic |
 | mesokurtic |
 | leptokurtic |
An estimator 
for the kurtosis excess 
is given by
 |
(3)
|
where the 
s
are k-statistics (Kenney and Keeping 1961,
p. 103). For a normal distribution, the
variance of this estimator is
 |
(4)
|
The following table lists the kurtosis excess for a number of common distributions.
![(6[a^3+a^2(1-2b)+b^2(1+b)-2ab(2+b)])/(ab(2+a+b)(3+a+b))](/images/equations/KurtosisExcess/Inline21.svg)
