Kurtosis
The excess kurtosis of a univariate population is defined by the following formula, where μ2 and μ4 are respectively the second and fourth central moments.
Intuitively, the excess kurtosis describes the tail shape of the data distribution. The normal distribution has zero excess kurtosis and thus the standard tail shape. It is said to be mesokurtic. Negative excess kurtosis would indicate a thin-tailed data distribution, and is said to be platykurtic. Positive excess kurtosis would indicate a fat-tailed distribution, and is said to be leptokurtic.
Problem
Find the excess kurtosis of eruption duration in the data set faithful.
Solution
We apply the function kurtosis from the e1071 package to compute the excess kurtosis of eruptions. As the package is not in the core R library, it has to be installed and loaded into the R workspace.
> duration = faithful$eruptions # eruption durations
> kurtosis(duration) # apply the kurtosis function
[1] -1.5116
Answer
The excess kurtosis of eruption duration is -1.5116, which indicates that eruption duration distribution is platykurtic. This is consistent with the fact that its histogram is not bell-shaped.
Exercise
Find the excess kurtosis of eruption waiting period in faithful.
Note
The default algorithm of the function kurtosis in e1071 is based on the formula g2 = m4∕s4 - 3, where m4 and s are the fourth central moment and sample standard deviation respectively. See the R documentation for selecting other types of kurtosis algorithm.
