# Kurtosis

The kurtosis of a univariate population is defined by the following formula, where
μ_{2} and μ_{4} are the second and fourth central moments.

Intuitively, the kurtosis is a measure of the peakedness of the data distribution. Negative kurtosis would indicates a flat data distribution, which is said to be platykurtic. Positive kurtosis would indicates a peaked distribution, which is said to be leptokurtic. Incidentally, the normal distribution has zero kurtosis, and is said to be mesokurtic.

#### Problem

Find the kurtosis of eruption duration in the data set faithful.

#### Solution

We apply the function kurtosis from the e1071 package to compute the 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 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 kurtosis of eruption waiting period in faithful.

#### Note

The default algorithm of the function kurtosis in e1071 is based on the formula
g_{2} = m_{4}∕s^{4} - 3, where m_{4} and s are the fourth central moment and sample standard
deviation respectively. See the R documentation for selecting other types of kurtosis
algorithm.