Kurtosis (K)

Kurtosis is a measure of the peakedness of a distribution. Like skewness statistics, it is not of much use in general risk analysis. The (normalized) kurtosis statistic is calculated from the generated output values with the following formula:

In a similar manner to skewness, the s4 factor is used to make the kurtosis a pure number. Kurtosis is often known as the fourth moment about the mean (with the symbol 4) and is very sensitive to the values of the data points in the tails of the distribution. Stable values for the kurtosis of a risk analysis result therefore require many more iterations than for other statistics.

Kurtosis is sometimes used in conjunction with the skewness statistics to determine whether an output is approximately Normally distributed. A Normal distribution has a kurtosis of 3 so any output that looks symmetric and bell-shaped, has zero skewness and a kurtosis of 3 can be considered approximately Normal.

A Uniform distribution has a kurtosis of 1.8 (so this is not a very intuitive scale!), a Triangle distribution has a kurtosis of 2.387, the kurtosis of a Lognormal distribution goes from 3.0 to infinity as its mean approaches zero, and an Exponential distribution has a kurtosis of 9.0.

There is some specialised terminology you will sometimes see associated with kurtosis:

• Leptokurtic - kurtosis > 3

• Mesokurtic - kurtosis = 3

• Platykurtic - kurtosis < 3

The kurtosis statistic is sometimes (in Excel, for example, with the KURT( ) function) calculated as

.

This is then called the excess kurtosis, as it depicts the "excess" of kurtosis compared to a Normal distribution.