# Kurtosis

March 29, 2018

Back to: Random Testing

Kurtosis describes the deviation of a data set’s peak values from the mean. It calculates the signal’s average deviation from the mean to the fourth power divided by the standard deviation to the fourth power. Equation 8 gives the kurtosis of a set of numbers, *x _{n}*,

*n*= 1, …,

*N*.

(1)

Equation 8

Kurtosis is dimensionless. For a random variable with normal distribution, the kurtosis value is 3. For example, the turbulent pressure signal in Figure 3.3 has a kurtosis value of 2.6, which is near the expected value. Some computer programs calculate the excess kurtosis value as *k *– 3. The value of the excess kurtosis for a normal distribution is zero.

#### Probability Distribution

Kurtosis is a ratio of statistical moments: parameters that measure data distribution. More specifically, it is the fourth statistical moment divided by the square of the second statistical moment (variance). A data set’s statistical moments define its probability distribution.

On a graph of a data set’s distribution, the kurtosis measures the distribution “tails.” A data set with a high kurtosis value will have a distribution curve with a higher peak value at the mean and longer tails or, in other words, more data points at the extreme values from the mean.