Kurtosis

March 29, 2018

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) $\begin{equation*} \kappa=\frac{1}{N\sigma^2}\sum_{n=1}^{N}(x_{n}-\mu)^4 \cong \frac{\frac{1}{N} \sum_{n=1}^{N} (x_{n}-\bar{x})^4}{(\frac{1}{N} \sum_{n=1}^{N} (x_{n}-\bar{x})^2)^2} \end{equation*}$

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.

comparison of PDFs with different kurtosis values

Comparison of two data sets’ distribution with different kurtosis values.