Skewness and Tailedness

Skewness describes how symmetric distribution is. It is defined as the third moment of the distribution after normalization:

$\mbox{Skew}[X]:= \mathbb{E}[(\frac{X-\mu}{\sigma})^3]$

Kurtosis is a measure of tailedness. The heavier the tail is, the larger its Kurtosis is. Mathematically, it is defined as

$\mbox{Kurt}[X]:= \mathbb{E}[(\frac{X-\mu}{\sigma})^4]=\frac{\mu_4}{\sigma^4}=\frac{\mu_4}{\mu_2^2},$

where $\mu$ is the mean, and $\sigma$ is the standard deviation of the distribution of $X$ ; and $\mu_k$ is $k$ -th central moment.

It can be shown that the Kurtosis of Gaussian distribution is $\mbox{Kurt}[N(0,1)] = 3$ . People usually use excess Kurtosis as the extra Kurtosis of a distribution compared with standard Gaussian distribution. Namely,

$\mbox{Excess Kurt}[X] := \mbox{Kurt}[X] - 3.$

We can create a plot with the square of Skewness as its x-axis and Kurtosis as its y-axis. This plot is called Cullen and Frey graph.

This graph helps us to determine which distribution our data is closest to.