Kurtosis

Topics: Measure of Shape - Statistics

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Etymology

Kurtosis comes from from Greek κυρτός (kyrtos), which means “curved, arching”.

(definition)

Let $x_1,\dots ,x_n$ be observations of a variable $X$. The kurtosis of $X$ is defined as:

$$\begin{array}{rl}k(X)& =\frac{1}{s^4}\left[\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{X}\right)}^4\right]\\ & =\frac{m_4}{(m_2{)}^2}\end{array}$$

(see moment as a measure of shape)

Kurtosis describes how arched the variable’s mass is. See excess kurtosis’s meaning for more details.