## Moment coefficient of kurtosis for ungrouped data

Let $x_1, x_2,\cdots, x_n$ be $n$ observations. The mean of $X$ is denoted by $\overline{x}$ and is given by $$\begin{eqnarray*} \overline{x}& =\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*}$$

## Formula

The moment coefficient of kurtosis $\beta_2$ is defined as

### $\beta_2=\dfrac{m_4}{m_2^2}$

The moment coefficient of kurtosis $\gamma_2$ is defined as

### $\gamma_2=\beta_2-3$

where

• $n$ total number of observations
• $\overline{x}$ sample mean
• $m_2 =\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2$ is second central moment
• $m_4 =\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4$ is fourth central moment