## 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`$m_2 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2$`

is second sample central moment`$m_4 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4$`

is fourth sample central moment

## Example

The hourly earning (in dollars) of sample of 7 workers are :

`26, 21, 24, 22, 25, 24, 23.`

Compute coefficient of kurtosis based on moments.

### Solution

The mean of $X$ is

```
$$
\begin{aligned}
\overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\
&=\frac{175}{7}\\
&=25 \text{ dollars}
\end{aligned}
$$
```

$x$ | $(x-xb)$ | $(x-xb)^2$ | $(x-xb)^4$ | |
---|---|---|---|---|

27 | 2 | 4 | 16 | |

27 | 2 | 4 | 16 | |

24 | -1 | 1 | 1 | |

26 | 1 | 1 | 1 | |

25 | 0 | 0 | 0 | |

24 | -1 | 1 | 1 | |

22 | -3 | 9 | 81 | |

Total | 175 | 0 | 20 | 116 |

**Second sample central moment**

The second sample central moment is

```
$$
\begin{aligned}
m_2 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2\\
&=\frac{20}{7}\\
&=2.8571
\end{aligned}
$$
```

**Fourth sample central moment**

The fourth sample central moment is

```
$$
\begin{aligned}
m_4 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4\\
&=\frac{116}{7}\\
&=16.5714
\end{aligned}
$$
```

**Coefficient of Kurtosis**

The coefficient of kurtosis based on moments ($\beta_2$) is
```
$$
\begin{aligned}
\beta_2 &=\frac{m_4}{m_2^2}\\
&=\frac{(16.5714)}{(2.8571)^2}\\
&=\frac{16.5714}{8.163}\\
&=2.0301
\end{aligned}
$$
```

The coefficient of kurtosis based on moments ($\gamma_2$) is
```
$$
\begin{aligned}
\gamma_2 &=\beta_2-3\\
&=2.0301 -3\\
&=-0.9699
\end{aligned}
$$
```

As the value of $\gamma_2 < 0$, the data is $\text{platy-kurtic}$.