## Moment coefficient of skewness 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 skewness $\beta_1$ is defined as

`$\beta_1=\dfrac{m_3^2}{m_2^3}$`

The moment coefficient of skewness $\gamma_1$ is defined as

`$\gamma_1=\sqrt{\beta_1}=\dfrac{m_3}{m_2^{3/2}}$`

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`$m_3 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3$`

is third 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 skewness based on moments.

### Solution

The sample mean of $X$ is

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

$x$ | $(x-xb)$ | $(x-xb)^2$ | $(x-xb)^3$ | |
---|---|---|---|---|

27 | 2 | 4 | 8 | |

27 | 2 | 4 | 8 | |

24 | -1 | 1 | -1 | |

26 | 1 | 1 | 1 | |

25 | 0 | 0 | 0 | |

24 | -1 | 1 | -1 | |

22 | -3 | 9 | -27 | |

Total | 175 | 0 | 20 | -12 |

**Second sample central moment**

The second sample central moment is

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

**Third sample central moment**

The third sample central moment is

```
$$
\begin{aligned}
m_3 &=\frac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^3\\
&=\frac{-12}{7}\\
&=-1.7143
\end{aligned}
$$
```

**Coefficient of Skewness**

The coefficient of skewness based on moments ($\beta_1$) is
```
$$
\begin{aligned}
\beta_1 &=\frac{m_3^2}{m_2^3}\\
&=\frac{(-1.7143)^2}{(2.8571)^3}\\
&=\frac{2.9388}{23.3226}\\
&=0.126
\end{aligned}
$$
```

The coefficient of skewness based on moments ($\gamma_1$) is
```
$$
\begin{aligned}
\gamma_1 &=\frac{m_3}{m_2^{3/2}}\\
&=\frac{-1.7143}{(2.8571)^{3/2}}\\
&=\frac{-1.7143}{4.8293}\\
&=-0.355
\end{aligned}
$$
```

As the value of $\gamma_1 < 0$, the data is $\text{negatively skewed}$.