## 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 $\text{negatively skewed}$.