## Moment coefficient of skewness for grouped data

Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The mean of $X$ is denoted by $\overline{x}$ and is given by $$\begin{eqnarray*} \overline{x}& =\frac{1}{N}\sum_{i=1}^{n}f_ix_i \end{eqnarray*}$$

## 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

• $N=\sum_i f_i$ total number of observations
• $\overline{x}$ sample mean
• $m_2 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^2$ is second central moment
• $m_3 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^3$ is third central moment