## Central moments 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 first four central moments are as follows

`$m_1=0$`

`$m_2 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^2$`

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

`$m_4 =\frac{1}{N}\sum_{i=1}^n f_i(x_i-\overline{x})^4$`

where,

`$N$`

total number of observations`$\overline{x}$`

sample mean