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
