Question

If x_i' s are the midpoints of the class intervals of a grouped data, f_i' s are the corresponding frequency and $\overline{x}$ is the mean, then $\sum f_i(x_i-\overline{x})=?$
(a) 1
(b) 0
(c) −1
(d) 2

Answer 1

$\begin{array}{l}\text{We know that}\overline{x}=\frac{{\displaystyle \sum f_i{x}_i}}{{\displaystyle \sum f_i}}\\ \Rightarrow \overline{x}\sum f_i=\sum f_i{x}_i...\left(\text{i}\right)\\ \text{Now,}\sum f_i(x_i-\overline{x})=\sum f_i{x}_i-\overline{x}\sum f_i\\ \Rightarrow \sum f_i(x_i-\overline{x})=\sum f_i{x}_i-\sum f_i{x}_i[\mathrm{U}\text{sing}\left(\text{i}\right)]\\ \Rightarrow \sum f_i(x_i-\overline{x})=0\\ \text{Hence, the correct option is}\left(\text{b}\right).\end{array}$