Question

Mean deviation about the mean $\overline{x}$ of the variate x is $\frac{2}{N}[\overline{x}\sum _{x_i<\overline{x}}{f}_i-\sum _{x_i<\overline{x}}{f}_i{x}_i]$. If this is true enter 1 , else enter 0

Answer 1

$S=$M.D.about the mean $\frac{1}{N}\sum _i{f}_i|x_i-\overline{x}|$
$Since\sum _i{f}_i(x-\overline{x})=0$ $\Rightarrow \sum _{x_i<\overline{x}}{f}_1\left(x_i\overline{x}\right)+\sum _{x_i<\overline{x}}{f}_i(x_i-\overline{x})=0$
$\Rightarrow \sum _{x_i>\overline{x}}{f}_1\left(x_i\overline{x}\right)=\sum _{x_i<\overline{x}}{f}_i(\overline{x}-x_i)$
$\therefore S=\frac{2}{N}\sum _{x_i<\overline{x}}{f}_i\left(\overline{x-x_i}\right)$
where $N=\sum _i{f}_i$
$=\frac{2}{N}[\overline{x}\sum _{x_i<\overline{x}}{f}_i-\sum _{x_i<\overline{x}}{f}_i{x}_i]$