Question

If the variance of the following frequency distribution :
$\begin{array}{cccc}\text{Class}& 10-20& 20-30& 30-40\\ \text{Frequency}& 2& x& 2\end{array}$
is $50$, then $x$ is equal to

Answer 1

$\begin{array}{cccccc}{x}_i& f_i& f_i{x}_i& x_i-\overline{x}& (x_i-\overline{x}{)}^2& f_i(x_i-\overline{x}{)}^2\\ 15& 2& 30& -10& 100& 200\\ 25& x& 25x& 0& 0& 0\\ 35& 2& 70& 10& 100& 200\\ & \sum f_i=4+x& \sum f_i{x}_i=100+25x& & & \sum f_i(x_i-\overline{x}{)}^2=400\end{array}$
Mean :
$\overline{x}=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{100+25x}{4+x}$
$\Rightarrow \overline{x}=25$

Now, variance
$=\frac{\sum f_i(x_i-\overline{x}{)}^2}{\sum f_i}=\frac{400}{4+x}$
$\Rightarrow \frac{400}{4+x}=50$
$\Rightarrow 200+50x=400$
$\therefore x=4$