Question

Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is $50$.

$\begin{array}{cc}{x}_i& f_i\\ 10& 17\\ 30& f_1\\ 50& 32\\ 70& f_2\\ 90& 19\\ \text{Total}& 120\end{array}$

Answer 1

$\begin{array}{ccc}{x}_i& f_i& x_i{f}_i\\ 10& 17& 170\\ 30& f_1& 30f_1\\ 50& 32& 1600\\ 70& f_2& 70f_2\\ 90& 19& 1710\\ \text{Total}& 120& 3480+30f_1+70f_2\end{array}$

Mean $=\frac{\sum f_i{x}_i}{\sum f_i}$

$\therefore 50=\frac{3480+30f_1+70f_2}{120}$

$\Rightarrow 6000=3480+30f_1+70f_2$

$\Rightarrow 30f_1+70f_2=2520$

$\Rightarrow 3f_1+7f_2=252$...(i)

Also,

$\sum f_i=120$

$\Rightarrow 68+f_1+f_2=120$

$\Rightarrow f_1+f_2=52$

$\Rightarrow f_1=52-f_2$...(ii)

Substituting from (ii) in (i), we get,

$\Rightarrow 3(52-f_2)+7f_2=252$

$\Rightarrow 156-3f_2+7f_2=252$

$\Rightarrow 4f_2=96$

$\therefore f_2=24$

Hence, $f_1=52-24=28$