Find the missing frequencies in the following frequency distribution, whose mean is 50.

Question:

Find the missing frequencies in the following frequency distribution, whose mean is 50.

Solution:

We will prepare the following table:

Thus, we have:

Mean $=\frac{\sum f_{i} x_{i}}{\sum x_{i}}$

$\Rightarrow 50=\frac{3480+30 f_{1}+70 f_{2}}{120}$

$\Rightarrow 6000=3480+30 f_{1}+70 f_{2}$

$\Rightarrow 30 f_{1}+70 f_{2}=2520 \ldots \ldots$ (i)

Also,
Given:

$17+f_{1}+32+f_{2}+19=120$

$\Rightarrow 68+f_{1}+f_{2}=120$

$\Rightarrow f_{1}+f_{2}=52$

or, $f_{2}=52-f_{1} \ldots$ (ii)

By putting the value of f2 in (i), we get:

$2520=30 f_{1}+70\left(52-f_{1}\right)$

$\Rightarrow 2520=30 f_{1}+3640-70 f_{1}$

$\Rightarrow 40 f_{1}=1120$

$\Rightarrow f_{1}=28$

Substituting the value in (ii), we get:

$f_{2}=52-f_{1}=52-28=24$