Question

Find the missing frequencies in the following frequency distribution whose mean is 50.
$\begin{array}{ccccccc}\text{x}& 10& 30& 50& 70& 90& \text{Total}\\ \text{f}& 17& f_1& 32& f_2& 19& 120\end{array}$

Answer 1

Sol:

Let,

\( f_1 ~and~ f_2\) be the missing frequencies.

We prepare the following frequency distribution table:

xi	fi	fixi
10	17	170
30	f1	30f1
50	32	1600
70	f2	70f2
90	19	1710
Total	120	3480+30f1+70f2

Here
Thus, we have:
Mean =$\frac{\sum f_i{x}_i}{\sum x_i}$
⇒$50=\frac{(3480+30f_1+70f_2)}{120}$
⇒$6000=3480+30f_1+70f_2$
⇒$30f_1+70f_2=2520$ ..... (i)
Also,
Given:
$17+f_1+32+f_2+19=120$
⇒ $68+f_1+f_2=120$
⇒ $f_1+f_2=52$
or, $f_2=52-f_1$ ...(ii)
By putting the value of f 2 in (i), we get:
$2520=30f_1+70(52-f_1)$
⇒ $2520=30f_1+3640-70f_1$
⇒ $40f_1=1120$
⇒ $f_1=28$
Substituting the value in (ii), we get:
$f_2=52-f_1=52-28=24$