Question

If the mean of the distribution is $62.8$ and sum of frequency is $50$, find $p$ and $q$.

Class	Frequency
0-20	5
20-40	P
40-60	10
60-80	Q
80-100	7
100-120	8

Answer 1

Given that

mean of distribution is $62.8$

sum of frequency is $50$

Class	$0-20$	$20-40$	$40-60$	$60-80$	$80-100$	$100-120$
frequency	$5$	$P$	$10$	$Q$	$7$	$8$

Class	frequency (f)	mid point (x)	fx
$0-20$	$5$	$10$	$50$
$20-40$	$P$	$30$	$30P$
$40-60$	$10$	$50$	$500$
$60-80$	$Q$	$70$	$70Q$
$80-100$	$7$	$90$	$630$
$100-120$	$8$	$110$	$880$

Here

$\sum f=30+P+Q$

$\sum fx=2060+30P+70Q$

We have $\sum f=50$

$\therefore 30+P+Q=50\Rightarrow P+Q=20$ __(1)

we know $\overline{x}=\frac{\sum fx}{\sum f}=(2060+30P+70Q)/50$

$62.8\times 50=2060+30P+70Q$

$30P+70Q=3140-2060=1080$ __(2)

$\left(1\right)\times 70\Rightarrow$ $70P+70Q=1400$

$\left(2\right)\Rightarrow$ $\underset{–}{30P+70Q=1050}$

$40P=320$ $\Rightarrow P=8$

$P=8\Rightarrow \left(1\right)\Rightarrow P+Q=20\Rightarrow Q=20-P=20-8=12$

$Q=12$