Question

An incomplete distribution is given below:

Variable:	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Frequency:	$12$	$30$	--	$65$	---	$25$	$18$

You are given that the median value is $46$ and the total number of items is $230$.
(i) Using the median formula and fill up the missing frequencies.
(ii) Calculate the AM of the completed distribution.

Answer 1

$\left(i\right)$

Class interval	Frequency	Cumulative Frequency
$10-20$	$12$	$12$
$20-30$	$30$	$42$
$30-40$	$f_1$	$42+f_1$
$40-50$	$65$	$107+f_1$
$50-60$	$f_2$	$107+f_1+f_2$
$60-70$	$25$	$132+f_1+f_2$
$70-80$	$18$	$150+f_1+f_2$

$\Rightarrow$ We have, $Median=46$

Clearly, it lies in the class interval $40-50$. So, $40-50$ is the median class.

$\therefore$ $l=40,h=10,f=65,F=42+f_1$ and $N=230,\frac{N}{2}=115$

Now, $Median=l+\frac{\frac{N}{2}-F}{f}\times h$

$\Rightarrow$ $46=40+\frac{115-42-f_1}{65}\times 10$

$\Rightarrow$ $3=\frac{73-f_1}{13}$

$\therefore$ $f_1=34$

We have, $N=230$

$\therefore$ $150+f_1+f_2=230$

$\therefore$ $f_2=46.$

$\left(ii\right)$

Class interval	$f_i$	$x_i$	$f_i{x}_i$
$10-20$	$12$	$15$	$180$
$20-30$	$30$	$25$	$750$
$30-40$	$34$	$35$	$1190$
$40-50$	$65$	$45$	$2925$
$50-60$	$46$	$55$	$2530$
$60-70$	$25$	$65$	$1625$
$70-80$	$18$	$75$	$1350$
	$\sum f_i=230$		$\sum f_i{x}_i=10550$

$\Rightarrow$ $Mean=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{10550}{230}=45.87$