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 $230.$

Using the median formula fill up the missing frequencies.

Answer 1

Class Interval	Frequency	Cumulative frequency
$10-20$	$12$	$12$
$20-30$	$30$	$30+12=42$
$30-40$	$x$	$42+x$
$40-50$	$65$	$42+x+65=107+x$
$50-60$	$y$	$107+x+y$
$60-70$	$25$	$107+x+y+25=132+x+y$
$70-80$	$18$	$132+x+y+18=150+x+y$
	$N=230$

Given median $=46$

Then, median class$=40-50$

$l=40$,$h=50-40=10$,$f=65$,$F=42+x$

Median$=l+\frac{\left(\frac{N}{2}\right)-F}{f}\times h$

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

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

$\Rightarrow 6=\frac{73-x}{65}\times 10$

$\Rightarrow \frac{6\times 65}{10}=73-x$

$\Rightarrow \frac{390}{10}=73-x$

$\Rightarrow 39=73-x$

$\therefore x=73-39=34$

Given: $N=230$

$\Rightarrow 12+30+x+65+y+25+18=230$

$\Rightarrow 12+30+34+65+y+25+18=230$

$\Rightarrow 184+y=230$

$\Rightarrow y=230-184=46$

$\therefore x=34$ and $y=46$