Question

If the median of the following data is $32.5$, find the missing frequencies.

Class interval:	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	Total
Frequency:	$f_1$	$5$	$9$	$12$	$f_2$	$3$	$2$	$40$

Answer 1

Class interval	Frequency	Cumulative frequency
$0-10$	$f_1$	$f_1$
$10-20$	$5$	$5+f_1$
$20-30$	$9$	$14+f_1$
$30-40$	$12$	$26+f_1$
$40-50$	$f_2$	$26+f_1+f_2$
$50-60$	$3$	$29+f_1+f_2$
$60-70$	$2$	$31+f_1+f_2$
	$N=40$

$\Rightarrow$ $Median=32.5,$ then $30-40$ is class interval.

$\Rightarrow$ $l=30,cf=14+f_1,f=12$ and $h=40-30=10$.

$\Rightarrow$ $N=40,\frac{N}{2}=20$

$\Rightarrow$ $31+f_1+f_2=40$.

$\Rightarrow$ Then, $f_1+f_2=9$ ----- ( 1 )

$\Rightarrow$ $Median=l+\frac{\frac{N}{2}-cf}{f}\times h$

$\Rightarrow$ $32.5=30+\frac{20-(14+f_1)}{12}\times 10$

$\Rightarrow$ $2.5=\frac{6-f_1}{6}\times 5$

$\Rightarrow$ $15=30-5f_1$

$\Rightarrow$ $5f_1=15$

$\therefore$ $f_1=3$

Substituting value in equation ( 1 ) we get,

$\Rightarrow$ $3+f_2=9$

$\therefore$ $f_2=6$

$\therefore$ $f_1=3,f_2=6$