Question

If the median of the following frequency distribution is $28.5$, then find the missing frequencies.

Class interval:	0-10	10-20	20-30	30-40	40-50	50-60	Total
Frequency:	$5$	$f_1$	$20$	$15$	$f_2$	$5$	$60$

Answer 1

Class interval	Frequency	Cumulative Frequency
$0-10$	$5$	$5$
$10-20$	$f_1$	$5+f_1$
$20-30$	$20$	$25+f_1$
$30-40$	$15$	$40+f_1$
$40-50$	$f_2$	$40+f_1+f_2$
$50-60$	$5$	$45+f_1+f_2$
	$N=60$

From the table, it can be observed that $N=60$, then $\frac{N}{2}=30$.

$45+f_1+f_2=60$

$\therefore$ $f_1+f_2=15$ -------- ( 1 )

Median of the data is given as $28.5$ which lies in interval $20-30$.

$\therefore$ Median class =$20-30$

$\Rightarrow$ $l=20,cf=5+f_1,f=20,h=30-20=10$.

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

$\Rightarrow$ $28.5=20+\frac{30-(5+f_1)}{20}\times 10$

$\Rightarrow$ $8.5=\frac{25-f_1}{2}$

$\Rightarrow$ $17=25-f_1$

$\therefore$ $f_1=8$

From equation ( 1 ),

$\Rightarrow$ $8+f_2=15$

$\therefore$ $f_2=7$

$\therefore$ The values of $f_1$ and $f_2$ are $8$ and $7$