Question

The median of the following data is 32.5:

Class interval	Frequency
0-10	x
10-20	5
20-30	9
30-40	12
40-50	y
50-60	3
60-70	2
Total	40

Find the value of x and y.

• A. $x=2,y=7$
• B. $x=3,y=6$
• C. $x=5,y=4$
• D. $x=7,y=9$

Answer 1

The correct option is B $x=3,y=6$

Frequency distribution table:

$ci$	$f_i$	$cf$
$0-10$	$x$	$x$
$10-20$	$5$	$x+5$
$20-30$	$9$	$x+14$
$30-40$	$12$	$x+26$
$40-50$	$y$	$x+y+26$
$50-60$	$3$	$x+y+29$
$60-70$	$2$	$x+y+31$

Median $=l+\frac{(\frac{N}{2}-cf)}{f_m}\times c$

Here,

Class Interval $c=10$

$N=\Sigma f_i=40$ ...(given)

$N=\Sigma f_i=x+y+31$

$\Rightarrow x+y+31=40$

$\Rightarrow y=9-x$ ...(1)

$\because Median=32.5$

$\Rightarrow$ Median class $=30-40$

$\Rightarrow$Frequency of median class$f_m=12$

Lower boundary of median class $l=30$

previous cumulative frequency of median class $cf=x+14$

$\therefore$ 32.5$=30+\frac{(\frac{40}{2}-(x+14))}{12}\times 10$

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

$\therefore 195=180+100-5x-70$

$\therefore$ $5x=15$

$\therefore$ $x=3$

substituting $x$ in (1), we get

$y=9-3$

$\therefore$ $y=6$

Hence. option $B$ is correct.