Question

Find the value of $x$ and $y$ if the median for the following data is $31$

$CI$	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	Total
Frequency	$5$	$x$	$6$	$y$	$6$	$5$	$40$

Answer 1

$Classes$	$Frequency$	$c.f$
$0-10$	$5$	$5$
$10-20$	$x$	$5+x$
$20-30$	$6$	$11+x$
$30-40$	$y$	$11+x+y$
$40-50$	$6$	$17+x+y$
$50-60$	$5$	$22+x+y$
$Total$	$40$

$\Rightarrow$ $22+x+y=40$

$\therefore$ $x+y=18$ ----- ( 1 )

$\Rightarrow$ Here, median class $=30-40$

$\Rightarrow$ $l=30,c.f=11+x,f=y,h=10,\frac{n}{2}=\frac{40}{2}=20$

$\Rightarrow$ Median $=31$

$Median=l+\left[\frac{\frac{n}{2}-cf}{f}\right]\times h$

$31=30+\left[\frac{20-(11+x)}{y}\right]\times 10$

$1=\frac{9-x}{y}\times 10$

$\therefore$ $10x+y=90$ ---- ( 2 )

Now, subtracting equation ( 1 ) from ( 2 ), we get

$x=8$ and

$y=10$