Question

Class	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$
Frequency	$7$	$12$	$x$	$28$	$y$	$9$

Mode of above data is $33\frac{1}{3}$ and total frequency is $100$. Find $x$ and $y$.

Answer 1

$\sum f_i=100$ [ Given ]

$\therefore$ $56+x+y=100$

$\therefore$ $x=44-y$ ----- ( 1 )

Modal class $=30-40$

So we have, $l=30,f_0=x,f_1=28,f_2=y,h=10$

$\Rightarrow$ Mode $=33\frac{1}{3}$ [ Given ]

$\Rightarrow$ $l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h=\frac{100}{3}$

$\Rightarrow$ $30+\frac{28-x}{2\times 28-x-y}\times 10=\frac{100}{3}$

$\Rightarrow$ $30+\frac{28-(44-y)}{56-(44-y)-y}\times 10=33.33$ [ From ( 1 ) ]

$\Rightarrow$ $\frac{-16+y}{12}\times 10=3.33$

$\Rightarrow$ $(-16+y)\times 5=3.33\times 6$

$\Rightarrow$ $-80+5y=19.98$

$\Rightarrow$ $5y=99.98$

$\Rightarrow$ $y=19.99\approx 20$

$\therefore$ $y=20$

Substituting $y=20$ in equation ( 1 ),

$\Rightarrow$ $x=44-20=24$

$\therefore$ $x=24$ and $y=20$