Question

Find the mean, mode and median of marks obtained by $20$ students in an examination. The marks are given below.

Marks	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$
No. of students	$1$	$4$	$5$	$8$	$2$

Answer 1

$Marks$	$Mid-value$ $x_i$	$No.offrequency$ $f_i$	$c.f$	$f_i{x}_i$
$0-10$	$5$	$1$	$1$	$5$
$10-20$	$15$	$4$	$5$	$60$
$20-30$	$25$	$5$	$10$	$125$
$30-40$	$35$	$8$	$18$	$280$
$40-50$	$45$	$2$	$20$	$90$
		$\sum f_i=20$		$\sum f_i{x}_i=560$

$\Rightarrow$ $Mean\left(\overline{x}\right)=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{560}{20}=28$

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

So, $l=30,f_0=5,f_1=8,f_2=2$ and $h=10$

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

$=30+\frac{8-5}{2\times 8-5-2}\times 10$

$=30+3.33$

$\therefore$ $Mode=33.33$

$\Rightarrow$ Here, $n=20$, then $\frac{n}{2}=\frac{20}{2}=10$

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

So, $c.f=5,f=5,l=20$ and $h=10$

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

$=20+\frac{10-5}{5}\times 10$

$=20+10$

$\therefore$ $Median=30$