Question

Find the mean, median and mode of the following data:

Classes:	$0-20$	$20-40$	$40-60$	$60-80$	$80-100$	$100-120$	$120-140$
Frequency:	$6$	$8$	$10$	$12$	$6$	$5$	$3$

Answer 1

$Classes$	$x_i$	$f_i$	$f_i{x}_i$	$c.f$
$0-20$	$10$	$6$	$60$	$6$
$20-40$	$30$	$8$	$240$	$14$
$40-60$	$50$	$10$	$500$	$24$
$60-80$	$70$	$12$	$840$	$36$
$80-100$	$90$	$6$	$540$	$42$
$100-120$	$110$	$5$	$550$	$47$
$120-140$	$130$	$3$	$390$	$50$
		$\sum f_i=50$	$\sum f_i{x}_i=3120$

$\Rightarrow$ $Mean=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{3120}{50}=62.4$

$\Rightarrow$ $N=50$ and $\frac{N}{2}=25$

$\therefore$ Median class $=60-80$

$l=$ lower limit of the modal class

$h=$ size of the class intervals

$f=$ frequency of the modal class

$f_1=$ frequency of the class preceding the modal class

$f_2=$ frequency of the class succeed in the modal class.

$\Rightarrow$ Here, $l=60,f=12,cf=24,h=20$

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

$\Rightarrow$ $Median=60+\frac{25-24}{12}\times 20$

$\Rightarrow$ $Median=60+\frac{20}{12}=61.66$

$\Rightarrow$ $Modalclass=60-80$

$\Rightarrow$ $l=30,f=12,f_1=10,f_2=6,h=20$

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

$\Rightarrow$ $Mode=60+\frac{12-10}{2\times 12-10-6}\times 20$

$\Rightarrow$ $Mode=60+5=65$