Question

Compare the modal ages of two groups of students appearing for an entrance test:

Age (in years)	$16-18$	$18-20$	$20-22$	$22-24$	$24-26$
Group A:	$50$	$78$	$46$	$28$	$23$
Group B:	$54$	$89$	$40$	$25$	$17$

Answer 1

We know,

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

$l=$ 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.

For Group $A$

Here the maximum frequency is $78$, then the corresponding class $18-20$ is modal class.

$\Rightarrow$ $l=18,h=20-18=2,f=78,f_1=50,f_2=46$

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

$\Rightarrow$ $Mode=18+\frac{78-50}{2\times 78-50-46}\times 2$

$\Rightarrow$ $Mode=18+\frac{56}{60}$

$\Rightarrow$ $Mode=18+0.93=18.93years$

For Group $B$

Here the maximum frequency is $89$, then the corresponding class $18-20$ is modal class.

$\Rightarrow$ $l=18,h=20-18=2,f=89,f_1=54,f_2=40$

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

$\Rightarrow$ $Mode=18+\frac{89-54}{2\times 89-54-40}\times 2$

$\Rightarrow$ $Mode=18+\frac{70}{84}$

$\Rightarrow$ $Mode=18+0.83=18.83years$

Hence, Modal age for the group $A$ is higher than that for group $B$.