Question

The following is the distribution of the height of students of a certain class in a certain city:

Height (in cms):	160-162	163-165	166-168	169-171	172-174
No. of students:	$15$	$118$	$142$	$127$	$18$

Find the average height of the maximum number of students.

Answer 1

Heights	No. of students
$159.5-162.5$	$15$
$162.5-165.5$	$118$
$165.5-168.5$	$142$
$168.5-171.5$	$127$
$171.5-174.5$	$18$

Here, the maximum frequency is $142$, then the corresponding class $165.5-168.5$ is the modal class.

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.

$\Rightarrow$ $l=165.5,h=168.5-165.5=3,f=142,f_1=118,f_2=127$

$\Rightarrow$ $Mode=165.5+\frac{142-118}{2\times 142-118-127}\times 3$

$\Rightarrow$ $Mode=165.5+\frac{72}{39}$

$\Rightarrow$ $Mode=165.5+1.85=167.35cm$