Question

Calculate the mode of the following data:

Marks	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$	$45 - 50$
Number of students	$4$	$8$	$18$	$30$	$20$	$10$	$5$	$2$

Answer 1

First of all we need to find the modal class which is that interval whose frequency is maximum and the mode is given by the formula
${\text{mode}} = l + \left( {\dfrac{{f - {f_1}}}{{2f - {f_1} - {f_2}}}} \right)h$
Here $l$ is the lower limit of the modal class and $f$ is the frequency of the modal class and ${f_1}$ is the preceding frequency of the modal class interval and similarly the succeeding one is ${f_2}$ and $h$ is the class size.

Complete step by step solution:
So here we are given that which class interval represents the marks earned by the number of students which is termed as the frequency and we need to calculate the mode. So basically mode is the value which appears the maximum time in the set of the observations. For example: if we have the set of the observations like $1, 1, 2, 3, 4, 4, 4, 4$ so here $4$ appears most of the time so it will be the mode of the given set of the observations.
Similarly we are given the set of the observations like

Marks	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$40 - 45$	$45 - 50$
Number of students	$4$	$8$	$18$	$30$	$20$	$10$	$5$	$2$

So here number of maximum items is $30$ and this belongs to the class interval $25 - 30$
So here we can say that mode lies in the interval $25 - 30$
${\text{mode}} = l + \left( {\dfrac{{f - {f_1}}}{{2f - {f_1} - {f_2}}}} \right)h$
Here $l$ is the lower limit of the modal class which is $l = 25$ and $f$ is the frequency of the modal class which is $30$ and ${f_1}$ is the preceding frequency of the modal class interval which is $18$and similarly the succeeding one is ${f_2}$which is $20$ and $h$ is the class size which is $h = 5$
${\text{mode}} = l + \left( {\dfrac{{f - {f_1}}}{{2f - {f_1} - {f_2}}}} \right)h$
${\text{mode}} = 25 + \left( {\dfrac{{30 - 18}}{{60 - 18 - 20}}} \right)5 = 27.73$

So here the mode is $27.73$

Note:
Means are the averages of the given data, median is the central value of the data set and mode is the number which has appeared most of the time in the data set. So these three terms are related by the formula:
${\text{mode}} = 3{\text{median}} - 2{\text{mean}}$
If any two of the above values are given then we can find the third value easily by using this formula.