Question

In the class test, $50$ students obtained the marks as follows:

Marks obtained	$0 - 20$	$20 - 40$	$40 - 60$	$60 - 80$	$80 - 100$
Number of students	$4$	$6$	$25$	$10$	$5$

Find the modal class and the median class.

Answer 1

Modal class is that class interval whose frequency is maximum and here the class interval represents the marks of the students and the frequency is denoted by the number of students and for median we need to calculate the value of cumulative frequency.

Complete step by step solution:
As here in the question we are given the marks of the class test of the students which are as follows

Marks obtained	$0 - 20$	$20 - 40$	$40 - 60$	$60 - 80$	$80 - 100$
Number of students	$4$	$6$	$25$	$10$	$5$

And we need to find the modal and the median class of these class marks we are given and the number of students here represent the frequency and marks range as the class intervals.
So for the median class we need to find the cumulative frequency which is the sum of all the preceding frequencies of that interval.
So we can write as

Marks obtained	Number of students(frequency)	Cumulative frequency
$0 - 20$	$4$	$4$
$20 - 40$	$6$	$10$
$40 - 60$	$25$	$35$
$60 - 80$	$10$	$45$
$80 - 100$	$5$	$50$

Here the cumulative frequency of the first interval remains the same as there is no preceding frequency to it but for next ones the preceding frequencies are added to that as shown above.
Here half the total number of students$= 25$
So $25$ is nearer to the $35$ in cumulative frequency. So that class interval is known as the median class whose cumulative frequency is nearer to $25$ which is $35$. So $40 - 60$ is the interval known as the median class and now for the modal class we need to see the maximum frequency and here we can see that $25$ is the maximum frequency of the class interval $40 - 60$. Hence is the modal class also.

Hence median class is $40 - 60$
Modal class is also $40 - 60$.

Note:
If we are given the modal class as $a - b$ then mode is given by the formula:
${\text{mode}} = a + \left( {\dfrac{{f - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}} \right)h$
Here $a$ is the lower limit and $b$ is the upper limit of the modal class.
Here ${f_1}$ is the frequency of the modal class
Here ${f_0},{f_2}$ are the frequencies of the class preceding and exceeding the modal class.
And $h$ is the class size.