Question

For the following distribution:

Marks	No. of Students
Less than 20	$4$
Less than 40	$12$
Less than 60	$25$
Less than 80	$56$
Less than 100	$74$
Less than 120	$80$

The Modal class is:
A) 20-40
B) 40-60
C) 60-80
D) 80-100
E) 100-120

Answer 1

Here we will use the formula for the modal class which states that, in a group of frequency distribution, it is very hard to determine the mode by just looking at the values of frequencies. So, we can locate a class with the maximum frequency which will be our modal class. The mode is a value inside the modal class, and below is the formula for the same:
${\text{Mode}} = l + \left( {\dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}} \right) \times h$
Where, $l =$a lower limit of the class, $h =$ size of the class interval ( by assuming all to be equal), ${f_1} =$frequency of the modal class, ${f_0} =$frequency of the class preceding the modal class, and ${f_2} =$ frequency of the class succeeding the modal class.

Complete step-by-step answer:
Step 1: First, for finding the range of the class, we will write the above table given in the question as below:

Marks Range	0-20	20-40	40-60	60-80	80-100	100-120
No. of Students	$4$	$8$	$9$	$35$	$18$	$6$

Step 2: Now, from the above table, it is clear that the maximum class frequency which is the maximum number of students is $35$ and the class corresponding to this is 60-80. So, the modal class will be 60-80.
Answer/Conclusion: Option (C) 60-80 is the correct one.

Note: Students should remember that in any set of data having different frequencies, the modal class will be the one with the highest frequency. Modal means the one that occurs most often.
We can also calculate the mode which is a part of the modal class as shown below:
For finding the mode of the modal class (100-120), we will use the formula ${\text{Mode}} = l + \left( {\dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}} \right) \times h$ where, frequency of the modal class ${f_1} = 80$, frequency of the class preceding the modal class ${f_0} = 74$, frequency of the class succeeding the modal class ${f_2} = 0$ because the modal class is the last observation, size of the class interval $h = 20$, and the lower limit of the modal class $l = 80$.
By substituting these values in the formula ${\text{Mode}} = l + \left( {\dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}} \right) \times h$, we get:
$\Rightarrow {\text{Mode}} = 80 + \left( {\dfrac{{80 - 74}}{{2\left( {80} \right) - 74 - 0}}} \right) \times 20$
Solving inside the brackets, we get:
$\Rightarrow {\text{Mode}} = 80 + \left( {\dfrac{6}{{86}}} \right) \times 20$
Multiplying the term inside the brackets with $20$, we get:
$\Rightarrow {\text{Mode}} = 80 + 1.39$
Finally, adding the RHS side we get the mode of the modal class (100-120):
$\Rightarrow {\text{Mode}} = 81.39$