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Question: For the following distribution: Marks less than \({\text{10 20 30 40 50 60}}\) No. of students \...

For the following distribution:
Marks less than 10 20 30 40 50 60{\text{10 20 30 40 50 60}}
No. of students 3 12 27 57 75 803{\text{ 12 27 57 75 80}}
The modal class is:
A. 10−2010 - 20
B. 20−3020 - 30
C. 30−4030 - 40
D. 50−6050 - 60

Explanation

Solution

Modal class is the class interval which has the maximum or the highest frequency. Here frequency is represented by the number of the students and the class interval is represents by the marks intervals which can be written as 0−10,10−20,20−30.........0 - 10,10 - 20,20 - 30.........

Complete step by step solution:
Here we are given the distribution of the number of students and their marks which is as follows:
Marks less than 10 20 30 40 50 60{\text{10 20 30 40 50 60}}
No. of students 3 12 27 57 75 803{\text{ 12 27 57 75 80}}
Now here we are given the intervals in the form of the marks less than but we need to convert it into the interval form. So we can write the less than 10 as 0−10{\text{less than 10 as }}0 - 10 and similarly the other ones.
So in the interval 0−100 - 10 we have only 33 students and similarly as given in the interval 10−2010 - 20 we have total of 12−3=912 - 3 = 9students because the students whose marks are less than 1010 are also included in the students whose marks are less than 2020 and therefore we need to subtract the values to get the desired range frequency.
So for 20−3020 - 30 we have 27−12=1527 - 12 = 15 students.
For 30−4030 - 40 we have 57−27=3057 - 27 = 30 students.
For 40−5040 - 50 we have 75−57=1875 - 57 = 18 students.
For 50−6050 - 60 we have 80−75=580 - 75 = 5 students.
So we have got all the values of the students who have secured the marks in the desired range. Hence we can form the table as follows of the above distribution:
Marks (class interval) 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60{\text{0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60}}
No. of students 3 9 15 30 18 53{\text{ 9 15 30 18 5}}
Hence here we can see that the maximum frequency is 3030 which is of the class interval 30−4030 - 40
Hence 30−4030 - 40 is the modal class

Hence option C is the correct option.

Note:
If we are given the modal class as (a−b)(a - b)then its mode lies between a and ba{\text{ and }}b and for example: If we are given the modal class as (30−40)(30 - 40) then the value of mode can be any number between 30 and 4030{\text{ and }}40 and therefore mode cannot be less than 3030 and not greater than 4040