Question

Calculate mode for the following data:

Marks	0-20	20-40	40-60	60-80	80-100
Number of students	8	10	12	6	3

Answer 1

Hint: Mode is the number which appears most often. Since the marks have been grouped, we will first find the midpoints of the given groups. Then we will look at the range of marks in which the most number of students have scored. We will make a table that has the groups, its midpoint and the frequency. The midpoint for this range will be the mode for the given data.

Complete step by step answer:
The mode is the number or observation that appears most often in a collected data. Now, here we have a range of marks and the number of students who have scored in that range. Let us calculate the midpoint for each range. We know that the midpoint for a given range of values is calculated as $\dfrac{\text{upper limit + lower limit}}{\text{2}}$. So, the midpoints of the given range of marks will be as follows,

Marks	Midpoint
0-20	10
20-40	30
40-60	50
60-80	70
80-100	90

Let us make a table with three columns. The first column will have the range of marks, the second column will have the mean of that range of marks and the third column will have the number of students who have scored marks in that range. So the table looks like the following,

Marks	Mean of the range of marks	Number of students
0-20	10	8
20-40	30	10
40-60	50	12
60-80	70	6
80-100	90	3

Now, the most frequent observations lie in the range 40-60. But a range cannot be the value of mode. Therefore, the midpoint of the range is the mode. Hence, the mode for the given data is 50 marks.
Note:
The mode can be a value in the range 40-60. It can be closer to the lower limit or to the upper limit, we cannot say for sure because we don't know the marks for individual students. Hence, for data that is grouped and we don't know the individual observations, the mode is the midpoint of the group that has the most frequency.