In a class of 40 students, marks obtained by the students in a class test (out of 10) are given below :

Marks	Number of students
1	1
2	2
3	3
4	3
5	6
6	10
7	5
8	4
9	3
10	3

Calculate the following for the given distribution :

(i) Median

(ii) Mode

The variates are already in ascending order. We construct the cumulative frequency table as under:

Total number of observations = 40, which is even.

(i) By formula,

Median = $\dfrac{\dfrac{\text{n}}{2} \text{ th observation} + \Big(\dfrac{\text{n}}{2} + 1\Big) \text{ th observation}}{2}$

$= \dfrac{\dfrac{40}{2} \text{ th observation} + \Big(\dfrac{40}{2} + 1\Big) \text{ th observation}}{2} \\[1em] = \dfrac{20 \text{ th observation} + \Big(20 + 1\Big) \text{ th observation}}{2} \\[1em] = \dfrac{20 \text{ th observation} + 21 \text{ st observation}}{2} \\[1em]$

All observations from 16th to 25th are equal, each = 6

Then,

Median = $\dfrac{6 + 6}{2} = \dfrac{12}{2}$ = 6.

Hence, median = 6.

(ii) As the variate 6 has maximum frequency 10, so mode = 6.

Hence, mode = 6.