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.

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

Total number of observations = 40, which is even.

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

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

Hence, median

$= \dfrac{6 + 6}{2} \\[1em] = \dfrac{12}{2} \\[1em] = 6.$

Hence, median = 6.

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

Hence, mode = 6.