The data given below shows the marks of 12 students in a test, arranged in ascending order :

2, 3, 3, 3, 4, x, x + 2, 8, p, q, 8, 9.

If the given value of the median and mode is 6 and 8 respectively, then find the values of x, p, q.

By formula,

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

Substituting values we get :

$\Rightarrow 6 = \dfrac{\dfrac{12}{2}\text{ th term} + \Big(\dfrac{12}{2} + 1\Big)\text{ th term}}{2} \\[1em] \Rightarrow 6 \times 2 = \text{6th term + 7th term} \\[1em] \Rightarrow 12 = x + x + 2 \\[1em] \Rightarrow 12 = 2x + 2 \\[1em] \Rightarrow 12 - 2 = 2x \\[1em] \Rightarrow 2x = 10 \\[1em] \Rightarrow x = \dfrac{10}{2} = 5.$

Numbers :

2, 3, 3, 3, 4, 5, 7, 8, p, q, 8, 9.

Since, mode = 8, it means 8 occurs for the most times in the series.

Since, 3 occurs 3 times in the series,

∴ 8 must occur for atleast 4 times in order to be the mode.

∴ p = q = 8.

Hence, x = 5, p = 8 and q = 8.