Question

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.

Answer 1

Given: The ordered data is 2, 3, 3, 3, 4, x, x + 2, 8, p, q, 8, 9, with median = 6 and mode = 8.

Step-wise calculation:

1. Count there are 12 observations (even), so median = average of 6th and 7th observations.

6th = x

7th = x + 2

`(x + (x + 2))/2 = 6`

⇒ `(2x + 2)/2 = 6`

⇒ 2x + 2 = 12

⇒ 2x = 10

⇒ x = 5

Then x + 2 = 7.

2. Mode = 8 (unique). Current counts before p, q:

3 appears 3 times (positions 2 and 4).

8 already appears twice (positions 8 and 11). For 8 to be the unique mode, its frequency must exceed 3, i.e., be ≥ 4.

The only places to increase 8’s frequency are p and q (positions 9 and 10).

Because the list is in nondecreasing order and there is an 8 at position 11, p and q cannot be > 8, otherwise the 8 at position 11 would be out of order.

Thus, p = 8 and q = 8.

That gives four 8’s in total, so 8 becomes the unique mode.