The median of the observations 11, 12, 14, (x - 2), (x + 4), (x + 9), 32, 38, 47 arranged in ascending order is 24. Find the value of x and hence find the mean.

Observations arranged in ascending order are :

11, 12, 14, (x - 2), (x + 4), (x + 9), 32, 38, 47.

Here, n (no. of observations) = 9, which is odd.

$\therefore \text{Median} = \dfrac{n + 1}{2} \text{th observation} \\[1em] = \dfrac{9 + 1}{2} \\[1em] = \dfrac{10}{2} \\[1em] = 5 \text{th observation}$

Given, median = 24 = 5th observation = x + 4.

⇒ 24 = x + 4
⇒ x = 24 - 4 = 20.

Putting the value of x in observations :

11, 12, 14, 18, 24, 29, 32, 38, 47.

Sum of terms = 11 + 12 + 14 + 18 + 24 + 29 + 32 + 38 + 47 = 225.

By definition,

$\text{Mean } = \dfrac{\text{Sum of terms}}{\text{No. of terms}} \\[1em] = \dfrac{225}{9} \\[1em] = 25.$

Hence, the value of x = 20 and mean = 25.