The median of the observations $11, \; 12, \; 14, \; \left(x - 2\right), \; \left(x + 4\right), \; \left(x + 9\right), \; 32, \; 38, \; 47$, arranged in ascending order, is $24$. Find the value of $x$ and hence find the mean.
Given observations: $\;$ $11, \; 12, \; 14, \; \left(x - 2\right), \; \left(x + 4\right), \; \left(x + 9\right), \; 32, \; 38, \; 47$
Number of observations $= n = 9 $ (odd number of observations)
Median value $= \left(\dfrac{n + 1}{2}\right)^{th}$ term
$\therefore \;$ Median $= \left(\dfrac{9 + 1}{2}\right)^{th}$ term $= 5^{th}$ term $= x + 4$
Given: $\;$ Median $= 24$
$\implies$ $x + 4 = 24$ $\implies$ $x = 20$
$\therefore \;$ The given observations are: $\;$ $11, \; 12, \; 14, \; 18, \; 24, \; 29, \; 32, \; 38, \; 47$
Mean $= \dfrac{11 + 12 + 14 + 18 + 24 + 29 + 32 + 38 + 47}{9} = \dfrac{225}{9} = 25$