The median of the following observations $\;$ $9$, $\;$ $10$, $\;$ $12$, $\;$ $\left(x - 4\right)$, $\;$ $\left(x + 2\right)$, $\;$ $\left(x + 7\right)$, $\;$ $30$, $\;$ $36$, $\;$ $45$ $\;$ arranged in ascending order is $22$.
Find the value of $x$ and hence find the mean of the observations.
Number of observations $= n = 9 $ $\;$ [odd number of observations]
Median $= \left(\dfrac{n + 1}{2}\right)^{th}$ term $= \left(\dfrac{9 + 1}{2}\right)^{th}$ term $= 5^{th}$ term $= \left(x + 2\right)$
Given: $\;$ Median $= 22$
i.e. $\;$ $x + 2 = 22$ $\implies$ $x = 20$
$\therefore \;$ The given observations are:
$9, \; 10, \; 12, \; 16, \; 22, \; 27, \; 30, \; 36, \; 45$
Mean of observations $= \dfrac{\Sigma x}{n}$
i.e. $\;$ Mean $= \dfrac{9 + 10 + 12 + 16 + 22 + 27 + 30 + 36 + 45}{9} = \dfrac{207}{9} = 23$