Find '$x$' from the following marks obtained by $9$ students if the mean mark is equal to the median:
$5, \; 7, \; 9, \; 10, \; x, \; 15, \; 21, \; 26,\; 27$.
The marks are arranged in ascending order.
Number of students $= N = 9$ (Odd)
The marks obtained by the $9$ students are (in ascending order): $\;$ $5, \; 7, \; 9, \; 10, \; x, \; 15, \; 21, \; 26,\; 27$
Median mark $= \left(\dfrac{N + 1}{2}\right)^{th}$ term $= \left(\dfrac{9 + 1}{2}\right)^{th}$ term $= 5^{th}$ term
i.e. $\;$ Median mark $= x$ $\;\;\; \cdots \; (1)$
Mean mark $= \dfrac{\Sigma x_i}{N} = \dfrac{5 + 7 + 9 + 10 + x + 15 + 21 + 26 + 27}{9} = \dfrac{120 + x}{9}$ $\;\;\; \cdots \; (2)$
Given: $\;$ Mean mark $=$ Median mark
$\therefore \;$ We have from equations $(1)$ and $(2)$,
$x = \dfrac{120 + x}{9}$
i.e. $\;$ $9x = 120 + x$
i.e. $\;$ $8x = 120$ $\implies$ $x = 15$