Statistics

Find the mean and the median for the following frequency distribution:

$x$	$8$	$9$	$10$	$11$	$12$
$f$	$5$	$4$	$2$	$6$	$3$

$x_i$	$f_i$	$f_i x_i$	Cumulative frequency
$8$	$5$	$40$	$5$
$9$	$4$	$36$	$9$
$10$	$2$	$20$	$11$
$11$	$6$	$66$	$17$
$12$	$3$	$36$	$20$

$\Sigma f_i = N = 20$ (Even), $\;$ $\Sigma f_i x_i = 198$

Mean $= \dfrac{\Sigma f_i x_i}{\Sigma f_i} = \dfrac{198}{20} = 9.9$

$\because$ $N$ is even, median $= \dfrac{\left(\dfrac{N}{2}\right)^{th} \text{term} + \left(\dfrac{N}{2} + 1\right)^{th} \text{term}}{2}$

i.e. $\;$ Median $= \dfrac{\left(\dfrac{20}{2}\right)^{th} \text{term} + \left(\dfrac{20}{2} + 1\right)^{th} \text{term}}{2}$

i.e. $\;$ Median $= \dfrac{10^{th} \; \text{term} + 11^{th} \; \text{term}}{2}$

i.e. $\;$ Median $= \dfrac{10 + 10}{2} = \dfrac{20}{2} = 10$