Question

Find the mean deviation about the median for the following data :

$x_i$	3	6	9	12	13	15	21	22
$f_i$	3	4	5	2	4	5	4	3

Answer 1

$x_i$	$f_i$	$c.f$	$|x_i-M|$	$f_i|x_i-M|$
$3$	$3$	$3$	$10$	$30$
$6$	$4$	$7$	$7$	$28$
$9$	$5$	$12$	$4$	$20$
$12$	$2$	$14$	$1$	$2$
$13$	$4$	$18$	$0$	$0$
$15$	$5$	$23$	$2$	$10$
$21$	$4$	$27$	$8$	$32$
$22$	$3$	$30$	$9$	$27$
	$\sum f_i=30$			$\sum f_i|x_i-M|=149$

$\Rightarrow$ $N=30$ and $\frac{N}{2}=15$

$\Rightarrow$ $Median=Meanof{\left(\frac{N}{2}\right)}^{th}term$ and ${(\frac{N}{2}+1)}^{th}term$

$=\frac{(15{)}^{th}term+(16{)}^{th}term}{2}$

$\Rightarrow$ As both observation lie in the $c.f.$ of $18$

$\therefore$ $Median=13$

$\Rightarrow$ Mean deviation about median $=\frac{\sum f_i|x_i-M|}{\sum f_i}$

$=\frac{149}{30}$

$=4.97$