Question

Find the mean deviation about mean and median for the following data.

$CI$	$1-5$	$6-10$	$11-15$	$16-20$	$21-25$
$f$	$2$	$9$	$5$	$4$	$10$

Answer 1

$CI$	$f$	$cf$	$x$	$xf$	$|x-\overline{x}|$	$f|x-\overline{x}|$	$|x-M|$	$f|x-M|$
$0.5-5.5$	$2$	$2$	$3$	$6$	$11.83$	$23.66$	$11.5$	$23$
$0.5-10.5$	$9$	$11$	$8$	$72$	$6.83$	$61.47$	$6.5$	$58.5$
$10.5-15.5$	$5$	$16$	$13$	$65$	$1.83$	$9.15$	$1.5$	$7.5$
$15.5-20.5$	$4$	$20$	$18$	$72$	$3.17$	$12.68$	$3.5$	$14$
$20.5-25.5$	$10$	$30$	$23$	$230$	$8.17$	$81.7$	$8.5$	$85$
	$\sum f=30$			$\sum xf=445$		$\sum f|x-\overline{x}|=188.66$		$\sum f|x-M|=$ $188$

$\Rightarrow$ $Mean\left(\overline{x}\right)=\frac{\sum xf}{\sum f}=\frac{445}{30}=14.83$

$\Rightarrow$ Mean deviation about mean $=\frac{\sum f|x-\overline{x}|}{\sum f}=\frac{188.66}{30}=6.29$

$\Rightarrow$ Here, $N=\sum f=30$

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

$\therefore$ Median class $=10.5-15.5$

$\Rightarrow$ $L=10.5,f=5,cf=11$ and $h=15.5-10.5=5$

$Median\left(M\right)=L+\frac{\frac{N}{2}-cf}{f}\times h$

$=10.5+\frac{15-11}{5}\times 5$

$=10.5+4$

$\therefore$ $Median\left(M\right)=14.5$

$\therefore$ Mean deviation about median $=\frac{\sum f|x-M|}{\sum f}=\frac{188}{30}=6.27$