Question

Find the variance of the following distribution.

Class interval	$3.5-4.5$	$4.5-5.5$	$5.5-6.5$	$6.5-7.5$	$7.5-8.5$
Frequency	$9$	$14$	$22$	$11$	$17$

Answer 1

Let the assumed mean A be $6$.

Class interval	x mid values	f	$d=x-6$	fd	$f{d}^2$
$3.5-4.5$	$4$	$9$	$-2$	$-18$	$36$
$4.5-5.5$	$5$	$14$	$-1$	$-14$	$14$
$5.5-6.5$	$6$	$22$	$0$	$0$	$0$
$6.5-7.5$	$7$	$11$	$1$	$11$	$11$
$7.5-8.5$	$8$	$17$	$2$	$34$	$68$
		$\sum f=73$		$\sum fd=13$	$\sum f{d}^2=129$

Now variance, ${\sigma }^2=\frac{\sum f{d}^2}{\sum f}-{\left(\frac{\sum fd}{\sum f}\right)}^2$
$=\frac{129}{73}={\left(\frac{13}{73}\right)}^2=\frac{129}{73}-\frac{169}{5329}$
$=\frac{9417-169}{5329}=\frac{9248}{5329}$
Thus, the variance is ${\sigma }^2\simeq 1.74$.