Question

Find the variance of the following distribution.

Class interval	$20-24$	$25-29$	$30-34$	$35-39$	$40-44$	$45-49$
Frequency	$15$	$25$	$28$	$12$	$12$	$8$

• A. $54.19$.
• B. $34.19$.
• C. $14.19$.
• D. None of these

Answer 1

The correct option is A $54.19$.

$Class$	$Mid-point$ $\left(x\right)$	$Frequency$ $\left(f\right)$	$xf$	$(x-\overline{x})$	$(x-\overline{x}{)}^2$	$(x-\overline{x}{)}^{2}f$
$20-24$	$22$	$15$	$330$	$-10.25$	$105.0625$	$1575.9375$
$25-29$	$27$	$25$	$675$	$-5.25$	$27.5625$	$689.0625$
$30-34$	$32$	$28$	$896$	$-0.25$	$0.0625$	$1.75$
$35-39$	$37$	$12$	$444$	$4.75$	$22.5625$	$270.75$
$40-44$	$42$	$12$	$504$	$9.75$	$95.0625$	$1140.75$
$45-49$	$47$	$8$	$376$	$14.75$	$217.5625$	$1740.5$
		$\sum f=100$	$\sum xf=3225$			$\sum (x-\overline{x}{)}^{2}f=5418.75$

$\Rightarrow$ $Mean\left(\overline{x}\right)=\frac{\sum xf}{\sum f}=\frac{3225}{100}=32.25$

$\Rightarrow$ $Variance\left({\sigma }^2\right)=\frac{\sum (x-\overline{x}{)}^{2}f}{\sum f}=\frac{5418.75}{100}=54.19$