Question

A group of $45$ house owners contributed money towards green environment of their street. The amount of money collected is shown in the table below.

Amount	$0-20$	$20-40$	$40-60$	$60-80$	$80-100$
No. of house owners	$2$	$7$	$12$	$19$	$5$

Calculate the variance and standard deviation.

• A. $416,20.4$.
• B. $6.76,6.06$.
• C. $36.76,6.76$.
• D. None of these

Answer 1

The correct option is A $416,20.4$.

$Amount$	$Mid-value$ $x$	$f$	$xf$	$(x-\overline{x})$	$(x-\overline{x}{)}^2$	$f(x-\overline{x}{)}^2$
$0-20$	$10$	$2$	$20$	$-48$	$2304$	$4608$
$20-40$	$30$	$7$	$210$	$-28$	$784$	$5488$
$40-60$	$50$	$12$	$600$	$-8$	$64$	$768$
$60-80$	$70$	$19$	$1330$	$12$	$144$	$2736$
$80-100$	$90$	$5$	$450$	$32$	$1024$	$5120$
$Total$		$45$	$2610$			$18720$

$\sum xf=2610$ and $\sum f=45$

$\Rightarrow$ $\overline{x}=\frac{\sum xf}{\sum f}=\frac{2610}{45}=58$

Now, $\sum f(x-\overline{x}{)}^2=18720$

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

$=\frac{18720}{45}$

$=416$

$\Rightarrow$ Standard deviation $\left(\sigma \right)=\sqrt{\frac{\sum f(x-\overline{x}{)}^2}{\sum f}}$

$=\sqrt{\frac{18720}{45}}$

$=20.39$