Question

A group of $45$ house owners contributed money towards the 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 the standard deviation.
${\text{A) 416, 20}}{\text{.4}}$
${\text{B) 6}}{\text{.76, 6}}{\text{.06}}$
${\text{C) 36}}{\text{.76, 6}}{\text{.76}}$
${\text{D) None of these}}$

Answer 1

Hint: First, we have to find the mid- value of the class interval.
With the help of that, we need to find the mean and \[\sum {{\text{f}}{{({\text{x - }}\overline {\text{x}} )}^2}} \].
Using the below given formula, we can find the required answer.

Formula used: \[\overline {\text{x}} = {\text{ }}\dfrac{{\sum {{\text{xf}}} }}{{\sum {\text{f}} }}\]
$\sigma _{}^2 = \dfrac{{\sum {{\text{f (x - }}\overline {\text{x}} {)^2}} }}{{\sum {\text{f}} }}$
${{\text{\sigma }}_{}}{\text{ = }}\sqrt {\dfrac{{\sum {{\text{f (x - }}\overline {\text{x}} {{\text{)}}^{\text{2}}}} }}{{\sum {\text{f}} }}} $

Complete step-by-step solution:
In order to find the variance and standard deviation, we have to find \[\sum {{\text{f}}{{({\text{x - }}\overline {\text{x}} )}^2}} \]
First, we will find the mean \[\overline {\text{x}} \].
For that, we have to find the mid value of the intervals.
Mid-value = $\dfrac{{{\text{upper limit + lower limit}}}}{2}$

Amount	Mid-value(X)	${\text{f}}$	${\text{xf}}$
$0 - 20$	$\dfrac{{0 + 20}}{2} = 10$	$2$	$20$
$20 - 40$	$\dfrac{{20 + 40}}{2} = 30$	$7$	$210$
$40 - 60$	$\dfrac{{40 + 60}}{2} = 50$	$12$	$600$
$60 - 80$	$\dfrac{{60 + 80}}{2} = 70$	$19$	$1330$
$80 - 100$	$\dfrac{{80 + 100}}{2} = 90$	$5$	$450$
	Total	$45$	$2610$

Now, we will find the mean to compute \[\sum {{\text{f}}{{({\text{x - }}\overline {\text{x}} )}^2}} \]
\[\overline {\text{x}} = {\text{ }}\dfrac{{\sum {{\text{xf}}} }}{{\sum {\text{f}} }}\]
Putting the values and we get
\[\overline {\text{x}} = {\text{ }}\dfrac{{2610}}{{45}}\]
Let us divide the terms and we get
\[\overline {\text{x}} = {\text{ 58}}\]
Now, we will put into the table format

Amount	Mid-value	${\text{f}}$	${\text{xf}}$	${\text{x - }}\overline {\text{x}} $$({\text{x - 58}})$	\[{({\text{x - }}\overline {\text{x}} )^2}\]	\[{\text{f}}{({\text{x - }}\overline {\text{x}} )^2}\]
$0 - 20$	$10$	$2$	$20$	$ - 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	$45$	$2610$			$18720$

\[\sum {{\text{f}}{{({\text{x - }}\overline {\text{x}} )}^2}} \]= $18720$
Now we have to find variance,
$\sigma _{}^2 = \dfrac{{\sum {{\text{f (x - }}\overline {\text{x}} {)^2}} }}{{\sum {\text{f}} }}$
Putting the values and we get
$\sigma _{}^2 = \dfrac{{18720}}{{45}}$
Let us divide the terms and we get
$\sigma _{}^2 = 416$
Also, we have to find standard deviation,
${{{\sigma }}_{}}{\text{ = }}\sqrt {\dfrac{{\sum {{\text{f (x - }}\overline {\text{x}} {{\text{)}}^{\text{2}}}} }}{{\sum {\text{f}} }}} $
Putting the values and we get
${{{\sigma }}_{}}{\text{ = }}\sqrt {\dfrac{{18720}}{{45}}} $
On dividing the terms and we get
${{{\sigma }}_{}}{\text{ = }}\sqrt {416} $
On squaring the value and we get
${{{\sigma }}_{}}{\text{ = 20}}{\text{.39}}$
Again we conclude the value,
${{{\sigma }}_{}}{\text{ = 20}}{\text{.4}}$
Therefore variance is $416$ and standard deviation is ${\text{20}}{\text{.4}}$

So the required answer is option ${\text{(A)}}$

Note: Another way to check whether the answer is correct when asked in Multiple Choice Question type is that, square of standard deviation is variance.
With that logic we can easily find or check the answer.
Here, we have an option
Option ${\text{A) is 416, 20}}{\text{.4}}$
Here we have to find \[{\left( {{\text{20}}{\text{.4}}} \right)^2}\] and check whether it is equal to \[{\text{ 416}}\]
${\text{20}}{\text{.4 }} \times {\text{20}}{\text{.4 = 416}}$
Option ${\text{B) is 6}}{\text{.76, 6}}{\text{.06}}$
Here, \[6.06 \times 6.06 = 36.72\]
\[ \ne 6.76\]
So cannot be the answer.
Option ${\text{C) is 36}}{\text{.76, 6}}{\text{.76}}$
And \[6.76 \times 6.76 = 45.697\]
\[ \ne 36.76\]
Thus, it cannot be the answer.
You can check in this method if you are not sure about the answer received.