From the data given below state which group is more variable, G_1 or G_2? Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Group G_1 9 17 32 33 40 10 9 Group G_2 10 20 30 25 43 15 7

From the data given below state which group is more variable, G_1…

Question

From the data given below state which group is more variable, $G_1$ or $G_2$?

Marks	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$
Group $G_1$	$9$	$17$	$32$	$33$	$40$	$10$	$9$
Group $G_2$	$10$	$20$	$30$	$25$	$43$	$15$	$7$

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Answer

Let's first find the coefficient of variable for Group $G_1$

CI	f	x	$\text{u}=\frac{\text{x}-\text{A}}{\text{h}}$	fu	$u^2$	$fu^2$
$10-20$	$9$	$15$	$-3$	$-27$	$9$	$81$
$20-30$	$17$	$25$	$-2$	$-34$	$4$	$68$
$30-40$	$32$	$35$	$-1$	$-32$	$1$	$32$
$40-50$	$33$	$45$	$0$	$0$	$0$	$0$
$50-60$	$40$	$55$	$1$	$40$	$1$	$40$
$60-70$	$10$	$65$	$2$	$20$	$4$	$40$
$70-80$	$9$	$75$	$3$	$27$	$9$	$81$
	$150$			$-6$		$342$

Here, $N = 150, A = 45$, $\sum\text{f}_\text{i}\text{u}_\text{i}=-6,\ \sum\text{f}_\text{i}\text{u}_\text{i}^2=342$ and $h = 10$
$\therefore\text{Mean}=\overline{\text{x}}=\text{A+h}\Big(\frac{1}{\text{N}}\sum\text{f}_\text{i}\text{u}_\text{i}\Big)$
$\Rightarrow\overline{\text{x}}=45+10\Big(\frac{-6}{150}\Big)=44.6$
$\text{Var}(\text{X})=\text{h}^2\bigg[\frac{1}{\text{N}}\sum\text{f}_\text{i}\text{u}_\text{i}^2-\Big(\frac{1}{\text{N}}\sum\text{f}_\text{i}\text{u}_\text{i}\Big)^2\bigg]$
$\text{Var}(\text{X})=100\bigg[\frac{342}{150}-\Big(\frac{-6}{150}\Big)^2\bigg]=227.84$
$\therefore\text{S.D.}=\sqrt{\text{Var}(\text{X})}=\sqrt{227.84}=15.09$
Coefficient of variation $=\frac{\text{S.D.}}{\overline{\text{x}}}\times100=\frac{15.09}{44.6}\times100=33.83$
Now, Let's first find the coefficient of variable for Group $G_2$

CI	f	x	$\text{u}=\frac{\text{x}-\text{A}}{\text{h}}$	$fu$	$u^2$	$fu^2$
$10-20$	$10$	$15$	$-3$	$-30$	$9$	$902$
$20-30$	$20$	$25$	$-2$	$-40$	$4$	$80$
$30-40$	$30$	$35$	$-1$	$-30$	$1$	$30$
$40-50$	$25$	$45$	$0$	$0$	$0$	$0$
$50-60$	$43$	$55$	$1$	$43$	$1$	$43$
$60-70$	$15$	$65$	$2$	$30$	$4$	$60$
$70-80$	$7$	$75$	$3$	$21$	$9$	$63$
	$150$			$-6$		$366$

Here, $N = 150, A = 45$, $\sum\text{f}_\text{i}\text{u}_\text{i}=-6,\ \sum\text{f}_\text{i}\text{u}_\text{i}^2=366$ and $h = 10$
$\therefore\text{Mean}=\overline{\text{x}}=\text{A+h}\Big(\frac{1}{\text{N}}\sum\text{f}_\text{i}\text{u}_\text{i}\Big)$
$\Rightarrow\overline{\text{x}}=45+10\Big(\frac{-6}{150}\Big)=44.6$
$\text{Var}(\text{X})=\text{h}^2\bigg[\frac{1}{\text{N}}\sum\text{f}_\text{i}\text{u}_\text{i}^2-\Big(\frac{1}{\text{N}}\sum\text{f}_\text{i}\text{u}_\text{i}\Big)^2\bigg]$
$\text{Var}(\text{X})=100\bigg[\frac{366}{150}-\Big(\frac{-6}{150}\Big)^2\bigg]=243.84$
$\therefore\text{S.D.}=\sqrt{\text{Var}(\text{X})}=\sqrt{227.84}=15.62$
Coefficient of variation $=\frac{\text{S.D.}}{\overline{\text{x}}}\times100=\frac{15.09}{44.6}\times100=35.02$
$\therefore$ Group $G_2$ is more variable.

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