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.