Question

From the data given below state which group is more variable, G₁ or G₂?

Marks	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Group G1	9	17	32	33	40	10	9
Group G2	10	20	30	25	43	15	7

Answer 1

Marks	$f_i$	Midpoint $\left(x_i\right)$	$u_i=\frac{x_i-45}{10}$	$f_i{u}_i$	$f_i{u_i}^2$
10−20	9	15	−3	−27	81
20−30	17	25	−2	−34	68
30−40	32	35	−1	−32	32
40−50	33	45	0	0	0
50−60	40	55	1	40	40
60−70	10	65	2	20	40
70−80	9	75	3	27	81
	N=150			$\sum f_i{u}_i=-6$	$\sum f_i{u_i}^2=342$

$h=5,a=45$
For group 1:
$\overline{X}=a+h\left(\frac{\sum f_i{u}_i}{N}\right)=45+10\left(\frac{-6}{150}\right)=44.6$
$$\begin{gathered} {\sigma }^2=h^2\left[\frac{\sum f_i{u_i}^2}{N}-{\left(\frac{\sum f_i{u}_i}{N}\right)}^2\right]=100\left[\frac{342}{150}-\frac{36}{22500}\right]=227.84 \\ \sigma =\sqrt{227.84}=15.09 \end{gathered}$$

Marks	$f_i$	Midpoint$\left(x_i\right)$	$u_i=\frac{x_i-45}{10}$	$f_i{u}_i$	$f_i{u_i}^2$
10−20	10	15	−3	−30	90
20−30	20	25	−2	−40	80
30−40	30	35	−1	−30	30
40−50	25	45	0	0	0
50−60	43	55	1	43	43
60−70	15	65	2	30	60
70−80	7	75	3	21	63
	$\sum f_i=150$			$\sum f_i{u}_i=-6$	$\sum f_i{u_i}^2=366$

For group 2:
$\overline{X}=a+h\left(\frac{\sum f_i{u}_i}{N}\right)=45+10\left(\frac{-6}{150}\right)=44.6$
$$\begin{gathered} {\sigma }^2=h^2\left[\frac{\sum f_i{u_i}^2}{N}-{\left(\frac{\sum f_i{u}_i}{N}\right)}^2\right]=100\left[\frac{366}{150}-\frac{36}{22500}\right]=243.84 \\ \sigma =\sqrt{243.84}=15.62 \end{gathered}$$
Mean of both the groups are same and SD of group 2 is greater than that of group 1.
So, group 2 will be more variable.