Question

From the data given below state which group is more variable $A$ or $B$

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

Answer 1

Class	$f_i$	$x_i$	$u_i=\frac{x_i-A}{h}$	$u_i^2$	$f_i{u}_i$	$f_i{u}_i^2$
10-20	9	15	-3	9	-27	81
20-30	17	25	-2	4	-34	68
30-40	32	35	-1	1	-32	32
40-50	33	45	0	0	0	0
50-60	40	55	1	1	40	40
60-70	10	65	2	4	20	40
70-80	9	75	3	9	27	81
Total	150				-6	342

Here $h=10,N=150$

Let assumed mean $A=45$
Mean $\overline{x}=A+\frac{{\sum }_{i=1}^7{f}_i{u}_i}{N}\times h$

$=45+\frac{(-6)\times 10}{150}=45-0.4=44.6$

Variance $\left({\sigma }_1^2\right)=\frac{h^2}{N^2}[N\sum _{i=l}^7{f}_i{u}_i^2-{\left(\sum _{i=l}^7{f}_i{u}_i\right)}^2]$
$=\frac{100}{22500}[150\times 342-{\left(6\right)}^2]$
$=\frac{1}{225}\left(51264\right)$
$=227.84$
$\therefore$ Standard deviation $\left({\sigma }_1\right)=\sqrt{227.84}=15.09$

Group B:

Class	$f_i$	$x_i$	$u_i=\frac{x_i-A}{h}$	$u_i^2$	$f_i{u}_i$	$f_i{u}_i^2$
10-20	10	15	-3	9	-30	90
20-30	20	25	-2	4	-40	80
30-40	30	35	-1	1	-30	30
40-50	25	45	0	0	0	0
50-60	43	55	1	1	43	43
60-70	15	65	2	4	30	60
70-80	7	75	3	9	21	63
Total	150				-6	366

Here $h=10,N=150$

Let assumed mean $A=45$
Mean $\overline{x}=A+\frac{{\sum }_{i=1}^7{f}_i{u}_i}{N}\times h$

$=45+\frac{(-6)\times 10}{150}=45-0.4=44.6$

Variance $\left({\sigma }_2^2\right)=\frac{h^2}{N^2}[N\sum _{i=l}^7{f}_i{u}_i^2-{\left(\sum _{i=l}^7{f}_i{u}_i\right)}^2]$
$=\frac{100}{22500}[150\times 366-{\left(6\right)}^2]$
$=\frac{1}{225}\left(54864\right)$
$=243.84$
$\therefore$ Standard deviation $\left({\sigma }_2\right)=\sqrt{243.84}=15.61$

Since, the mean of both the groups is same the group with greater standard deviation will be more variable.
Thus group $B$ has more variability in the marks.