(Each question 4 marks)

Question

From the following data, 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

Vidyadip · Accepted Answer

For group A,
Let assumed mean a = 45

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

Here,
$\sum f_i$ = N = 150, $\sum f_{i} u_{i}$ = -6, and $\sum f_{i} u_{i}^{2}$ = 342, b = 10
$\therefore$ $\overline x$ = a + $\frac{\sum f_{i} u_{i}}{\sum f_{i}} \times$ b = 45 + $\frac{(-6)}{150}$ $\times$ 10 = 45 - 0.4 = 44.6
Variance, $\sigma_{A}=\frac{b^{2}}{N^{2}}\left[N \sum f_{i} u_{i}^{2}-\left(\sum f_{i} u_{i}\right)^{2}\right]$
= $\frac{100}{22500}$[150 $\times 342 - (-6)^2$]
= $\frac{1}{225}$ (51300 - 36) = $\frac{51264}{225}$ = 227.84
$\therefore$ Standard deviation, $\sigma _A$ = $\sqrt{227.84}$ = 15.09
Coefficient of variation (CV) = $\frac{\sigma_{{A}}}{\overline{x}} \times$ 100 = $\frac{15.09}{44.6}$ $\times$ 100 = 33.83
For group B,
Let the assumed mean a = 45

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

$\sum F_i$ = N = 150,
$\sum f_{i} u_{i}$ = -6 and $\sum f_{i} u_{i}^{2}$ = 366
$\therefore$ $\overline{x}=a+\frac{\sum f_{i} u_{i}}{\sum f_{i}} \times b$
= 45 + $\frac{(-6)}{150}$ $\times$ 10 = 45 - 0.4 = 44.6
Variance, $\sigma_{B}^{2}=\frac{b^{2}}{N^{2}}\left[N \sum f_{i} u_{i}^{2}-\left(\sum f_{i} u_{i}\right)^{2}\right]$
= $\frac{100}{22500}$ [150 $\times 366 - (-6)^2$]
= $\frac{1}{225}$(54900 - 36) = $\frac{1}{225}$ $\times$ 54864 = 243.84
$\therefore$ Standard deviation, $\sigma _ B$ = $\sqrt{243.84}$ = 15.61
$\therefore$ Coefficient of variation (CV) = $\frac{\sigma_{B}}{\overline{x}} \times$ 100 = $\frac{15.61}{44.6}$ $\times$ 100 = 35
Since, CV(B) > CV(A)
So, group B is more variable than group A.

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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

STATISTICS — MATHS STD 11 Science — Question

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