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

Solution:

Standard deviation of Group A is calculated as follows

Marks	Group A fi	Mid-point xi	yi = (xi - 45)/10	yi²	fiyi	fiyi²
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
	150				- 6	342

Here, 

N = 150, h = 10, A = 45

Mean,

x = A + [∑⁷_{i = 1} f_iy_i] / N × h

= 45 + (- 6) / 150 × 10

= 45 - 0.4

= 44.6

Variance,

(σ ²) = h²/N² [N ∑⁷_{i = 1}(f_iy_i)² - (∑⁷_{i = 1}f_iy_i)²]

= (100)/(22500) [150 × 343 - (- 6)²]

= (1)/(225) × 51264

= 227.84

Standard deviation,

(σ) = √ 227.84

= 15.09

Standard deviation of Group B is calculated as follows.

Marks	Group B fi	Mid-point xi	yi = (xi - 45)/10	yi²	fiyi	fiyi²
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
	150				- 6	366

Mean,

x = A + [∑⁷_{i = 1} f_iy_i]/N × h

= 45 + (- 6)/150 × 10

= 45 - 0.4

= 44.6

Variance,

(σ ²) = h²/N² [N ∑⁷_{i = 1}(f_iy_i)² - (∑⁷_{i = 1}f_iy_i)²]

= (100)/(22500) [150 × 366 - (- 6)²]

= (1)/(225) × 54864

= 243.84

Standard deviation,

(σ) = √ 243.84

= 15.61

Since the mean of both the groups is the same, the group with a greater standard deviation will be more variable.

Thus, group B has more variability in the marks

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NCERT Solutions Class 11 Maths Chapter 15 Exercise 15.3 Question 1

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

Summary:

From the given data, we have observed that group B has more variability in marks