NCERT Solutions For Class 11 Maths Chapter 15 Statistics Exercise 15.3

Class 11 Maths NCERT Solutions for Chapter 15 Statistics Exercise 15.3

Statistics Exercise 15.3 Solutions

1. 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

Firstly, the standard deviation of group A is calculated as follows. 

Marks	Group A fi	Mid – point xi		yi2	fi yi	fi yi2
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, h = 10, N = 150, A = 45 

The standard deviation of group B is calculated as follows. 

Marks	Group A fi	Mid – point xi		yi2	fi yi	fi yi2
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

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.

2. From the prices of shares X and Y below, find out which is more stable in value:

X	35	54	52	53	56	58	52	50	51	49
Y	108	107	105	105	106	107	104	103	104	101

Solution

The prices of the shares X are 
35, 54, 52, 53, 56, 58, 52, 50, 51, 49 
Here, the number of observations, N = 10 

The following table is obtained corresponding to shares X. 

xi
35	-16	256
54	3	9
52	1	1
53	2	4
56	5	25
58	7	49
52	1	1
50	-1	1
51	0	0
49	-2	4
		350

The prices of share Y are 
108, 107, 105, 105, 106, 107, 104, 103, 104, 101 

The following table is obtained corresponding to shares Y. 

yi
108	3	9
107	2	4
105	0	0
105	0	0
106	1	1
107	2	4
104	-1	1
103	-2	4
104	-1	1
101	-4	16
		40

C.V. of prices of shares X is greater than the C.V. of prices of shares y.

Thus, the prices of shares Y are more stable than the prices of shares X.

3. An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results::

	Firm A	Firm B
No. of wage earners	586	648
Mean of monthly wages	Rs 5253	Rs 5253
Variance of the distribution of wages	100	121

(i) Which firm A or B pays larger amount as monthly wages?
(ii) Which firm, A or B, shows greater variability in individual wages?

Solution

(i) Monthly wages of firm A = Rs 5253
Number of wage earners in firm A = 586
∴Total amount paid = Rs 5253 × 586
Monthly wages of firm B = Rs 5253
Number of wage earners in firm B = 648
∴Total amount paid = Rs 5253 × 648
Thus, firm B pays the larger amount as monthly wages as the number of wage earners in firm B are more than the number of wage earners in firm A.

(ii) Variance of the distribution of wages in firm A (σ₁²) = 100
∴ Standard deviation of the distribution of wages in firm
A ((σ₁) = √100 = 10
Variance of the distribution of wages in firm B (σ₂²) = 121
∴ Standard deviation of the distribution of wages in firm B (σ₂²) = √121 = 11
The mean of monthly wages of both the firms is same i.e., 5253. Therefore, the firm with greater
standard deviation will have more variability.
Thus, firm B has greater variability in the individual wages.

4. The following is the record of goals scored by team A in a football session:

No. of goals scored	0	1	2	3	4
No. of matches	1	9	7	5	3

For the team B, mean number of goals scored per match was 2 with a standard deviation 1.25 goals. Find which team may be considered more consistent?

Solution

The Mean and the standard deviation of goals scored by team A are calculated as follows. 

No. of goals scored	No. of matches	fi xi	xi2	fixi2
0	1	0	0	0
1	9	9	1	9
2	7	14	4	28
3	5	15	9	45
4	3	12	16	48
	25	50		130

The standard deviation of team B is 1.25 goals.
The average number of goals scored by both the teams is same i.e., 2. Therefore, the team with lower standard deviation will be more consistent.
Thus, team A is more consistent than team B.

5. The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below:

Which is more varying, the length or weight?

Solution

Thus, C.V. of weights is greater than the c.v. of lengths. Therefore, weights vary more than the lengths.