Class 11 Maths NCERT Solutions for 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 (in cm) and weight (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.
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