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NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Exercise - 13.1

Q1: Find the mean deviation about the mean for the data   4, 7, 8, 9, 10, 12, 13, 17
Ans: The given data is
4, 7, 8, 9, 10, 12, 13, 17
Mean of the data,  NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The deviations of the respective observations from the mean  NCERT Solutions Class 11 Maths Chapter 13 - Statistics are
–6, – 3, –2, –1, 0, 2, 3, 7
The absolute values of the deviations, i.e. NCERT Solutions Class 11 Maths Chapter 13 - Statistics , are
6, 3, 2, 1, 0, 2, 3, 7
The required mean deviation about the mean is

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q2: Find the mean deviation about the mean for the data  38, 70, 48, 40, 42, 55, 63, 46, 54, 44
Ans: The given data is
38, 70, 48, 40, 42, 55, 63, 46, 54, 44
Mean of the given data,

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The deviations of the respective observations from the mean  NCERT Solutions Class 11 Maths Chapter 13 - Statistics are
–12, 20, –2, –10, –8, 5, 13, –4, 4, –6
The absolute values of the deviations, i.e.  NCERT Solutions Class 11 Maths Chapter 13 - Statistics , are
12, 20, 2, 10, 8, 5, 13, 4, 4, 6
The required mean deviation about the mean is

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q3: Find the mean deviation about the median for the data.
13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
Ans: The given data is
13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
Here, the numbers of observations are 12, which is even.
Arranging the data in ascending order, we obtain
10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The deviations of the respective observations from the median, i.e. NCERT Solutions Class 11 Maths Chapter 13 - Statistics are
–3.5, –2.5, –2.5, –1.5, –0.5, –0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5
The absolute values of the deviations, NCERT Solutions Class 11 Maths Chapter 13 - Statistics , are
3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5
The required mean deviation about the median is

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q4: Find the mean deviation about the median for the data
36, 72, 46, 42, 60, 45, 53, 46, 51, 49
Ans: The given data is
36, 72, 46, 42, 60, 45, 53, 46, 51, 49
Here, the number of observations is 10, which is even.
Arranging the data in ascending order, we obtain
36, 42, 45, 46, 46, 49, 51, 53, 60, 72

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The deviations of the respective observations from the median, i.e. NCERT Solutions Class 11 Maths Chapter 13 - Statistics are
–11.5, –5.5, –2.5, –1.5, –1.5, 1.5, 3.5, 5.5, 12.5, 24.5
The absolute values of the deviations, NCERT Solutions Class 11 Maths Chapter 13 - Statistics , are
11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5
Thus, the required mean deviation about the median is

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q5: Find the mean deviation about the mean for the data.

NCERT Solutions Class 11 Maths Chapter 13 - StatisticsAns:

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q6: Find the mean deviation about the mean for the data

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Ans:

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q7: Find the mean deviation about the median for the data.

NCERT Solutions Class 11 Maths Chapter 13 - StatisticsAns: The given observations are already in ascending order.
Adding a column corresponding to cumulative frequencies of the given data, we obtain the following table.

NCERT Solutions Class 11 Maths Chapter 13 - StatisticsHere, N = 26, which is even.
Median is the mean of 13th and 14th observations. Both of these observations lie in the cumulative frequency 14, for which the corresponding observation is 7.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The absolute values of the deviations from median, i.e. NCERT Solutions Class 11 Maths Chapter 13 - Statistics are

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

 NCERT Solutions Class 11 Maths Chapter 13 - Statisticsand  NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q8: Find the mean deviation about the median for the data

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Ans: The given observations are already in ascending order.
Adding a column corresponding to cumulative frequencies of the given data, we obtain the following table.

NCERT Solutions Class 11 Maths Chapter 13 - StatisticsHere, N = 29, which is odd.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics observation = 15th observation
This observation lies in the cumulative frequency 21, for which the corresponding observation is 30.
∴ Median = 30
The absolute values of the deviations from median, i.e. NCERT Solutions Class 11 Maths Chapter 13 - Statistics are

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

∴  NCERT Solutions Class 11 Maths Chapter 13 - StatisticsNCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q9: Find the mean deviation about the mean for the data.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Ans: The following table is formed

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

 Here, NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q10: Find the mean deviation about the mean for the data

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

 Ans: The following table is formed.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

 Here, NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q11: Find the mean deviation about median for the following data:

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Ans: The following table is formed.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

The class interval containing the NCERT Solutions Class 11 Maths Chapter 13 - Statistics or 25th item is 20 – 30.
Therefore, 20 – 30 is the median class.
It is known that,

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
Here, l = 20, C = 14, f = 14, h = 10, and N = 50
∴ Median = NCERT Solutions Class 11 Maths Chapter 13 - StatisticsNCERT Solutions Class 11 Maths Chapter 13 - Statistics
Thus, mean deviation about the median is given by,

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q12: Calculate the mean deviation about median age for the age distribution of 100 persons given below:

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

 Ans: The given data is not continuous. Therefore, it has to be converted into continuous frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval.
The table is formed as follows.

Age

Number fi

Cumulative frequency (c.f.)

Mid-point xi

|xi – Med.|

fi |xi – Med.|

15.5-20.5

5

5

18

20

100

20.5-25.5

6

11

23

15

90

25.5-30.5

12

23

28

10

120

30.5-35.5

14

37

33

5

70

35.5-40.5

26

63

38

0

0

40.5-45.5

12

75

43

5

60

45.5-50.5

16

91

48

10

160

50.5-55.5

9

100

53

15

135

 

100

 

 

 

735

 The class interval containing the NCERT Solutions Class 11 Maths Chapter 13 - Statistics or 50th item is 35.5 – 40.5.
Therefore, 35.5 – 40.5 is the median class.
It is known that,

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
Here, l = 35.5, C = 37, f = 26, h = 5, and N = 100

NCERT Solutions Class 11 Maths Chapter 13 - Statistics   NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Thus, mean deviation about the median is given by,

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Exercise - 13.2

Q1: Find the mean and variance for the data 6, 7, 10, 12, 13, 4, 8, 12
Ans: 6, 7, 10, 12, 13, 4, 8, 12
Mean,  NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The following table is obtained.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q2: Find the mean and variance for the first n natural numbers
Ans: The mean of first n natural numbers is calculated as follows.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q3: Find the mean and variance for the first 10 multiples of 3
Ans: The first 10 multiples of 3 are
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Here, number of observations, n = 10

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The following table is obtained.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q4: Find the mean and variance for the data

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Ans: The data is obtained in tabular form as follows.

xi

f i

fixi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

6

2

12

–13

169

338

10

4

40

–9

81

324

14

7

98

–5

25

175

18

12

216

–1

1

12

24

8

192

5

25

200

28

4

112

9

81

324

30

3

90

11

121

363

 

40

760

  

1736

Here, N = 40,  NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q5: Find the mean and variance for the data

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

 Ans: The data is obtained in tabular form as follows.

xi

f i

fixi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

92

3

276

–8

64

192

93

2

186

–7

49

98

97

3

291

–3

9

27

98

2

196

–2

4

8

102

6

612

2

4

24

104

3

312

4

16

48

109

3

327

9

81

243

 

22

2200

  

640

 Here, N = 22,  NCERT Solutions Class 11 Maths Chapter 13 - Statistics 

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q6: Find the mean and standard deviation using short-cut method.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Ans: The data is obtained in tabular form as follows.

xi

fi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

yi2

fiyi

fiyi2

60

2

–4

16

–8

32

61

1

–3

9

–3

9

62

12

–2

4

–24

48

63

29

–1

1

–29

29

64

25

0

0

0

0

65

12

1

1

12

12

66

10

2

4

20

40

67

4

3

9

12

36

68

5

4

16

20

80

 

100

220

 

0

286

Mean,  NCERT Solutions Class 11 Maths Chapter 13 - StatisticsNCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q7: Find the mean and variance for the following frequency distribution.

Classes

0-30

30-60

60-90

90-120

120-150

150-180

180-210

Frequencies

2

3

5

10

3

5

2

Ans: 

Class

Frequency fi

Mid-point xi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

yi2

fiyi

fiyi2

0-30

2

15

–3

9

–6

18

30-60

3

45

–2

4

–6

12

60-90

5

75

–1

1

–5

5

90-120

10

105

0

0

0

0

120-150

3

135

1

1

3

3

150-180

5

165

2

4

10

20

180-210

2

195

3

9

6

18

 

30

   

2

76

Mean,  NCERT Solutions Class 11 Maths Chapter 13 - StatisticsNCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q8: Find the mean and variance for the following frequency distribution.

Classes

0-10

10-20

20-30

30-40

40-50

Frequencies

5

8

15

16

6

Ans:

Class

Frequency fi

Mid-point xi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

yi2

fiyi

fiyi2

0-10

5

5

–2

4

–10

20

10-20

8

15

–1

1

–8

8

20-30

15

25

0

0

0

0

30-40

16

35

1

1

16

16

40-50

6

45

2

4

12

24

 

50

   

10

68

 Mean,  NCERT Solutions Class 11 Maths Chapter 13 - StatisticsNCERT Solutions Class 11 Maths Chapter 13 - Statistics

 

NCERT Solutions Class 11 Maths Chapter 13 - Statistics


Q9: Find the mean, variance and standard deviation using short-cut method

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Ans:

Class Interval

Frequency fi

Mid-point xi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

yi2

fiyi

fiyi2

70-75

3

72.5

–4

16

–12

48

75-80

4

77.5

–3

9

–12

36

80-85

7

82.5

–2

4

–14

28

85-90

7

87.5

–1

1

–7

7

90-95

15

92.5

0

0

0

0

95-100

9

97.5

1

1

9

9

100-105

6

102.5

2

4

12

24

105-110

6

107.5

3

9

18

54

110-115

3

112.5

4

16

12

48

 

60

   

6

254

Mean,  NCERT Solutions Class 11 Maths Chapter 13 - StatisticsNCERT Solutions Class 11 Maths Chapter 13 - Statistics

 

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q10: The diameters of circles (in mm) drawn in a design are given below:

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Ans:

Class Interval

Frequency fi

Mid-point xi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

fi2

fiyi

fiyi2

32.5-36.5

15

34.5

–2

4

–30

60

36.5-40.5

17

38.5

–1

1

–17

17

40.5-44.5

21

42.5

0

0

0

0

44.5-48.5

22

46.5

1

1

22

22

48.5-52.5

25

50.5

2

4

50

100

 

100

 

 

 

25

199

Here, N = 100, h = 4
Let the assumed mean, A, be 42.5.

Mean,  NCERT Solutions Class 11 Maths Chapter 13 - StatisticsNCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics


Old NCERT Questions

Q1: 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

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

Marks

Group A fi

Mid-point xi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

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, h = 10, N = 150, A = 45

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

The standard deviation of group B is calculated as follows.

Marks

Group B

fi

Mid-point

xi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

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

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

 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.

Q2: 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

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

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The following table is obtained corresponding to shares X.

xi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

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

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The prices of share Y are
108, 107, 105, 105, 106, 107, 104, 103, 104, 101

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The following table is obtained corresponding to shares Y.

yi

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

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

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
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.

Q3: 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

  • Which firm A or B pays larger amount as monthly wages?
  • Which firm, A or B, shows greater variability in individual wages?

Ans: (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 NCERT Solutions Class 11 Maths Chapter 13 - Statistics  = 100
∴ Standard deviation of the distribution of wages in firm
A ((σ1) =  NCERT Solutions Class 11 Maths Chapter 13 - Statistics
Variance of the distribution of wages in firm  NCERT Solutions Class 11 Maths Chapter 13 - Statistics = 121
∴ Standard deviation of the distribution of wages in firm  NCERT Solutions Class 11 Maths Chapter 13 - Statistics
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.

Q4: The following is the record of goals scored by team A in a football session:

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

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?
Ans: The mean and the standard deviation of goals scored by team A are calculated as follows.

No. of goals scored

No. of matches

fixi

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

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Thus, the mean of both the teams is same.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

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.

Q5: The sum and sum of squares corresponding to length x (in cm) and weight (in gm) of 50 plant products are given below:

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Which is more varying, the length or weight?
Ans: NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Here, N = 50

∴ Mean,  NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Mean,  NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

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

Q6: The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
Ans: Let the remaining two observations be x and y.

Therefore, the observations are 6, 7, 10, 12, 12, 13, x, y.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
From (1), we obtain
x2 +  y2 + 2xy = 144 …(3)
From (2) and (3), we obtain
2xy = 64 … (4)
Subtracting (4) from (2), we obtain
x2 +  y2 – 2xy = 80 – 64 = 16
xy = ± 4 … (5)
Therefore, from (1) and (5), we obtain
x = 8 and y = 4, when xy = 4
x = 4 and y = 8, when xy = –4
Thus, the remaining observations are 4 and 8.

Q7: The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12 and 14. Find the remaining two observations.
Ans: 
Let the remaining two observations be x and y.

The observations are 2, 4, 10, 12, 14, x, y.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
From (1), we obtain
x2 +  y2 + 2xy = 196 … (3)
From (2) and (3), we obtain
2xy = 196 – 100
⇒ 2xy = 96 … (4)
Subtracting (4) from (2), we obtain
x2 +  y2 – 2xy = 100 – 96
⇒ (xy)2 = 4
xy = ± 2 … (5)
Therefore, from (1) and (5), we obtain
x = 8 and y = 6 when xy = 2
x = 6 and y = 8 when xy = – 2
Thus, the remaining observations are 6 and 8.

Q8: The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.
Ans: Let the observations be x1, x2, x3, x4, x5, and x6.
It is given that mean is 8 and standard deviation is 4.

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
If each observation is multiplied by 3 and the resulting observations are yi, then

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
From (1) and (2), it can be observed that,

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
Substituting the values of xi and  NCERT Solutions Class 11 Maths Chapter 13 - Statistics  in (2), we obtain

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Therefore, variance of new observations =  NCERT Solutions Class 11 Maths Chapter 13 - Statistics
Hence, the standard deviation of new observations is  NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q9: Given that  NCERT Solutions Class 11 Maths Chapter 13 - Statistics is the mean and σ2 is the variance of n observations x1, x2xn. Prove that the mean and variance of the observations ax1, ax2, ax3axn are  NCERT Solutions Class 11 Maths Chapter 13 - Statistics  and a2 σ2, respectively (a ≠ 0).
Ans:
The given n observations are x1, x2xn.

Mean =  NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Variance = σ2

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

If each observation is multiplied by a and the new observations are yi, then

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Therefore, mean of the observations, ax1, ax2axn, is  NCERT Solutions Class 11 Maths Chapter 13 - Statistics .

Substituting the values of xiand  NCERT Solutions Class 11 Maths Chapter 13 - Statistics in (1), we obtain

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Thus, the variance of the observations, ax1, ax2axn, is a2 σ2.

Q10: The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:
(i) If wrong item is omitted.
(ii) If it is replaced by 12.
Ans: (i) Number of observations (n) = 20
Incorrect mean = 10
Incorrect standard deviation = 2

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

That is, incorrect sum of observations = 200
Correct sum of observations = 200 – 8 = 192
∴ Correct mean  NCERT Solutions Class 11 Maths Chapter 13 - Statistics   NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

(ii) When 8 is replaced by 12,
Incorrect sum of observations = 200
∴ Correct sum of observations = 200 – 8 + 12 = 204

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Q11: The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below:

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

Which of the three subjects shows the highest variability in marks and which shows the lowest?
Ans: Standard deviation of Mathematics = 12
Standard deviation of Physics = 15
Standard deviation of Chemistry = 20
The coefficient of variation (C.V.) is given by  NCERT Solutions Class 11 Maths Chapter 13 - Statistics .

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
The subject with greater C.V. is more variable than others.
Therefore, the highest variability in marks is in Chemistry and the lowest variability in marks is in Mathematics.

Q12: The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.
Ans: Number of observations (n) = 100
Incorrect mean  NCERT Solutions Class 11 Maths Chapter 13 - Statistics
Incorrect standard deviation  NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
∴ Incorrect sum of observations = 2000
⇒ Correct sum of observations = 2000 – 21 – 21 – 18 = 2000 – 60 = 1940

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
Standered deviation

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics
Correct Standered deviation

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

NCERT Solutions Class 11 Maths Chapter 13 - Statistics

The document NCERT Solutions Class 11 Maths Chapter 13 - Statistics is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on NCERT Solutions Class 11 Maths Chapter 13 - Statistics

1. What are the different types of statistics?
Ans. Descriptive statistics and inferential statistics are the two main types of statistics. Descriptive statistics involve organizing, summarizing, and presenting data, while inferential statistics involve making predictions or inferences about a population based on a sample.
2. How is central tendency calculated in statistics?
Ans. Central tendency in statistics is typically measured using the mean, median, and mode. The mean is calculated by adding up all the values and dividing by the number of values. The median is the middle value when the data is arranged in order, and the mode is the most frequently occurring value.
3. What is the importance of statistics in research?
Ans. Statistics plays a crucial role in research by helping researchers analyze and interpret data. It allows researchers to draw meaningful conclusions, make predictions, and test hypotheses based on the data collected.
4. How can outliers affect statistical analysis?
Ans. Outliers are data points that significantly differ from the rest of the data. They can skew the results of statistical analysis, leading to inaccurate conclusions. It is important to identify and address outliers to ensure the reliability of the analysis.
5. What are the different measures of variability in statistics?
Ans. The range, variance, and standard deviation are common measures of variability in statistics. The range is the difference between the highest and lowest values, while variance and standard deviation measure the spread or dispersion of data around the mean.
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