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14. STATISTICS

CBSE Class 10 Chapter 14 STATISTICS

Chapter 14. STATISTICS

Chapter 14 . Statistics

Exercise 14.1 complete solution

Exercise 14.2 complete solution

Exercise 14.3 complete solution

1. Class mark

2. There is a empirical relationship between the three measures of central tendency :
       3 Median = Mode + 2 Mean

3. (a) If  is odd, then the median   observation and,

 (b) If n is even, then the median observations.

4. The mean for grouped data can be found by :
(i) the direct method :

Mean

(ii) the assume mean method :

Mean

(iii) the step deviation method :

Mean

with the assumption that the frequency of a class is centred at its mid-point, called its class mark.
5. The mode for grouped data can be found by using the formula:

Mode

 where  lower limit of the modal class,
   size of the class interval (assuming all class sizes to be equal),
   frequency of the modal class,
   frequency of the class preceding the modal class,
   frequency of the class succeeding the modal class.

6. The median for grouped data is formed by using the formula:

Median

 where  lower limit of median class,
  number of observations,
  cumulative frequency of class preceding the median class,
   frequency of median class,
   class size (assuming class size to be equal)

 Class 10 Maths Chapter 14. STATISTICS Exercise 14.1 Solutions

1. A survey was conducted by a group of students as apart of their environment  awareness  program , in which they collected the following data regarding the number of plants in 20 house in a locality . Find the mean number of plants per house .

Number of plants

  0 – 2

   2 – 4

  4 – 6 

  6 – 8

  8 – 10

  10 – 12

  12 – 14

Number of houses

    1

    2

    1

    5

     6

     2

  3

Which method did you use for finding the mean , and why ?   

Solution: We construct the table ,

Numbers of plants

Class mark         

Number of houses

         

      0 – 2

      2 – 4

      4 – 6

      6 – 8

     8 – 10

    10 – 12

    12 – 14 

         1

        3

        5

        7

        9

       11

       13

        1

        2

        1

        5

        6

        2

        3

          1

           6

          5

        35

        54

        22

        39

      Total

 

    

   

      Mean

Because, class mark and frequency are sufficiently small . So, the direct method is an appropriate choice .

2. Consider the following distribution of daily wages of 50 workers of a factory .

Daily wages (in Rs)

100 – 120 

120 – 140

140 – 160

160 – 180

180 – 200

Number of workers

12

14

8

6

10

 Find the mean daily wages of the workers of the factory by using an appropriate method .

Solution:  We construct the distribution table ,

Daily wages

(in Rs)

         

No. of workers           

      

100 – 120

120 – 140

140 – 160

160 – 180

180 – 200 

110

130

150

170

190

 – 40

 – 20

0

20

40  

12

14

8

6

10

 – 480

 – 280

0

120

400 

Total

 

 

      

   

Using the assumed mean method ,

   Mean

    

   

  

  Therefore, the wages of the workers of the factory is Rs 145.20 .

3. The following distribution shows the daily pocket allowance of children of a locality . The mean pocket allowance is Rs. 18 . Find the missing frequency  .

Daily pocket allowance (in Rs)

Number of children

       11 – 13

       13 – 15

       15 – 17

       17 – 19

       19  - 21

       21 – 23

       23 – 25

              7

              6

              9

             13

             

              5

              4

Solution:  We construct the table ,

Daily pocket allowance (in Rs)

     

Number of children 

     

       11 – 13

      13 – 15

      15 – 17

      17 – 19

      19  - 21

      21 – 23

      23 – 25

     12

     14

    16

    18

    20

    22

    24

      7

      6

      9

     13

     

      5

      4

       84

       84

      144

       234

        

        110

          96

Total

 

      

   

  Mean

 

 

 

 

 

Therefore, the missing frequency is 20 .

4. Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were recorded and summarized as follows . Find the mean heart beats per minute for these women , choosing a suitable method .

Number of heart beats per minute

Number of women

                 65 – 68

                 68 – 71

                 71 – 74

                 74 – 77

                 77 – 80

                 80 – 83

                 83 – 86

            2

            4

            3

            8

            7

           4

           2

Solution: We construct the table ,

Number of heart beats per minute

     

 

Number of women 

          

65 – 68

68 – 71

71 – 74

74 – 77

77 – 80

80 – 83

83 – 86

66.5

69.5

72.5

75.5

78.5

81.5

84.5

 – 9

 – 6

 – 3

0

3

6

9  

2

4

3

8

7

4

2

 – 18

 – 24

 – 9

0

21

24

18  

Total

 

 

    

    

Using the assumed mean method , we have

  Mean

                    

 Therefore, the heart beats per minute for the women is 75.9 .

5.  In a retail market , fruit vendors were selling mangoes kept in packing boxes . These boxes contained varying number of mangoes . The following was the distribution of mangoes according to the number of boxes .

Number of mangoes

50 – 52

53 – 55

56 – 58

59 – 61

62 – 64

Number of boxes

15

110

135

115

25

Find the number of mangoes kept in a packing box .Which method of finding the mean did you choose ?

Solution: We construct the table :

No. of mangoes

    

   

      

   

50 – 52

53 – 55

56 – 58

59 – 61

62 – 64

51

54

57

60

63

 – 6

 – 3

0

3

6

15

110

135

115

25

 – 90

 – 330

0

345

150 

Total

 

 

400

75

Here ,   ,     and

Using the assumed mean method ,

Mean

              

             

             

               

6. The table below shows the daily expenditure on food of 25 households in a locality .

Daily expenditure (Rs)

100 – 150

150 – 200

200 – 250

250 – 300

300 – 350

Number of households

4

5

12

2

2

Find the mean daily expenditure on food by a suitable method .

Solution: We construct the table , we have

Daily exp. (Rs)

     

   

No. of house

-holds  

     

100 – 150

150 – 200

200 – 250

250 – 300

300 – 350

125

175

225

275

325

 – 2

 – 1

0

1

4

5

12

2

2

 – 8

 – 5

0

2

4

Total

 

 

     

    

     Here ,

Using  the step-deviation method , we have

Mean

    

    

    

Therefore, the daily expenditure on food is Rs 211 .

7. To find out the concentration of   in the air (in parts per million, i.e., ppm) , the data was collected for 30 localities in a certain city and is presented below :

Concentration of  (in ppm)

Frequency

0.00 – 0.04

0.04 – 0.08

0.08 – 0.12

0.12 – 0.16

0.16 – 0.20

0.20 – 0.24

4

9

9

2

1

2

Find the mean concentration of  , in the air .

Solution : We construct the table :

Concentration of

     

   

   

0.00 – 0.04

0.04 – 0.08

0.08 – 0.12

0.12 – 0.16

0.16 – 0.20

0.20 – 0.24

0.02

0.06

0.10

0.14

0.18

0.22

4

9

9

2

1

2

0.08

0.54

0.90

0.28

0.18

0.44

Total

 

27

2.42

Using the direct method , we have

Mean

Therefore, the concentration of  in the air is 0.09 ppm .

8. A class teacher has the following absentee record of 40 students of a class for the whole term . Find the mean number of days a student was absent .

Number of days

Number of students

0 – 6

6 – 10

10 – 14

14 – 20

20 – 28

28 – 38

38 – 40

              11

              10

               7

               4

               4

               3

               1

Solution: We construct the table :

No. of days

   

    

   

0 – 6

6 – 10

10 – 14

14 – 20

20 – 28

28 – 38

38 – 40   

3

8

12

17

24

33

39

11

10

7

4

4

3

1

33

80

84

68

96

99

39

Total

 

40

499

Using the direct method , we have

Mean

     

Therefore, the number of days is 12.475 .  

9. The following table gives the literacy rate (in percentage) of 35 cities . Find the mean literacy rate .

Literacy rate (in %)

45 – 55

55 – 65

65 – 75

75 – 85

85 – 95

Number of cities

3

10

11

8

3

Solution: We construct the table :

Literacy rate

   

  

   

  

45 – 55

55 – 65

65 – 75

75 – 85

85 – 95

50

60

70

80

90

 – 20

 – 10

0

10

20 

3

10

11

8

3

 – 60

 – 100

0

80

60 

Total

 

 

35

 – 20 

Using the assumed mean method ,

Mean

    

Therefore , the mean literacy rate is 69.4%  . 

 Class 10 Maths Chapter 14. STATISTICS Exercise 14.2 Solutions 

1. The following table shows the ages of the patients admitted in a hospital during a year :

Age (in years)

Number of patients

5 – 15

15 – 25

25 – 35

35 – 45

45 – 55

55 – 65

6

11

21

23

14

5

Find the mode and the mean of the data given above . Compare and interpret the two measures of central tendency .

Solution : We construct the table :

Age

Class mark

No. of patients

 

5 – 15

15 – 25

25 – 35

35 – 45

45 – 55

55 – 65

10

20

30

40

50

60

6

11

21

23

14

5

60

220

630

920

700

300

Total

 

80

2830

     Here,  

Using the formula ,

  Mode

            

              years

Using the direct method ,

    Mean years

Therefore, the maximum number of patients admitted in the hospital are of the age 36.82 years , while on an average the age of a patient admitted to the hospital is 35.37 years .

2. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components :

Lifetimes (in hours)

Frequency

0 – 20

20 – 40

40 – 60

60 – 80

80 – 100

100 – 120

10

35

52

61

38

29

Determine the modal lifetimes of the components .

Solution : We construct the table :

Lifetimes (in hours)

Frequency

0 – 20

20 – 40

40 – 60

60 – 80

80 – 100

100 – 120

10

35

52

61

38

29

   Here,  

Using the formula ,

  Mode

  

Therefore, the modal lifetimes of the components is 65.63 (approx.) .

3. The following data gives the distribution of total monthly household expenditure of 200 families of a villages . Find the modal monthly expenditure of the families . Also , find the mean monthly expenditure :

Expenditure (in Rs)

Number of families

1000 – 1500

1500 – 2000

2000 – 2500

2500 – 3000

3000 – 3500

3500 – 4000

4000 – 4500

4500 – 5000

24

40

33

28

30

22

16

7

Solution : We construct the distribution table :

Expenditure

Class mark

  

     

      

1000 – 1500

1500 – 2000

2000 – 2500

2500 – 3000

3000 – 3500

3500 – 4000

4000 – 4500

4500 – 5000

1250

1750

2250

2750

3250

3750

4250

4750

  – 2000

 – 1500

 – 1000

 – 500

0

500

1000

1500   

24

40

33

28

30

22

16

7

 – 48000

 – 60000

 – 33000

 – 14000

             0

    11000

    16000

    10500   

Total

 

 

200

 – 117500

   Here,  

Using the formula ,

Mode

Using direct method :  Here ,

Mean

Therefore, the modal monthly expenditure is Rs 1847.83 and the mean monthly expenditure is Rs 2662.5  .

4. The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India . Find the mode and mean of this data . Interpret the two measures .

Number of students per teacher

Number of states/U.T.

15 – 20

20 – 25

25 – 30

30 – 35

35 – 40

40 – 45

45 – 50

50 – 55

3

8

9

10

3

0

0

2

Solution : We construct the distribution table :

No. of students per teacher

      

    

   

15 – 20

20 – 25

25 – 30

30 – 35

35 – 40

40 – 45

45 – 50

50 – 55

17.5

22.5

27.5

32.5

37.5

42.5

47.5

52.5

3

8

9

10

3

0

0

2

52.5

180

247.5

325

112.5

0

0

105

Total

 

35

1022.5

  Here,  

Using the formula ,

  Mode

Using the direct method ,

    Mean

Therefore, the most states/U.T. have a student teacher ratio of 30.6 and on an average, this ratio is 29.2 .

5. The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches .

Runs scored

Number of batsmen

3000 – 4000

4000 – 5000

5000 – 6000 

6000 – 7000

7000 – 8000

8000 – 9000

9000 – 10000

10000 – 11000

4

18

9

7

6

3

1

1

Find the mode of the data .

Solution : We construct the table :

Runs scored

No. of batsmen

3000 – 4000

4000 – 5000

5000 – 6000

6000 – 7000

7000 – 8000

8000 – 9000

9000 – 10000

10000 – 11000

4

18

9

7

6

3

1

1

Using the formula ,     Here,  

  Mode

 

6. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised  it in the table given below . Find the mode of the data :

Number of cars

Frequency

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

60 – 70

70 – 80  

7

14

13

12

20

11

15

8

Solution : We construct the table :

Number of cars

Frequency

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

60 – 70

70 – 80  

7

14

13

12

20

11

15

8

   Here,  

Using the formula ,

  Mode

 Class 10 Maths Chapter 14. STATISTICS Exercise 14.3 Solutions

1. The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality . Find the median , mean and mode of the data and compare them .

Monthly consumption (in units)

Number of consumers

65 – 85

85 – 105

105 – 125

125 – 145

145 – 165

165 – 185

185 – 205

4

5

13

20

14

8

4

Solution : We the frequency distribution table :

Monthly Con.

   

  

    

C.F.

     

65 – 85

85 – 105

105 – 125

125 – 145

145 – 165

165 – 185

185 – 205

75

95

115

135

155

175

195

 – 60

 – 40

 – 20

0

20

40

60

4

5

13

20

14

8

4

4

9

22

42

56

64

68

 – 240

 – 200

 – 260

0

280

320

240  

Total

 

 

68

 

 140

Here,

This observation lies in the class 125 – 145 . Then,

     

Using the formula ,

Median

              

                units

Using assumed mean method ,

Here,

   Mean

 

   units

Using the formula ,

Here,  

          

 Mode

 units

The three measures are approximately the same in this case .

2. If median of the distribution given below is 28.5 , find the values of  and .

       Class interval

       Frequency

           0 – 10

         10 – 20

         20 – 30

         30 – 40

         40 – 50

         50 – 60

            5

            

         20

         15

            

            5

         Total

        60

Solution : We construct the distribution table :

Class interval

Frequency

       C.F.

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

          5

         

        20

       15

        

        5

        5

         

                  

  

  

Total

60

 

   Given ,  

             

            

The median is 28.5 , which lies in the class 20 – 30 .

Here,  

Using the formula ,

Median  

 

 

 

 

 

Putting the value of  in  , we get

      

    

      

     The value of  and  .

3. A life insurance agent found the following data for distribution of ages of 100 policy holders . Calculate the median age , if policies are given only to persons having age 18 years onwards but less than 60 year .

Age (in years)

Number of policy holders

Below 20

Below 25

Below 30

Below 35

Below 40

Below 45

Below 50

Below 55

Below 60

2

6

24

45

78

89

92

98

100

Solution : We construction the distribution table :

Age

 (in years)

    

Number of policy holders (c.f.)

15 – 20

20 – 25

25 – 30

30 – 35

35 – 40

40 – 45

45 – 50

50 – 55

55 – 60

2

4

18

21

33

11

3

6

2

2

6

24

45

78

89

92

98

100

Here,   

This observation lies in the class 35 – 40 . Then,

    

Using the formula ,

Median  

              

              

              

              

                years

4. The lengths of 40 leaves of a plat are measured correct to the nearest millimeter, and  the data obtained is represented in the following table :

Length (in mm)

Number of leaves

118 – 126

127 – 135

136 – 144

145 – 153

154 – 162

163 – 171

172 – 180

              3

              5

              9

            12

             5

             4

             2

Find the median length of the leaves .  

Solution : We have ,

The lower class limit of 127 – 135 is 127 .

The upper class limit of 118 – 126 is 126 .

  Difference 

New Lower limit of a class

      

New upper limit of a class

 

So, New the class interval are :  

  

We construct the table,

Length (in mm)

Number of leaves

C.F.

117.5 – 126.5

126.5 – 135.5

135.5 – 144.5

144.5 – 153.5

153.5 – 162.5

162.5 – 171.5

171.5 – 180.5

3

5

9

12

5

4

2

3

8

17

29

34

38

40

Here,   

This observation lies in the class 144.5 – 153.5 . Then,

    

Using the formula ,

Median  

              

              

              

              

                m.m.

5. The following table gives the distribution of the life time of 400 neon lamps :

Life time (in hours)

 Number of lamps

1500 – 2000

2000 – 2500

2500 – 3000

3000 – 3500

3500 – 4000

4000 – 4500

4500 – 5000

14

56

60

86

74

62

48

Find the median life time of a lamp .

Solution : We construct the distribution table :

Life time

 (in hours)

 Number of lamps

C.F.

1500 – 2000

2000 – 2500

2500 – 3000

3000 – 3500

3500 – 4000

4000 – 4500

4500 – 5000

14

56

60

86

74

62

48

14

70

130

216

290

352

400

Here,   

This observation lies in the class 3000 – 3500 . Then,

     

Using the formula ,

Median

            

            

            

            

             hours .

6. 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surname

was obtained as follows :

Number of letters

Number of surnames

1 – 4

4 – 7

7 – 10

10 – 13

13 – 16

16 – 19 

6

30

40

16

4

4

Determine the median number of letters in the surnames . Find the mean number of letters in the surname ? Also ,find the modal size of the surnames .

Solution : We construct the table :

Number of letters

      

       

  C.F.

    

1 – 4

4 – 7

7 – 10

10 – 13

13 – 16

16 – 19 

2.5

5.5

8.5

11.5

14.5

17.5

6

30

40

16

4

4

6

36

76

92

96

100

15

165

340

184

58

70

Total

 

100

 

832

Here,   

This observation lies in the class 7 – 10 . Then,

     

Using the formula ,

Median

              

              

              

              

               

Using the direct method,

  Mean 

                

Using the mode formula,

Here,

             

 Mode

            

            

            

            

            

7. The distribution below gives the weights of 30 students of a class . Find the median weight of the students .

Weight (in kg)

Number of students

40 – 45

45 – 50

50 – 55

55 – 60

60 – 65

65 – 70

70 – 75

2

3

8

6

6

3

2

Solution: We construct the distribution table :

Weight (in kg)

Number of students

C.F.

40 – 45

45 – 50

50 – 55

55 – 60

60 – 65

65 – 70

70 – 75

2

3

8

6

6

3

2

2

5

13

19

25

28

30

Here,   

This observation lies in the class 55 – 60 . Then,

   

Using the formula ,

Median  

             

             

             

             

              kg .