Chapter 23 : Graphical Representation (Histograms and Ogives) - Selina Solutions for Class 10 Maths ICSE

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Chapter 23 - Graphical Representation (Histograms and Ogives) Excercise Ex. 23

Question 1

Draw histogram for the following distributions:

(i)

Class Interval

0-10

10-20

20-30

30-40

40-50

50-60

Frequency

12

20

26

18

10

6

(ii)

Class Interval

10-16

16-22

22-28

28-34

34-40

Frequency

15

23

30

20

16

(iii)

Class Interval

30-39

40-49

50-59

60-69

70-79

Frequency

24

16

09

15

20

(iv)

Class Marks

16

24

32

40

48

56

64

Frequency

8

12

15

18

25

19

10

Solution 1

(i)

Class Interval

Frequency

0-10

12

10-20

20

20-30

26

30-40

18

40-50

10

50-60

06

Steps of construction:

(a) Taking suitable scales, mark class intervals on x-axis and frequency on y-axis.

(b)Construct rectangles with class intervals as bases and corresponding frequencies as heights.

(ii)

Class Interval

Frequency

10-16

15

16-22

23

22-28

30

28-34

20

34-40

16

Steps of construction:

(a) Taking suitable scales, mark class intervals on x-axis and frequency on y-axis.

(b) Construct rectangles with class intervals as bases and corresponding frequencies as heights.

(iii)

Class Interval

(Inclusive form)

Class Interval

(Exclusive Form)

Frequency

30-39

29.5-39.5

24

40-49

39.5-49.5

16

50-59

49.5-59.5

09

60-69

59.5-69.5

15

70-79

69.5-79.5

20

Steps of construction:

(a) Convert the data into exclusive form.

(b) Taking suitable scales, mark class intervals on x-axis and frequency on y-axis.

(c) Construct rectangles with class intervals as bases and corresponding frequencies as heights.

(iv)

Class Marks

Class Intervals

Frequency

16

12-20

08

24

20-28

12

32

28-36

15

40

36-44

18

48

44-52

25

56

52-60

19

64

60-68

10

Steps of construction:

(a) Convert the class marks into class intervals.

(b) Taking suitable scales, mark class intervals on x-axis and frequency on y-axis.

(c) Construct rectangles with class intervals as bases and corresponding frequencies as heights.

Question 2

Draw cumulative frequency curve (ogive) for each of the following distributions:

 

(i)

Class

Interval

10-15

15-20

20-25

25-30

30-45

35-40

Frequency

10

15

17

12

10

08

(ii)

Class Interval

10-19

20-29

30-39

40-49

50-59

Frequency

23

16

15

20

12

Solution 2

(i)

Class Interval

Frequency

10-15

10

15-20

15

20-25

17

25-30

12

30-35

10

35-40

08

 

Steps of construction:

(a) Taking suitable scales, mark class intervals on x-axis and frequency on y-axis.

(b) Construct rectangles with class intervals as bases and corresponding frequencies as heights.

( c) Join the mid-points of the rectangle to obtain the ogive.

(ii)

Class Interval

(Inclusive)

Class Interval

(Exclusive)

Frequency

Cumulative

Frequency

10-19

9.5-19.5

23

23

20-29

19.5-29.5

16

39

30-39

29.5-39.5

15

54

40-49

39.5-49.5

20

74

50-59

49.5-59.5

12

86

   

Total

86

 

Steps of construction:

(a) Convert the data into exclusive form.

(b) Taking suitable scales, mark class intervals on x-axis and frequency on y-axis.

(c) Construct rectangles with class intervals as bases and corresponding frequencies as heights.

(d) Join the mid-points of the rectangle to obtain the ogive.

Question 3

Draw an ogive for each of the following distributions:

(i)

Marks Obtained

less

than 10

less

than 20

less

than30

less

than 40

less

than 50

No. of

Students

8

25

38

50

67

(ii)

Age in years (less than)

10

20

30

40

50

60

70

Cumulative Frequency

0

17

32

37

53

58

65

Solution 3

(i)

Marks Obtained

No. of students (c.f.)

less than 10

8

less than 20

25

less than 30

38

less than 40

50

less than 50

67

 

Steps Of construction:

(a) Plot the points (10,8), (20, 25), (30, 38), (40, 50) and (50, 67) on the graph.

(b) Join them with free hand to obtain an ogive.

(ii)

Age in years

(less than)

Cumulative

Frequency

10

0

20

17

30

32

40

37

50

53

60

58

70

65

 

 

Steps Of construction:

(a) Plot the points (10, 0), (20, 17), (30, 32), (40, 37), (50, 53), (60, 58) and (70, 65) on the graph.

(b) Join them with free hand to obtain an ogive.

Question 4

Construct a frequency distribution table for the number given below, using the class intervals 21-30, 31-40 … etc.

75, 67, 57, 50, 26, 33, 44, 58, 67, 75, 78, 43, 41, 31, 21, 32, 40, 62, 54, 69, 48, 47, 51, 38, 39, 43, 61, 63, 68, 53, 56, 49, 59, 37, 40, 68, 23, 28, 36, 47

Use the table obtained to draw:

(i) a histogram (ii) an ogive

Solution 4


(i)

(ii)

Plot the points (30,4), (40,13), (50,22), (60,29), (70,37) and (80,40) on the graph and join them with free hand to obtain an ogive.

Question 5

(a) Use information given in the adjoining histogram to construct a frequency table.

(b) Use this table to construct an ogive.

Solution 5

(a)

Class Interval

Frequency

c.f.

8-12

9

9

12-16

16

25

16-20

22

47

20-24

18

65

24-28

12

77

28-32

4

81

(b) Now plot the points (12, 9), (16, 25), (20, 47), (24, 65), (28, 77), (32, 81) and join them to obtain an ogive.

Question 6

Class Mark

12·5

17·5

22·5

27·5

32·5

37·5

42·5

Frequency

12

17

22

27

30

21

16

(a) From the distribution, given above, construct a frequency table.

(b) Use the table obtained in part (a) to draw: (i) a histogram, (ii) an

ogive.

Solution 6

(a)

Difference in consecutive class marks = 17.5 - 12.5 = 5

first class interval will be 10-15 and so on.

Class Mark

Class Interval

Frequency

c.f.

12.5

10-15

12

12

17.5

15-20

17

29

22.5

20-25

22

51

27.5

25-30

27

78

32.5

30-35

30

108

37.5

35-40

21

129

42.5

40-45

16

145

Total = 145

(b)

Now plot the points (15,12), (20,29), (25,51), (30,78), (35,108), (40,129), (45,145) and join them to obtain an ogive.

Question 7

Use graph paper for this question.

The table given below shows the monthly wages of some factory workers.

(i) Using the table, calculate the cumulative frequencies of workers

(ii) Draw a cumulative frequency curve.

Use 2 cm = Rs 500, starting the origin at Rs 6500 on x-axis, and 2 cm = 10 workers on the y-axis.

Wages

(in Rs)

6500 - 7000

7000 - 7500

7500 - 8000

8000 - 8500

8500 - 9000

9000 - 9500

9500 - 10000

No. of workers

10

18

22

25

17

10

8

Solution 7

(i)

 

Wages

No. Of workers

c.f.

6500-7000

10

10

7000-7500

18

28

7500-8000

22

50

8000-8500

25

75

8500-9000

17

92

9000-9500

10

102

9500-10000

8

110

Total = 110

Now plot the points (7000,10), (7500,28), (8000,50), (8500,75), (9000,92), (9500,102) and (10000,110) and join them to obtain an ogive.

Question 8

The following table shows the distribution of the heights of a

group of factory workers :

Ht.(cm):

150 - 155

155 - 160

160 - 165

165 - 170

170 - 175

175 - 180

180 - 185

No. of

workers:

6

12

18

20

13

8

6

(i) Determine the cumulative frequencies.

(ii) Draw the 'less than' cumulative frequency curve on graph

paper. Use 2 cm = 5 cm height on one axis and 2 cm = 10

workers on the other.

Solution 8

Height (in cm)

No. Of workers

c.f.

150-155

6

6

155-160

12

18

160-165

18

36

165-170

20

56

170-175

13

69

175-180

8

77

180-185

6

83

We plot the points (155, 6), (160, 18), (165, 36), (170, 56), (175, 69),

(180, 77) and (185, 83) on the graph and join them in free hand to obtain an ogive.

Question 9

Construct a frequency distribution table for each of the following

distributions:

(i)

Marks (less than)

0

10

20

30

40

50

60

70

80

90

100

Cumulative

0

7

28

54

71

84

105

147

180

196

200

Frequency

(ii)

Marks (more than)

0

10

20

30

40

50

60

70

80

90

100

Cumulative

Frequency

100

87

65

55

42

36

31

21

18

7

0

Solution 9

(i)

Marks

(less than)

Cumulative frequency

Frequency

0-10

7

7

10-20

28

28-7=21

20-30

54

54-28=26

30-40

71

71-54=17

40-50

84

84-71=13

50-60

105

105-84=21

60-70

147

147-105=42

70-80

180

180-147=33

80-90

196

196-180=16

90-100

200

200-196=4

Total

200

(ii)

Marks

(more than)

Cumulative frequency

Frequency

0-10

100

13

10-20

87

22

20-30

65

10

30-40

55

13

40-50

42

6

50-60

36

5

60-70

31

10

70-80

21

3

80-90

18

11

90-100

7

7

Total

100

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