Class X Chapter 23 – Graphical Representation Maths
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Book Name: Selina Concise
Question 1:
Draw histograms for the following frequency distributions:
(i)
Class Interval 0 – 16 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 Interval 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.
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
(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 Class Interval
Frequency
(Inclusive form) (Exclusive Form)
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
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
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.
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
Question 2:
Draw a cumulative frequency curve (ogive) for each of the following distributions:
(i)
Class Interval 10 – 15 15 – 20 20 – 25 25 – 30 30 – 35 35 – 40
Frequency 10 15 17 12 10 8
(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 Class Interval Cumulative
Frequency
(Inclusive) (Exclusive) 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
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
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 than 30 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
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
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 Cumulative
(less than) 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.
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Class X Chapter 23 – Graphical Representation Maths
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Question 4:
Construct a frequency distribution table for the numbers given below, using the class intervals
21 – 30, 31 – 40 ………… etc.
75, 65, 57, 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 and 47.
Use the table obtained to draw: (i) a histrogram (ii) an ogive
Solution 4:
Class Interval Tally Frequency c.f.
21-30 ∥∥ 4 4
31- 40 ∦∦ ∥∥ 9 13
41 – 50 ∦∦ ∥∥ 9 22
51 – 60 ∦∦ ∥ 7 29
61 – 70 ∦∦ ∥l 8 37
71 – 80 ∥l 3 40
(i)
(ii)
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
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 the 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.
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
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
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Class X Chapter 23 – Graphical Representation Maths
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(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 = ₹ 500, starting the origin at ₹ 6500 on x-axis, and 2 cm = 10 workers on the y –
axis.
Wages 6500- 7000- 7500- 8000- 8500- 9000- 9500-
(in Rs) 7000 7500 8000 8500 9000 9500 10000
No of
10 18 22 25 17 10 8
workers
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.
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
Question 8:
The following table shows the distribution of the heights of a group of factory workers:
Ht. (cm) 180 –
150 - 155 155 – 160 160 - 165 165 – 170 170 – 175 175 - 180
185
No of
6 12 18 20 13 8 6
workers:
(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:
No. of
Height (in cm) c.f.
workers
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.
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
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 frequency 0 7 28 54 71 84 105 147 180 196 200
(ii)
Marks (less than) 0 10 20 30 40 50 60 70 80 90 100
Cumulative
100 87 65 55 42 36 31 21 18 7 0
frequency
Solution 9:
(i)
Marks Cumulative
Frequency
(less than) 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
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Class X Chapter 23 – Graphical Representation Maths
______________________________________________________________________________
(ii)
Marks Cumulative
Frequency
(more than) 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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