SELINA Solutions for Class 10 Maths Chapter 23  Graphical Representation (Histograms and Ogives)
Chapter 23  Graphical Representation (Histograms and Ogives) Exercise Ex. 23
Draw histogram for the following distributions:
(i)
Class Interval 
010 
1020 
2030 
3040 
4050 
5060 
Frequency 
12 
20 
26 
18 
10 
6 
(ii)
Class Interval 
1016 
1622 
2228 
2834 
3440 
Frequency 
15 
23 
30 
20 
16 
(iii)
Class Interval 
3039 
4049 
5059 
6069 
7079 
Frequency 
24 
16 
09 
15 
20 
(iv)
Class Marks 
16 
24 
32 
40 
48 
56 
64 
Frequency 
8 
12 
15 
18 
25 
19 
10 
(i)
Class Interval 
Frequency 
010 
12 
1020 
20 
2030 
26 
3040 
18 
4050 
10 
5060 
06 
Steps of construction:
(a) Taking suitable scales, mark class intervals on xaxis and frequency on yaxis.
(b)Construct rectangles with class intervals as bases and corresponding frequencies as heights.
(ii)
Class Interval 
Frequency 
1016 
15 
1622 
23 
2228 
30 
2834 
20 
3440 
16 
Steps of construction:
(a) Taking suitable scales, mark class intervals on xaxis and frequency on yaxis.
(b) Construct rectangles with class intervals as bases and corresponding frequencies as heights.
(iii)
Class Interval (Inclusive form) 
Class Interval (Exclusive Form) 
Frequency 
3039 
29.539.5 
24 
4049 
39.549.5 
16 
5059 
49.559.5 
09 
6069 
59.569.5 
15 
7079 
69.579.5 
20 
Steps of construction:
(a) Convert the data into exclusive form.
(b) Taking suitable scales, mark class intervals on xaxis and frequency on yaxis.
(c) Construct rectangles with class intervals as bases and corresponding frequencies as heights.
(iv)
Class Marks 
Class Intervals 
Frequency 
16 
1220 
08 
24 
2028 
12 
32 
2836 
15 
40 
3644 
18 
48 
4452 
25 
56 
5260 
19 
64 
6068 
10 
Steps of construction:
(a) Convert the class marks into class intervals.
(b) Taking suitable scales, mark class intervals on xaxis and frequency on yaxis.
(c) Construct rectangles with class intervals as bases and corresponding frequencies as heights.
Draw cumulative frequency curve (ogive) for each of the following distributions:
(i)
Class Interval 
1015 
1520 
2025 
2530 
3045 
3540 
Frequency 
10 
15 
17 
12 
10 
08 
(ii)
Class Interval 
1019 
2029 
3039 
4049 
5059 
Frequency 
23 
16 
15 
20 
12 
(i)
Class Interval 
Frequency 
1015 
10 
1520 
15 
2025 
17 
2530 
12 
3035 
10 
3540 
08 
Steps of construction:
(a) Taking suitable scales, mark class intervals on xaxis and frequency on yaxis.
(b) Construct rectangles with class intervals as bases and corresponding frequencies as heights.
( c) Join the midpoints of the rectangle to obtain the ogive.
(ii)
Class Interval (Inclusive) 
Class Interval (Exclusive) 
Frequency 
Cumulative Frequency 
1019 
9.519.5 
23 
23 
2029 
19.529.5 
16 
39 
3039 
29.539.5 
15 
54 
4049 
39.549.5 
20 
74 
5059 
49.559.5 
12 
86 
Total 
86 
Steps of construction:
(a) Convert the data into exclusive form.
(b) Taking suitable scales, mark class intervals on xaxis and frequency on yaxis.
(c) Construct rectangles with class intervals as bases and corresponding frequencies as heights.
(d) Join the midpoints of the rectangle to obtain the ogive.
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 
(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.
Construct a frequency distribution table for the number given below, using the class intervals 2130, 3140 … 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
(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.
(a) Use information given in the adjoining histogram to construct a frequency table.
(b) Use this table to construct an ogive.
(a)
Class Interval 
Frequency 
c.f. 
812 
9 
9 
1216 
16 
25 
1620 
22 
47 
2024 
18 
65 
2428 
12 
77 
2832 
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.
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.
(a)
Difference in consecutive class marks = 17.5  12.5 = 5
first class interval will be 1015 and so on.
Class Mark 
Class Interval 
Frequency 
c.f. 
12.5 
1015 
12 
12 
17.5 
1520 
17 
29 
22.5 
2025 
22 
51 
27.5 
2530 
27 
78 
32.5 
3035 
30 
108 
37.5 
3540 
21 
129 
42.5 
4045 
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.
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 xaxis, and 2 cm = 10 workers on the yaxis.
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 
(i)
Wages 
No. Of workers 
c.f. 
65007000 
10 
10 
70007500 
18 
28 
75008000 
22 
50 
80008500 
25 
75 
85009000 
17 
92 
90009500 
10 
102 
950010000 
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.
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.
Height (in cm) 
No. Of workers 
c.f. 
150155 
6 
6 
155160 
12 
18 
160165 
18 
36 
165170 
20 
56 
170175 
13 
69 
175180 
8 
77 
180185 
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.
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 























(i)
Marks (less than) 
Cumulative frequency 
Frequency 
010 
7 
7 
1020 
28 
287=21 
2030 
54 
5428=26 
3040 
71 
7154=17 
4050 
84 
8471=13 
5060 
105 
10584=21 
6070 
147 
147105=42 
7080 
180 
180147=33 
8090 
196 
196180=16 
90100 
200 
200196=4 
Total 

200 
(ii)
Marks (more than) 
Cumulative frequency 
Frequency 
010 
100 
13 
1020 
87 
22 
2030 
65 
10 
3040 
55 
13 
4050 
42 
6 
5060 
36 
5 
6070 
31 
10 
7080 
21 
3 
8090 
18 
11 
90100 
7 
7 
Total 

100 
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