ICSE Solutions for Selina Concise Chapter 23 Graphical Representation (Histograms and Ogives) Class 10 Maths
Exercise 23
1. Draw histograms for the following frequency 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
(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 the class intervals on x-axis and the frequencies 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 the 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. [Here, adjustment factor = 0.5]
(b) Taking suitable scales, mark the class intervals on x-axis and the frequency on y-axis.
(c) Construct rectangles with class intervals as bases and corresponding frequencies as heights.
(iv) From the given data:
| 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.
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
(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 the class intervals on x-axis and the frequencies 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. [Here, adjustment factor = 0.5]
(b) Taking suitable scales, mark the class intervals on x-axis and the frequencies 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.
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
(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.
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
(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.