Concise Mathematics Class 10 ICSE Solutions for Chapter 23 - Graphical Representation (Histograms, Frequency Polygon and Ogives)

Exercise 23(A)

1. Draw a histograms for the following frequency distributions:

(i) 

Ans: Steps of Construction:

1. Using appropriate scales, mark class intervals on the x-axis and frequency on the y-axis.

2. Draw rectangles with class intervals as bases and frequencies as heights.

(ii)

Ans: Steps of Construction:

1. Using appropriate scales, mark class intervals on the x-axis and frequency on the y-axis.

2. Draw rectangles with class intervals as bases and frequencies as heights.

(iii)

Ans: 

Steps of Construction:

1. Convert the data to its exclusive form. (Adjustment factor is 0.5)

2. Using appropriate scales, mark class intervals on the x-axis and frequency on the y-axis.

3. Draw rectangles with class intervals as bases and frequencies as heights.

(iv)

Ans: 

Steps of Construction:

1. Convert the class marks to intervals.

2. Using appropriate scales, mark class intervals on the x-axis and frequency on the y-axis.

3. Draw rectangles with class intervals as bases and frequencies as heights.

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

(i) 

Ans: 

Steps of Construction:

1. Using appropriate scales, trace class intervals on the x-axis and frequency on the y-axis.

2. Draw rectangles with class intervals as bases and frequencies as heights.

3. To make the ogive, join the rectangle's midpoints.

(ii)

Ans: 

Steps of Construction:

1. Convert the data into the exclusive form. (Adjustment factor is 0.5)

2. Using appropriate scales, draw class intervals on the x-axis and frequencies on the y-axis.

3. Draw rectangles with class intervals as bases and frequencies as heights.

4. To obtain ogive, join the rectangle's midpoints.

3. Draw an ogive for each of the following distributions:

(i) 

Ans: Steps of Construction:

1. On the graph, plot the points (10, 8), (20, 25), (30, 38), (40, 50), and (50, 67).

2. Obtain an ogive by joining them with your free hand.

(ii)

Ans: Steps of Construction:

1. On the graph, plot the points (10, 0), (20, 17), (30, 32), (40, 37), (50, 53), (60, 58) and (70, 65)

2. Obtain an ogive by joining them with your free hand.

4. Construct a frequency distribution table for the numbers 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, and 47.

Use the table obtained to draw:

1. A Histogram

Ans: Individual class intervals can be used to count the frequency.

Now, from the class interval 21 - 30, the numbers occurring in the above mentioned distributions are 21, 23, 26, 28 = 4.

31 - 40 $\to $ 31, 32, 33, 36, 37, 38, 39, 40

41 - 50 $\to $ 41, 43, 43, 44, 47, 47, 48, 49, 50

51 - 60 $\to $ 51, 53, 54, 56, 57, 58, 59

61 - 70 $\to $ 61, 62, 63, 67, 67, 68, 68, 69

71 - 80 $\to $ 75, 75, 78

2. An Ogive

Ans:

To draw an ogive, plot the points (30, 4), (40, 12), (50, 21), (60, 28), (70, 36), and (80, 39) on the graph and join them with your free hand.

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

Ans: (a) 

(b) Use this table to construct an ogive. 

Ans: To obtain an ogive, plot and connect the points (12, 9), (16, 25), (20, 47), (24, 65), (28, 77), and (32, 81).

6. 

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

Ans: Differences in consecutive class marks = 17.5 - 12.5 = 5

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

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

Ans: To make an ogive, join the points (15, 12), (20, 29), (25, 51), (30, 78), (35, 108), (40, 129) and (45, 145).

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. 

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

Ans: 

(ii) Draw a cumulative frequency curve. 

Ans: To make an ogive, join the points (7000, 10), (7500, 28), (8000, 50), (8500, 75), (9000, 92), (9500, 102) and (10000, 110).

8. The following table shows the distribution of the heights of a group of factory workers: 

(i) Determine the cumulative frequencies. 

Ans:  

(ii) Draw the 'less than' cumulative frequency curve on graph paper. Use 2cm = 5cm height on one axis and 2cm = 10 workers on the other.

Ans: Draw an ogive by connecting the points (155, 6), (160, 18), (165, 36), (170, 56), (175, 69), (180, 77), and (185, 83) on the graph.

9. Construct a frequency distribution table for each of the following distributions : 

(i) 

Ans:   

(ii) 

Ans:

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