Selina Concise Solutions for ICSE Class 10 Mathematics Chapter 23 Graphical Representation Histograms Frequency Polygon Ogives

Exercise 23

Question 1. Draw histogram for the following distributions:
(i)

(ii)

(iii)

(iv)

Answer:

(i)

[Diagram: This histogram shows a bar chart with class intervals on x-axis (0-10, 10-20, 20-30, 30-40, 40-50, 50-60) and frequency on y-axis. The bars have heights of 12, 20, 26, 18, 10, and 6 respectively.]

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)

[Diagram: This histogram shows a bar chart with class intervals on x-axis (10-16, 16-22, 22-28, 28-34, 34-40) and frequency on y-axis. The bars have heights of 15, 23, 30, 20, and 16 respectively.]

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)
First convert the data into exclusive form:

[Diagram: This histogram shows a bar chart with exclusive class intervals on x-axis (29.5-39.5, 39.5-49.5, 49.5-59.5, 59.5-69.5, 69.5-79.5) and frequency on y-axis. The bars have heights of 24, 16, 9, 15, and 20 respectively.]

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)
First convert the class marks into class intervals:

[Diagram: This histogram shows a bar chart with class intervals on x-axis (12-20, 20-28, 28-36, 36-44, 44-52, 52-60, 60-68) and frequency on y-axis. The bars have heights of 8, 12, 15, 18, 25, 19, and 10 respectively.]

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.

In simple words: A histogram is like a bar graph that shows data. We draw rectangles for each group. The width shows the range of numbers. The height shows how many times that group appears.

📝 Teacher's Note: Show students that histograms have bars touching each other. This is different from bar graphs where bars are separate. Use real examples like student heights or test scores to practice.

🎯 Exam Tip: Always convert inclusive form to exclusive form when needed. Mark the axes clearly with labels and scales. Draw neat rectangles with correct heights matching frequencies.

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

(i)

(ii)

Answer:

(i)

[Diagram: This shows a cumulative frequency histogram with class intervals on x-axis (0-15, 10-15, 15-20, 20-25, 25-30, 30-35, 35-40) and frequency on y-axis (0 to 18). Blue rectangles show frequencies: 10, 15, 17, 12, 10, 8. A smooth curve connects the midpoints of the tops of the rectangles to form the ogive.]

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)

[Diagram: This shows a cumulative frequency histogram with exclusive class intervals on x-axis (0-9.5, 9.5-19.5, 19.5-29.5, 29.5-39.5, 39.5-49.5, 49.5-59.5) and frequency on y-axis (0 to 25). Blue rectangles show frequencies: 23, 16, 15, 20, 12. A smooth curve connects the midpoints of the tops of the rectangles to form the ogive.]

In simple words: An ogive is a smooth curve that shows how the total count adds up as we go from left to right. We first draw rectangles for each group, then connect their top middle points with a smooth line.

📝 Teacher's Note: Show students how to add frequencies step by step to get cumulative frequency. Use a simple example like counting students in different classes - Class 1 has 10, Class 2 has 15, so total up to Class 2 is 25.

🎯 Exam Tip: Always make the cumulative frequency table first. Then plot points and join them with a smooth curve. Do not draw straight lines between points - make it a smooth curve.

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

(i)

(ii)

Answer:

(i)

[Diagram: This shows an ogive graph with Marks Obtained on x-axis (10, 20, 30, 40, 50) and Cumulative Frequency on y-axis (0 to 80). Points plotted are (10,8), (20,25), (30,38), (40,50), (50,67). A smooth S-shaped curve connects these points.]

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)

[Diagram: This shows an ogive graph with Age (in years) on x-axis (10, 20, 30, 40, 50, 60, 70) and Cumulative Frequency on y-axis (0 to 70). Points plotted are (10,0), (20,17), (30,32), (40,37), (50,53), (60,58), (70,65). A smooth S-shaped curve connects these points.]

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.

In simple words: When data is already given as "less than" type, we just plot the points and join them with a smooth curve. The curve always goes up because cumulative frequency keeps adding up.

📝 Teacher's Note: Explain that "less than" means all values below that number. For example, "less than 20" includes all students with marks from 0 to 19. This is why the curve always goes upward.

🎯 Exam Tip: For "less than" ogive, plot points directly as given. Always start from (0,0) or the first point given. Join with a smooth curve - never use straight lines between points.

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
Answer:
Step 1: First, we arrange the given numbers into class intervals.

Step 2: Count how many numbers fall in each class interval.

Step 3: For the histogram, draw bars with heights equal to the frequency of each class.

Step 4: For the ogive, plot the points (30,4), (40,13), (50,22), (60,29), (70,37) and (80,40) on the graph and join them with a smooth curve.

[Diagram: The histogram shows bars for each class interval with heights 4, 9, 9, 7, 8, 3. The ogive shows a smooth curve rising from (20,0) through the cumulative frequency points to (80,40).]
In simple words: We sort the numbers into groups and count how many are in each group. Then we draw a bar chart (histogram) and a rising curve graph (ogive) to show the data clearly.

📝 Teacher's Note: Show students how to use tally marks for counting. This makes it easy to avoid mistakes. Also explain that c.f. means "how many numbers we have counted so far."

🎯 Exam Tip: Always write the table first, then draw the graphs. For histogram, bars touch each other. For ogive, plot points at the upper limit of each class and join with smooth curves.

Question 5. (a) Use information given in the adjoining histogram to construct a frequency table. (b) Use this table to construct an ogive.
Answer:
Part (a): Reading the frequencies from the histogram:

Part (b): For the ogive, plot points (12, 9), (16, 25), (20, 47), (24, 65), (28, 77), (32, 81) and join them with a smooth curve.

[Diagram: The ogive shows a smooth curve starting from (12,9) and rising to (32,81), passing through all the cumulative frequency points.]
In simple words: We read the bar heights from the graph to make a table. Then we add up the frequencies to get cumulative frequency. The ogive shows how the total builds up step by step.

📝 Teacher's Note: Teach students to read histogram heights carefully. The cumulative frequency always keeps adding up - it never goes down. This is a good way to check their work.

🎯 Exam Tip: Read the histogram bar heights exactly. Write the frequency table first, then calculate c.f. by adding frequencies step by step. Plot ogive points at the upper limit of each class interval.

Question 6. (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.
Answer:
Part (a): First, we find the class width. The difference between consecutive class marks is \( 17.5 - 12.5 = 5 \), so the first class interval will be 10-15 and so on.

Total = 145

Part (b):
(i) For histogram, draw bars with heights 12, 17, 22, 27, 30, 21, 16 for respective class intervals.
(ii) For ogive, plot points (15,12), (20,29), (25,51), (30,78), (35,108), (40,129), (45,145) and join with smooth curve.

[Diagram: Histogram shows bars with the given frequencies. Ogive shows a smooth rising curve from (10,0) to (45,145) passing through all cumulative frequency points.]
In simple words: Class mark is the middle value of each class. We use this to find the actual class intervals. Then we draw the bar chart and rising curve graph as usual.

📝 Teacher's Note: Explain that class mark is always in the middle of the class interval. So if class mark is 12.5, the class goes from 10 to 15. Students often get confused about this.

🎯 Exam Tip: When given class marks, subtract and add half the class width to get the actual intervals. Always verify your total frequency matches the sum of individual frequencies. Show all working clearly.

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.

[Diagram: The question includes a table showing wages in Rs (6500-7000, 7000-7500, 7500-8000, 8000-8500, 8500-9000, 9000-9500, 9500-10000) and corresponding number of workers (10, 18, 22, 25, 17, 10, 8).]

Answer:
(i) Calculating cumulative frequencies:

Total = 110 workers

(ii) For drawing the cumulative frequency curve:
Plot the points (7000,10), (7500,28), (8000,50), (8500,75), (9000,92), (9500,102) and (10000,110) and join them to get a smooth curve.

[Diagram: The solution shows a cumulative frequency graph with wages on x-axis and cumulative frequency on y-axis, showing an ogive curve that rises from left to right.]

In simple words: Cumulative frequency means we keep adding up all the workers. First 10, then 10+18=28, then 28+22=50, and so on. The graph shows how many workers earn less than a certain amount.

📝 Teacher's Note: Show students how to add step by step. First group has 10 workers. Second group adds 18 more, so total becomes 28. This keeps going until all workers are counted.

🎯 Exam Tip: Always write the cumulative frequency table first. Then plot points using the upper limit of each class. Join the points with a smooth curve, not straight lines.

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

[Diagram: The question includes a table showing heights in cm (150-155, 155-160, 160-165, 165-170, 170-175, 175-180, 180-185) and corresponding number of workers (6, 12, 18, 20, 13, 8, 6).]

Answer:
(i) Calculating cumulative frequencies:

(ii) For drawing the curve:
Plot the points (155, 6), (160, 18), (165, 36), (170, 56), (175, 69), (180, 77) and (185, 83) on the graph and join them to get an ogive.

[Diagram: The solution shows a cumulative frequency graph with height on x-axis and number of workers on y-axis, displaying a smooth ogive curve.]

In simple words: This graph shows how many workers have height less than a certain value. For example, 36 workers have height less than 165 cm.

📝 Teacher's Note: Explain that "less than" means we use the upper limit of each class for plotting. The curve should be smooth, not made of straight line segments.

🎯 Exam Tip: Mark the scale clearly on both axes. Use the upper boundary of each class interval for x-coordinates. The curve should pass through all plotted points smoothly.

Question 9. Construct a frequency distribution table for each of the following distributions:
(i) [Table showing Marks (less than): 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 and Cumulative Frequency: 0, 7, 28, 54, 71, 84, 105, 147, 180, 196, 200]
(ii) [Table showing Marks (more than): 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 and Cumulative Frequency: 100, 87, 65, 55, 42, 36, 31, 21, 18, 7, 0]

Answer:
(i) Frequency distribution table from "less than" cumulative frequency:

(ii) For "more than" cumulative frequency:
The frequency for each class interval is calculated by subtracting consecutive cumulative frequencies from the higher value to the lower value.

In simple words: To find frequency from cumulative frequency, we subtract. For example, if 28 students scored less than 20 marks and 7 scored less than 10 marks, then 28-7=21 students scored between 10-20 marks.

📝 Teacher's Note: Draw a simple diagram showing how cumulative frequency builds up by adding. Then show how we reverse this process by subtracting to get individual frequencies.

🎯 Exam Tip: Always check that your individual frequencies add up to the total given in the last cumulative frequency. This helps you spot calculation mistakes.