Concise Mathematics Selina Solutions Class 10 ICSE Chapter 23 Graphical Representation (Histograms and Ogives)

This chapter on ICSE Class 10 Histograms and Ogives provides a detailed guide to visually representing statistical data, a fundamental skill in mathematics. In your Class 10 Concise Mathematics Selina textbook, Chapter 23, ‘Graphical Representation’, teaches you how to move beyond simple tables and charts. You will learn to construct histograms to understand the frequency distribution of continuous data and draw ogives, or cumulative frequency curves, which are essential for finding key statistical measures like the median and quartiles. Mastering these graphical techniques is crucial not just for your exams, but for interpreting real-world data in a more intuitive way. This chapter builds a strong foundation for higher-level statistics and data analysis.

If you are stuck on how to draw a histogram with unequal class intervals or how to find the median from an ogive, you’ve come to the right place. We understand that graphical questions require precise steps and clear interpretation. This page provides detailed, step-by-step solutions for all 17 questions in Exercise 23. Each solution is crafted to follow the exact method prescribed by the ICSE board, ensuring you learn the correct technique for plotting points, choosing scales, and interpreting the graphs. Here, you will find clear, easy-to-follow guides to master every problem in this chapter.

Exercise 23

Question 1(a)

From the given table the values of a, b and c are :

• (a) a = 16, b = 52 and c = 8
• (b) a = 16, b = 12 and c = 8
• (c) a = 40, b = 52 and c = 60
• (d) a = 40, b = 12 and c = 60

To determine the values of a, b, and c, observe the table:

For the class interval 40-50, the cumulative frequency is given as 40. Since the frequency for the 30-40 interval is 24, we have:

Solving for a, we find:

Next, for the class interval 50-60, we know the frequency is 12, and the cumulative frequency is b. Therefore:

Finally, for the class interval 60-70, the cumulative frequency is given as 60. Using the value of b, we have:

Substituting b = 52, we get:

Solving for c, we arrive at:

∴ The values are a = 16, b = 52, and c = 8. Hence, option 1 is the correct option.

Question 1(b)

The cumulative frequency of the class 70-90 is :

1. 80
2. 120
3. 180
4. 40

Analyzing the graph, we can compile the cumulative frequency table as follows:

Hence, Option 2 is the correct option.

Question 1(c)

For the given frequency distribution, the class mark is :

• (a) \dfrac{38 - 34}{2}
• (b) \dfrac{30 + 34}{2}
• (c) 10 – 8
• (d) 34 + 38

To find the class mark, use the formula:

Class mark = \dfrac{\text{Lower limit of class} + \text{Upper limit of class}}{2}

Applying this to the first class interval, we have:

Class mark = \dfrac{30 + 34}{2}.

Hence, Option 2 is the correct option.

Question 1(d)

The cumulative curve for a frequency distribution starts from :

• (a) 0
• (b) \dfrac{\text{upper limit + lower limit}}{2}
• (c) lower limit of 1st class
• (d) \dfrac{\text{lower limit - upper limit}}{2}

In a frequency distribution, the starting point for the cumulative curve, also known as the ogive, is the lower limit of the first class. This is because the cumulative frequency begins accumulating from this point. Hence, Option 3 is the correct option.

Question 1(e)

The cumulative curve for a frequency distribution terminates at :

• (a) 100
• (b) \dfrac{\text{upper limit + lower limit}}{2}
• (c) \dfrac{\text{lower limit + upper limit}}{2}
• (d) upper limit of last class

In a frequency distribution, the cumulative frequency curve, also known as an ogive, reaches its endpoint at the upper limit of the last class interval. This is because the cumulative frequency accounts for all data values up to the highest boundary of the dataset.

Hence, Option 4 is the correct option.

Question 2(i)

Draw histograms for the following frequency distributions :

To construct the histogram, follow these steps:

1. On the x-axis, use a scale where 2 cm represents 10 units. This will help in clearly marking the class intervals: 0-10, 10-20, 20-30, 30-40, 40-50, and 50-60.

2. On the y-axis, choose a scale where 1 cm corresponds to 5 units. This scale will allow us to accurately represent the frequencies: 12, 20, 26, 18, 10, and 6.

3. Draw rectangles for each class interval. The height of each rectangle should match the frequency of the corresponding class interval from the table.

The histogram you create will visually represent the frequency distribution given in the table.

Question 2(ii)

Draw histograms for the following frequency distributions :

To draw the histogram for the given frequency distribution, follow these steps:

1. Notice that the x-axis scale begins at 10. Therefore, place a kink near the origin on the x-axis to indicate that the scale starts at 10.
2. On the x-axis, assign 2 cm to represent 6 units.
3. On the y-axis, let 1 cm equal 5 units.
4. Construct rectangles for each class interval based on the continuous frequency distribution provided.

The completed histogram can be seen in the figure below:

Question 2(iii)

Draw histograms for the following frequency distributions :

To determine the class intervals, notice that the difference between consecutive class marks is 8 (24 – 16). Thus, by subtracting \dfrac{8}{2} = 4 from each class mark, we find the lower limit of the interval. Similarly, adding 4 to each class mark gives us the upper limit.

Here is the frequency distribution table:

To construct the histogram:

1. Since the x-axis starts from 12, introduce a kink near the origin to indicate the scale begins at 12.
2. Set the scale such that 1 cm on the x-axis represents 8 units.
3. On the y-axis, 1 cm should represent 5 units.
4. Draw rectangles for each class interval using the frequencies provided in the table.

The histogram you need is illustrated in the accompanying figure:

Question 3(i)

Draw a cumulative frequency curve (ogive) for the following distributions :

To create the cumulative frequency distribution, follow these steps:

Let’s go through the steps to draw the ogive:

1. On the x-axis, use a scale where 2 cm equals 5 units.
2. On the y-axis, let 1 cm represent 10 units.
3. Begin the ogive at the lower limit of the first class interval. Plot the initial point as (10, 0).
4. For the x-axis, use the upper limits of the class intervals, and for the y-axis, use the cumulative frequencies. Plot the following points: (15, 10), (20, 25), (25, 42), (30, 54), (35, 64), and (40, 72).
5. Connect these points smoothly using a freehand curve.

The ogive you need is illustrated in the accompanying figure:

Question 3(ii)

Draw a cumulative frequency curve (ogive) for the following distributions :

To begin with, we need to convert the given discontinuous class intervals into continuous ones. For this, calculate the adjustment factor using the formula:

Adjustment factor = \frac{\text{Lower limit of one class} - \text{Upper limit of previous class}}{2}

Plugging in the values, we get:

= \frac{20 - 19}{2} = \frac{1}{2}

= 0.5

Now, subtract 0.5 from each lower limit and add 0.5 to each upper limit to adjust the classes.

Now, let’s construct the ogive:

1. Since the x-axis scale begins at 9.5, indicate this by showing a break (kink) near the origin.
2. Choose a scale of 2 cm = 10 units for the x-axis.
3. Choose a scale of 1 cm = 10 units for the y-axis.
4. The ogive starts at the lower limit of the first class. Therefore, plot the point (9.5, 0) on the x-axis.
5. Next, plot the points using the upper class limits on the x-axis and the corresponding cumulative frequencies on the y-axis: (19.5, 23), (29.5, 39), (39.5, 54), (49.5, 74), and (59.5, 86).
6. Connect these points with a smooth, freehand curve.

The ogive you need is depicted in the diagram below:

Question 4(i)

Draw an ogive for the following distribution :

To create a cumulative frequency distribution table, arrange the data as follows:

To draw the ogive, follow these steps:

1. On the x-axis, represent 10 units with 2 cm.
2. On the y-axis, represent 10 units with 1 cm.
3. Start the ogive from the origin, point (0, 0), which corresponds to the lowest limit.
4. Plot the points using the upper class limits on the x-axis and the cumulative frequencies on the y-axis: (10, 8), (20, 25), (30, 38), (40, 50), and (50, 67).
5. Connect these points smoothly with a freehand curve.

The resulting curve is your desired ogive, as depicted in the accompanying figure.

Question 4(ii)

Draw an ogive for the following distribution :

To construct the ogive, we start by organizing the cumulative frequency data:

For the ogive construction, follow these steps:

1. Set the x-axis scale: 2 cm represents 10 units.
2. Set the y-axis scale: 1 cm represents 10 units.
3. Begin the ogive from the x-axis at the lowest boundary. Plot the starting point at (0, 0).
4. Plot the points using the upper class limits on the x-axis and their corresponding cumulative frequencies on the y-axis. These points are: (10, 0), (20, 17), (30, 32), (40, 37), (50, 53), (60, 58), and (70, 65).
5. Connect these points smoothly with a freehand curve.

The resulting curve is the desired ogive, which visually represents the cumulative frequency distribution.

Question 5

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

(b) Use this table to construct an ogive.

(a) Here’s how to create the cumulative frequency table based on the histogram data:

(b) To draw the ogive, follow these steps:

1. Notice that the x-axis scale begins at 8, so a break (kink) is shown near the origin to indicate this starting point.
2. Set the x-axis scale to 2 cm for every 4 units.
3. Set the y-axis scale to 1 cm for every 10 units.
4. The ogive begins at the lowest class limit on the x-axis, starting at the point (8, 0).
5. Plot the upper class limits on the x-axis against their corresponding cumulative frequencies on the y-axis. The points to plot are (12, 9), (16, 25), (20, 47), (24, 65), (28, 77), and (32, 81).
6. Connect these points smoothly with a freehand curve.

The resulting curve is the required ogive, as illustrated in the accompanying figure.

Question 6

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.

(i) Let’s create the cumulative frequency table:

(ii) To draw the cumulative frequency curve, follow these steps:

1. Notice that the x-axis begins at ₹ 6500, so we indicate a break near the origin to show this starting point.
2. Set the scale on the x-axis such that 2 cm represents ₹ 500.
3. On the y-axis, let 1 cm equal 10 workers.
4. Begin the ogive at the x-axis point for the lower limit of the first class, which is (6500, 0).
5. Plot the upper class limits on the x-axis against the cumulative frequencies on the y-axis. The points to plot are: (7000, 10), (7500, 28), (8000, 50), (8500, 75), (9000, 92), (9500, 102), and (10000, 110).
6. Connect these points using a smooth, freehand curve.

The resulting ogive is depicted in the accompanying figure.

Question 7

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.

(i) Let’s create the cumulative frequency table for the given data:

(ii) To draw the ‘less than’ cumulative frequency curve, follow these steps:

1. Note that the x-axis starts at 150 cm, so include a kink near the origin to indicate the scale begins at this point.
2. Set the scale on the x-axis as 2 cm for every 5 cm.
3. On the y-axis, use 1 cm to represent 10 workers.
4. The ogive begins at the lower limit of the first class, so start with the point (150, 0).
5. Use the upper class limits on the x-axis and the corresponding cumulative frequencies on the y-axis to plot: (155, 6), (160, 18), (165, 36), (170, 56), (175, 69), (180, 77), and (185, 83).
6. Connect these points with a smooth, freehand curve to form the ogive.

The resulting ogive is depicted in the accompanying graph.

Question 8(i)

Construct a frequency distribution table for the following distribution :

Here is the frequency distribution table derived from the given cumulative frequency data:

Notice that to find the frequency for each class interval, we subtract the cumulative frequency of the previous class from the current class’s cumulative frequency. For instance, for the class interval 10 – 20, the frequency is calculated as 28 – 7, which equals 21. Similarly, for 20 – 30, it is 54 – 28, resulting in 26. This method is used for all intervals to complete the table.

Question 8(ii)

Construct a frequency distribution table for the following distribution :

To create the frequency distribution table from the given data, we need to determine the frequency for each class interval. The cumulative frequency given is ‘more than,’ so we’ll calculate the frequency by subtracting the cumulative frequency of the next class from the current class. Here’s how it unfolds:

Notice that for each class interval, the frequency is calculated by finding the difference between consecutive cumulative frequencies. This approach ensures we accurately translate the cumulative data into a frequency distribution table.