ICSE Class 10 Maths Chapter 23 Graphical Representation

Unit 6: Statistics

Graphical Representation

Histograms, Frequency Polygon and Ogives

23.1 Graphical Representation

The statistical data can be represented by diagram, chart, etc., so that the significance attached to these data may immediately be grasped. Of course, the diagrams should be neatly and accurately drawn.

Out of several types of diagrams, charts, etc., we shall be studying only the following three types of diagrams:

1. Histogram

2. Frequency polygon

3. Ogive (cumulative frequency curve).

23.2 Histogram

A histogram is a two-dimensional graphical representation of continuous frequency distribution.

In this case, rectangles are drawn with bases proportional to class intervals and heights proportional to the frequencies of respective classes.

23.3 Histogram For Continuous Grouped Data

Steps:

1. Convert the data in the exclusive form, if it is in inclusive form.

2. Taking suitable scales, mark class intervals on x-axis and frequencies on y-axis.

The scales chosen for both the axes need not be the same.

3. Construct rectangles with class intervals as bases and corresponding frequencies as heights.

1 Draw a histogram to represent the following:

Solution:

Starting from 0, mark 8, 16, 24, 32 and 40 on x-axis at equal distances and 2, 4, 6, 8, 10 and 12 on y-axis at equal distances.

Now draw the rectangles to get the required histogram.

[Histogram image showing class intervals 0-8, 8-16, 16-24, 24-32, 32-40 on x-axis and frequencies 0-12 on y-axis with corresponding rectangular bars]

Teacher's Note

Histograms help us visualize data patterns at a glance, similar to how weather forecasts use bar charts to show rainfall across different months.

2 Draw a histogram to represent the following:

[2005]

Solution:

Note: In the given frequency distribution, the first class interval is 150 - 200; therefore, the scale on x-axis starts at 150. For this, in general, a kink (break) or a zig-zag curve is drawn near the origin to tell that the graph is drawn to scale beginning at 150 and not at the origin itself.

[Histogram image showing class intervals 150-200, 200-250, 250-300, 300-350, 350-400 on x-axis and frequencies 0-10 on y-axis with a break at origin]

Teacher's Note

When data doesn't start from zero, like pocket money starting at 150 rupees, we use a break symbol to show this, similar to how maps show breaks for non-continuous territory.

23.4 For Discontinuous Grouped Data

3 Draw a histogram for the following:

Solution:

In this case, the class intervals given are in inclusive form. So, first of all we have to convert them into exclusive form.

Since the adjustment factor

= 1/2 (difference between the upper limit of a class and the lower limit of next class)

= 1/2 (21 - 20) = 0.5.

Therefore, to convert given class intervals in exclusive form, subtract the adjustment factor from all the lower limits and add it to all upper limits.

The adjusted class intervals would then be as follows:

And the required histogram will be as shown alongside.

[Histogram image showing class intervals 10.5-20.5, 20.5-30.5, 30.5-40.5, 40.5-50.5, 50.5-60.5 on x-axis and frequencies 0-14 on y-axis]

Teacher's Note

Converting inclusive to exclusive form ensures histogram bars touch without gaps, like tiles fitting perfectly on a floor.

23.5 When Class Marks Are Given

First of all, find the class intervals and then draw the histogram.

4 Draw the histogram for the following:

Solution:

Since the difference between the values of any two consecutive class marks is 10, therefore, subtract 10/2 = 5 from each class mark to get the lower limit of the corresponding class interval and add 5 to each class mark to get the upper limit.

Thus, the given frequency distribution will be of the form:

and the required histogram will be as shown alongside.

[Histogram image showing class intervals 20-30, 30-40, 40-50, 50-60, 60-70 on x-axis and frequencies 0-18 on y-axis]

Teacher's Note

Class marks represent the middle value of each interval; this is how average scores in a test bracket can help us reconstruct the original class ranges.

23.6 Cumulative Frequency And Cumulative Frequency Table

The cumulative frequency of a class interval is the sum of frequencies of all the classes up to this class interval.

This is a preview of the first 3 pages. To get the complete book, click below.