12.1. Graphical Representation of Data

The representation of data by tables has already been discussed. Now let us turn our attention to another representation of data, i.e., the graphical representation. It is well said that one picture is better than a thousand words. Usually comparisons among the individual items are best shown by means of graphs. The representation then becomes easier to understand than the actual data. We shall study the following graphical representations in this section.

(A) Bar graphs (B) Histograms of uniform width, and of varying widths (C) Frequency polygons

This chunk introduces the concept of graphical representation of data. Unlike tables, which show data in rows and columns, graphical representation uses visual forms like graphs to make data easier to understand. It supports the idea that visual depictions can convey information more effectively than numerical data alone. The section will cover different types of graphs: bar graphs, histograms, and frequency polygons.

Imagine trying to show the number of books read by your friends in a month: if you list out each friend's name with numbers in a table, it might be hard to see who read the most. However, if you draw a bar graph where each bar represents a friend, you'll instantly see which bars are taller, indicating who read more books. This visual comparison is much quicker and easily understandable.

(A) Bar Graphs

In earlier classes, you have already studied and constructed bar graphs. Here we shall discuss them through a more formal approach. Recall that a bar graph is a pictorial representation of data in which usually bars of uniform width are drawn with equal spacing between them on one axis (say, the x-axis), depicting the variable. The values of the variable are shown on the other axis (say, the y-axis) and the heights of the bars depend on the values of the variable.

A bar graph visually represents data using bars, where the length or height of each bar corresponds to the size of the data it represents. Each bar has a uniform width and standard spacing. The x-axis typically shows the categories being compared, and the y-axis shows the values. For example, if you have data about students' favorite fruits, the x-axis would list the fruits, while the y-axis would represent the number of students who chose each fruit.

Think about a carnival where different games have different numbers of players. A bar graph could display each game like a frog jumping contest or a ring toss on the x-axis, and how many players participated on the y-axis. This way, it would be clear at a glance which game was the most popular!

Example 1: In a particular section of Class IX, 40 students were asked about the months of their birth and the following graph was prepared for the data so obtained: [Bar Graph Reference]

Observe the bar graph given above and answer the following questions: (i) How many students were born in the month of November? (ii) In which month were the maximum number of students born?

Solution: Note that the variable here is the ‘month of birth’, and the value of the variable is the ‘Number of students born’. (i) 4 students were born in the month of November. (ii) The Maximum number of students were born in the month of August.

This example illustrates how a bar graph can be used to analyze data. In this case, the data shows the birth months of students. By looking at the heights of the bars in the graph, one can quickly see how many students were born in each month and identify both the months with the fewest and most births.

Imagine you’re at a birthday party and you want to know when most of your friends were born. By using a bar graph, you can visualize how many friends have birthdays in each month just with a glance, easily spotting the busiest birthday month at the party.

Let us now recall how a bar graph is constructed by considering the following example. Example 2: A family with a monthly income of ₹20,000 had planned the following expenditures per month under various heads: [Expenditure Data Table]

Draw a bar graph for the data above.

This activity leads students through the steps of constructing a bar graph. The expenditure data provides various categories (like Grocery, Rent, etc.), and students learn how to represent these categories visually. They begin by labeling the axes, choosing a scale, and plotting the bars according to the expenditure values indicated.

Consider budgeting your pocket money for different activities like games, books, or snacks. If you make a bar graph showing how much money you plan to spend on each, it will help you visualize your budget effectively and make adjustments as needed.

(B) Histogram This is a form of representation like the bar graph, but it is used for continuous class intervals. For instance, consider the frequency distribution Table 12.2, representing the weights of 36 students of a class: [Weight Data Table].

A histogram is similar to a bar graph but specifically used for continuous data, where the data is grouped into ranges (or intervals) rather than distinct categories. The heights of the bars correspond to the frequency of data points within each interval, allowing for quick visual interpretation of the distribution of values.

Imagine measuring the heights of a class of students; instead of writing down every height, you group them into ranges (like 150-160 cm, 160-170 cm). A histogram would show how many students fall into each height range, giving you a clear picture of the height distribution in your class.

Now, consider a situation different from the one above. Example 3: A teacher wanted to analyse the performance of two sections of students in a mathematics test of 100 marks. [Test Score Data Table].

In this example, you see how to create a histogram when the intervals are not of uniform width. The key here is to ensure the areas of the bars represent the frequencies accurately even if the widths vary. This is vital to avoid any misleading interpretations of the data.

Think about measuring the time taken by runners over a race, where some runners take varying times to finish different segments. If you represent that on a graph, ensuring the length of each segment accurately represents the number of runners requires careful calculations, similar to the varying widths of intervals in this histogram.

(C) Frequency Polygon There is yet another visual way of representing quantitative data and its frequencies. This is a polygon. To see what we mean, consider the histogram represented by Fig. 12.3. Let us join the mid-points of the upper sides of the adjacent rectangles of this histogram by means of line segments.

A frequency polygon is formed by connecting the midpoints of the tops of the bars in a histogram with straight lines. This creates a shape that effectively visualizes how data values are distributed, similar to how the histogram does but often providing a clearer indication of trends and patterns.

Think of a frequency polygon like the path of a roller coaster in an amusement park, where each peak and trough represents the data (like the number of students) at various intervals (like scores in a test). Just as the highs and lows give you an exciting overview of the ride, the peaks and valleys of a frequency polygon can provide an exciting picture of data trends.

Now, the question arises: how do we complete the polygon when there is no class preceding the first class? Let us consider such a situation. Example 4: Consider the marks, out of 100, obtained by 51 students of a class in a test. [Marks Data Table].

When creating a frequency polygon where there's no class interval before the first, we extend the horizontal axis backward to create an imaginary class with zero frequency. This practice ensures that the polygon accurately reflects the overall shape and area relative to the original histogram.

Imagine you're drawing a line on a graph that represents temperatures over a day. If a certain hour of the day had no data recorded, you can still put a point at that hour to show that the temperature there was zero, maintaining the flow of your graph, just like how we manage continuity for our frequency polygons.

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