In our daily life we come across a huge amount of information. Newspapers, television, the internet and school records all provide us with data - about weather, marks, populations, cricket scores and so on. This raw information, when it is gathered, is called data. However, data by itself is often confusing and hard to understand. Data handling is the branch of mathematics which helps us organise, present and interpret data so that it becomes meaningful and useful for making decisions.
In this chapter we will learn how to collect and organise data in frequency distribution tables, how to represent data pictorially using bar graphs, histograms and pie charts, and how to interpret these pictures. We will also learn about probability, the chance or likelihood of an event happening, and how to calculate the probability of simple events. These skills are useful not only in mathematics but also in science, economics, sports and every field which deals with information.
Raw data means data which is collected in its original form, without any arrangement. To make sense of it, we organise it. One of the most common ways is to arrange it in ascending or descending order. We can also group the data into classes.
When data is grouped into classes, the number of items (observations) falling in each class is called the frequency of that class. A table showing the classes and their frequencies is called a frequency distribution table. The difference between the upper and lower class limits is called the class size, and the midpoint of a class is called its class mark.
For example, if marks of 20 students are grouped as 0-10, 10-20, 20-30 and so on, then the number of students in each group is its frequency.
A bar graph is a pictorial representation using rectangular bars. The bars are of equal width, and their heights (or lengths) are proportional to the values they represent. Bar graphs are useful to compare data of different categories.
A histogram is a graph for grouped (continuous) data. It consists of adjacent rectangles, where the class intervals are taken on the horizontal axis and the frequencies on the vertical axis. Unlike a bar graph, there is no gap between the bars in a histogram because the data is continuous.
A pie chart is a circle divided into sectors. Each sector shows the fraction of the total represented by that category. The central angle of a sector is calculated as: $$\text{Central angle} = \frac{\text{Value of the item}}{\text{Total value}} \times 360^\circ$$
For example, if a student spends 6 hours of a 24-hour day sleeping, the sector for sleeping has central angle (6/24) x 360 = 90 degrees.
Probability is the measure of the chance of an event happening. If there are several equally likely outcomes of an experiment, then: $$\text{Probability of an event} = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$$
The probability of an event always lies between 0 and 1. A probability of 0 means the event cannot happen, and a probability of 1 means the event is certain to happen.
When a coin is tossed, the total possible outcomes are Head (H) and Tail (T). The probability of getting a head is 1/2. When a die is rolled, the total possible outcomes are 1, 2, 3, 4, 5 and 6, so the probability of getting a 4 is 1/6.
To construct a histogram, mark the class intervals on the horizontal axis and the frequencies on the vertical axis, then draw adjacent rectangles for each class. To construct a pie chart, calculate the central angle of each sector using the formula above, divide the circle into the required sectors and label them with their percentages or values.
| Type of graph | Used for | Special feature |
|---|---|---|
| Bar graph | Comparing categories | Bars of equal width, gaps between bars |
| Histogram | Grouped continuous data | No gaps between adjacent rectangles |
| Pie chart | Parts of a whole | Sectors show central angles out of 360 degrees |
| Line graph | Trend over time | Points joined by straight lines |
| Experiment | Total outcomes | Example event | Probability |
|---|---|---|---|
| Toss of a coin | 2 (H, T) | Getting a tail | 1/2 |
| Roll of a die | 6 (1 to 6) | Getting an even number | 3/6 = 1/2 |
| Roll of a die | 6 | Getting a 5 | 1/6 |
| Tossing two coins | 4 (HH, HT, TH, TT) | Getting two heads | 1/4 |
Data handling teaches us how to deal with the information that surrounds us. In this chapter we learnt how to organise raw data into frequency distribution tables, how to present it graphically using bar graphs, histograms and pie charts, and how to interpret such pictures to draw conclusions. We also studied the basic ideas of probability, learning how to calculate the chance of an event using the ratio of favourable outcomes to total possible outcomes, and remembering that probability always lies between 0 and 1. These skills are widely used in statistics, science and daily decision-making, and they form the foundation of the statistical thinking that students will develop further in higher classes.