Every day we are surrounded by information. The number of students in a class, the marks obtained in a test, the number of cars passing a road, and the favourite games of children are all examples of data. Data is a collection of facts or information gathered for a purpose. The branch of mathematics that deals with collecting, organising, presenting and interpreting data is called statistics, and in this chapter we learn its basic tools.
Raw data, as it is first collected, is often difficult to understand. To make sense of it, we organise the data into tables, draw pictures and graphs, and summarise it. A pictograph uses pictures or symbols to represent data, and a bar graph uses rectangular bars of different heights or lengths. These visual representations make data easy to read and compare at a glance.
In this chapter, we also learn to summarise data using averages. The mean is found by adding all the values and dividing by the number of values. The median is the middle value when the data is arranged in order. And the mode is the value that appears most often. These measures help us understand the typical or representative value of a collection of data.
Data can be collected by counting, measuring or asking questions. For example, to know the favourite sport of students in a class, we can ask every student and record the answers. The information collected in this way, in its original form, is called raw data.
For data to be useful, it must be collected carefully. The questions we ask should be clear, and we should record the answers accurately. Data collection is the first step of statistics. After collection, we organise the data so that patterns become visible.
Raw data is usually not organised. To organise it, we often use tally marks and frequency tables.
A tally mark table is a quick way to count. Each item is represented by a vertical stroke. The fifth stroke is drawn diagonally across the previous four to make a group of five. Groups of five make counting easy: we count in groups of five.
The number of times a particular value occurs in the data is called its frequency. A table that shows how often each value occurs is called a frequency table or tally chart.
For example, if the heights of 10 students are recorded, we can group the data and note how many students fall in each height range. Grouping data into intervals makes it easier to handle when the data has many different values.
A pictograph is a way of representing data using pictures or symbols. Each picture stands for a certain number of items. A key explains what one picture represents.
For example, if one picture of a bus represents 10 buses, then two buses pictures represent 20 buses. The key is very important because it tells us the value of one symbol. When drawing a pictograph, the symbols should be equal in size, and we must always include the key.
To interpret a pictograph, we count the symbols and multiply by the value given in the key. Pictographs make data interesting and easy to understand, especially for large amounts of information.
A bar graph is a graphical representation of data using rectangular bars. The bars can be drawn vertically or horizontally, and their heights or lengths are proportional to the values they represent.
To read a bar graph, look at the height of each bar and read its value on the y-axis. The tallest bar represents the largest value and the shortest bar represents the smallest value. A bar graph allows us to compare data quickly.
The scale is the number of units represented by one small division on the axis. Choosing a suitable scale is important so that the graph is neither too small nor too big. For example, if the data values are 50, 120 and 200, a scale of 1 unit = 20 may be suitable.
The mean (or average) of a set of observations is obtained by dividing the sum of all observations by the number of observations.
Mean = Sum of all observations / Number of observations
For example, the mean of 5, 7 and 9 = (5 + 7 + 9) / 3 = 21 / 3 = 7.
The median is the middle value when the observations are arranged in ascending (or descending) order.
For example, the median of 3, 7, 9, 11, 15 is 9 (the third value in the ordered list). The median of 2, 4, 6, 8 is (4 + 6) / 2 = 5.
The mode is the observation that occurs the most number of times. A set of data may have one mode, more than one mode, or no mode at all.
For example, in the data 2, 3, 3, 5, 3, 7, the mode is 3 because it appears three times, more often than any other value.
Different situations need different measures. The mode is useful for finding the most common item, like the most popular colour. The mean is useful for finding a fair average, like the average marks of a class. The median is useful when a few extreme values would distort the average, like the median income of a group.
Data handling is used in many real-life situations: a school records the number of students in each class, a shopkeeper records the sales of different items, a weather station records daily temperature, and the government records population figures. Presenting this data through tables and graphs helps people understand it quickly and make decisions.
| Term | Meaning | Example |
|---|---|---|
| Data | Collection of facts or information | Marks of students |
| Frequency | Number of times a value occurs | 5 appears 3 times |
| Pictograph | Data shown using pictures | One symbol = 10 students |
| Bar graph | Data shown using bars | Heights of bars show values |
| Measure | How to find | Example |
|---|---|---|
| Mean | Sum divided by number of observations | (5 + 7 + 9) / 3 = 7 |
| Median | Middle value in ordered data | 3, 7, 9, 11, 15 gives 9 |
| Mode | Most frequent value | 2, 3, 3, 5, 3 gives 3 |
Data Handling teaches us how to turn raw information into organised and meaningful knowledge. We learnt to collect and organise data using tally marks and frequency tables, and to represent it through pictographs and bar graphs. We also learnt the three measures of central tendency: mean, median and mode, which summarise data with a single representative value. These skills help us understand information and make decisions in everyday life and in all subjects that use data.