Data Presentation with Tables, Charts and Graphs for SS 1

Data Presentation

Performance Objectives

By the end of the lesson, students should be able to:

1. collect, tabulate and present data to give understandable information;
2. prepare frequency distribution table;
3. plot line graph, bar graph and histogram with given data;
4. differentiate between bar chart and histograms;
5. calculate the sectorial component of pie chart;
6. draw pie chart correctly;
7. interpret the pie chart;
8. draw pie charts and frequency polygon of real-life data set.

Introduction to Data

Data refers to a collection of facts, figures, observations, or information that can be processed and analysed. These facts can be numbers, words, measurements, or descriptions of things.

In everyday life, people encounter data constantly. For example, the number of students in a class, the types of fruits sold in a market, or the scores obtained by a football team are all forms of data.

Why Present Data?

Raw data, when collected, is often disorganised and difficult to understand. Presenting data means arranging it in a clear, organised, and visual way so that it can be easily understood, analysed, and interpreted. This helps in making informed decisions and drawing conclusions.

For instance, if a school principal wants to know the most popular sports among students, simply listing every student’s favourite sport would be confusing. Presenting this information in a chart or table makes it easy to see which sports are most popular at a glance.

Steps in Data Presentation

The process of presenting data typically involves three main steps:

1. Collection: Gathering the raw facts and figures from various sources. This can involve surveys, observations, or experiments.
2. Tabulation: Organising the collected raw data into tables, often using tally marks and frequencies, to make it more structured.
3. Presentation: Displaying the tabulated data using various graphical methods such as graphs and charts, which make the information visually appealing and easy to interpret.

Data Collection and Tabulation

Data collection is the first step in any statistical analysis. It involves systematically gathering information relevant to a specific topic or question.

Methods of Data Collection

Common methods of collecting data include:

1. Observation: Watching and recording events or behaviours as they happen. For example, observing the number of cars passing a certain point in an hour.
2. Surveys/Questionnaires: Asking a set of questions to a group of people to gather their opinions or information. For example, asking students about their favourite subjects.
3. Experiments: Conducting controlled tests to gather data on cause-and-effect relationships.

Preparing a Frequency Distribution Table

After collecting raw data, the next step is to organise it into a frequency distribution table. This table shows how often each item or value appears in a dataset. It makes the data much easier to read and understand.

A frequency distribution table typically has three columns:

1. Category/Item: The different types of data or values.
2. Tally: Marks used to count the occurrences of each item. A vertical line is drawn for each occurrence, and every fifth line is drawn diagonally across the previous four to form a group of five.
3. Frequency: The total number of times each item or value appears, obtained by counting the tally marks.

The procedure for preparing a frequency distribution table involves the following steps:

1. List all the distinct categories or items observed in the data.
2. Go through the raw data item by item. For each item, place a tally mark ( | ) next to its corresponding category in the table.
3. For every fifth tally mark in a category, draw it diagonally across the previous four ( |||| ) to make counting easier.
4. After all data items have been tallied, count the tally marks for each category and write the total in the ‘Frequency’ column.
5. Sum the frequencies to ensure it matches the total number of data items collected.

Example 1: Preparing a Frequency Distribution Table

A teacher asked 20 students about their favourite colours. The responses were:

Red, Blue, Green, Red, Yellow, Blue, Red, Green, Blue, Red, Yellow, Green, Blue, Red, Green, Yellow, Blue, Red, Green, Blue.

Prepare a frequency distribution table for this data.

Solution:

First, list the distinct colours: Red, Blue, Green, Yellow.

Then, go through the data and make tally marks:

Types of Data Presentation

Once data is collected and tabulated, it can be presented visually using various graphs and charts. These visual representations help to highlight patterns, trends, and comparisons within the data.

Line Graph

A line graph is used to show how data changes over a continuous period, such as time. It connects data points with straight lines, making it easy to see trends, increases, or decreases.

The procedure for plotting a line graph involves the following steps:

1. Draw two perpendicular axes: a horizontal axis (x-axis) and a vertical axis (y-axis).
2. Label the horizontal axis with the independent variable (e.g., time, months, years).
3. Label the vertical axis with the dependent variable (e.g., temperature, sales, population).
4. Choose an appropriate scale for both axes so that all data points can fit and the graph is clear.
5. Plot each data point on the graph according to its x and y values.
6. Connect the plotted points with straight lines.

Example 2: Plotting a Line Graph

The table below shows the temperature in a city over 6 hours:

Plot a line graph for the given data.

Solution:

Bar Graph (Bar Chart)

A bar graph uses rectangular bars of equal width to represent data. The length or height of each bar is proportional to the value it represents. Bar graphs are useful for comparing discrete categories of data.

The procedure for plotting a bar graph involves the following steps:

1. Draw two perpendicular axes: a horizontal axis (x-axis) and a vertical axis (y-axis).
2. Label the horizontal axis with the categories (e.g., types of fruits, names of students).
3. Label the vertical axis with the frequency or value (e.g., number of fruits, scores).
4. Choose an appropriate scale for the vertical axis.
5. Draw bars of equal width for each category. The height of each bar corresponds to its frequency or value.
6. Leave equal spaces between the bars.

Example 3: Plotting a Bar Graph

Using the favourite colours data from Example 1:

Plot a bar graph for this data.

Solution:

Histogram

A histogram is similar to a bar graph but is used to represent continuous data, typically grouped into class intervals. In a histogram, the bars are drawn adjacent to each other without any gaps, indicating the continuous nature of the data. The width of each bar represents the class interval, and the height represents the frequency of that interval.

The procedure for plotting a histogram involves the following steps:

1. Draw two perpendicular axes: a horizontal axis (x-axis) and a vertical axis (y-axis).
2. Label the horizontal axis with the class boundaries or class intervals of the continuous data.
3. Label the vertical axis with the frequency.
4. Choose an appropriate scale for both axes.
5. Draw rectangular bars for each class interval. The width of each bar should correspond to the width of the class interval.
6. The height of each bar should correspond to the frequency of that class interval.
7. Ensure there are no gaps between adjacent bars, as the data is continuous.

Example 4: Plotting a Histogram

The table below shows the heights of 30 students in a class:

Plot a histogram for this data.

Solution:

First, determine the class boundaries. For the interval 140-149, the lower class boundary is 139.5 and the upper class boundary is 149.5. Similarly for other intervals.

The class boundaries are: 139.5, 149.5, 159.5, 169.5, 179.5.

Differentiating Between Bar Charts and Histograms

While both bar charts and histograms use bars to represent data, there are key differences between them:

Pie Chart

A pie chart is a circular graph divided into sectors, where each sector represents a proportion of the whole. The entire circle represents 100% of the data, and the size of each sector (its angle at the centre) is proportional to the frequency or value it represents. Pie charts are effective for showing the relative sizes of different categories within a whole.

Calculating the Sectorial Component of a Pie Chart

To draw a pie chart, the first step is to calculate the angle for each sector. The total angle in a circle is $360^\circ$. The angle for each category is found using the formula:

$$ \text{Angle for a category} = \frac{\text{Value of category}}{\text{Total value}} \times 360^\circ $$

Example 5: Calculating Sectorial Components

A survey of 100 students showed their favourite fruits:

1. Mango: 40 students
2. Orange: 25 students
3. Banana: 20 students
4. Apple: 15 students

Calculate the angle for each sector in a pie chart.

Solution:

Total number of students = $40 + 25 + 20 + 15 = 100$

1. Mango:
   $$ \text{Angle for Mango} = \frac{40}{100} \times 360^\circ = 0.4 \times 360^\circ = 144^\circ $$
2. Orange:
   $$ \text{Angle for Orange} = \frac{25}{100} \times 360^\circ = 0.25 \times 360^\circ = 90^\circ $$
3. Banana:
   $$ \text{Angle for Banana} = \frac{20}{100} \times 360^\circ = 0.2 \times 360^\circ = 72^\circ $$
4. Apple:
   $$ \text{Angle for Apple} = \frac{15}{100} \times 360^\circ = 0.15 \times 360^\circ = 54^\circ $$

Check: $144^\circ + 90^\circ + 72^\circ + 54^\circ = 360^\circ$. The sum of the angles should always be $360^\circ$.

Drawing a Pie Chart

The procedure for drawing a pie chart involves the following steps:

1. Calculate the angle for each category as shown above.
2. Draw a circle using a compass.
3. Draw a radius from the centre of the circle to the circumference. This will be the starting line for the first sector.
4. Using a protractor, measure the angle for the first category from the starting radius and draw another radius.
5. Continue measuring and drawing radii for each subsequent category, starting from the end of the previous sector.
6. Label each sector clearly with its category name and its percentage or value. Different colours can be used for each sector for better clarity.

Example 6: Drawing a Pie Chart

Draw a pie chart for the favourite fruits data from Example 5.

Solution:

The calculated angles are: Mango ($144^\circ$), Orange ($90^\circ$), Banana ($72^\circ$), Apple ($54^\circ$).

Interpreting a Pie Chart

Interpreting a pie chart involves understanding the proportions represented by each sector and drawing conclusions about the data. Key aspects of interpretation include:

1. Largest/Smallest Category: Identify which category has the largest or smallest share of the total. This corresponds to the largest or smallest sector.
2. Comparison: Compare the sizes of different sectors to understand their relative importance or frequency.
3. Overall Distribution: Get a quick overview of how the whole is divided among the different categories.

Example 7: Interpreting a Pie Chart

From the pie chart in Example 6 (Favourite Fruits):

1. Which fruit is the most popular?
2. Which fruit is the least popular?
3. How many times more popular is Mango than Apple?

Solution:

1. Most popular fruit: Mango, as it has the largest sector ($144^\circ$) and the highest frequency (40 students).
2. Least popular fruit: Apple, as it has the smallest sector ($54^\circ$) and the lowest frequency (15 students).
3. Comparison of Mango and Apple:

   Mango frequency = 40 students

   Apple frequency = 15 students

   Ratio = $\frac{40}{15} = \frac{8}{3} \approx 2.67$ times.

   Mango is approximately 2.67 times more popular than Apple.

Frequency Polygon

A frequency polygon is a line graph that represents the frequency distribution of continuous data. It is constructed by plotting points at the midpoints of the top of each bar in a histogram and then connecting these points with straight lines. To close the polygon, points are added on the x-axis at the midpoints of the class intervals immediately before the first and after the last, with zero frequency.

The procedure for drawing a frequency polygon involves the following steps:

1. Prepare a frequency distribution table with class intervals and frequencies.
2. Calculate the midpoint (class mark) for each class interval. The midpoint is found by adding the lower and upper class limits and dividing by 2.
3. Draw two perpendicular axes. Label the horizontal axis with the midpoints of the class intervals. Label the vertical axis with the frequency.
4. Plot points corresponding to the frequency of each class interval at its midpoint.
5. To close the polygon, add two extra midpoints on the x-axis, one before the first class interval and one after the last, each with a frequency of zero.
6. Connect all the plotted points (including the zero-frequency points) with straight lines.

Example 8: Drawing a Frequency Polygon

Using the heights of 30 students data from Example 4:

Draw a frequency polygon for this data.

Solution:

First, calculate the midpoints for each class interval:

1. 140-149: Midpoint = $\frac{140+149}{2} = \frac{289}{2} = 144.5$
2. 150-159: Midpoint = $\frac{150+159}{2} = \frac{309}{2} = 154.5$
3. 160-169: Midpoint = $\frac{160+169}{2} = \frac{329}{2} = 164.5$
4. 170-179: Midpoint = $\frac{170+179}{2} = \frac{349}{2} = 174.5$

To close the polygon, we need midpoints for the intervals before 140-149 and after 170-179. The class width is 10. So, the midpoint before 144.5 would be $144.5 – 10 = 134.5$. The midpoint after 174.5 would be $174.5 + 10 = 184.5$. These points will have a frequency of 0.

The points to plot are:

1. (134.5, 0)
2. (144.5, 5)
3. (154.5, 10)
4. (164.5, 8)
5. (174.5, 7)
6. (184.5, 0)

Drawing Pie Charts and Frequency Polygons of Real-Life Data Set

Applying these data presentation techniques to real-life data helps in understanding practical situations. For example, a teacher might collect data on the number of boys and girls in a class, or the types of soft drink corks collected by students, and then use this data to draw charts and graphs.

Example 9: Real-Life Data – Number of Boys and Girls in a Class

In an SS1 class, there are 25 boys and 35 girls.

1. Draw a pie chart to represent this data.
2. Draw a bar chart to represent this data.

Solution:

Total number of students = $25 + 35 = 60$

Part 1: Pie Chart

Calculate the angles for each sector:

1. Boys:
   $$ \text{Angle for Boys} = \frac{25}{60} \times 360^\circ = \frac{5}{12} \times 360^\circ = 5 \times 30^\circ = 150^\circ $$
2. Girls:
   $$ \text{Angle for Girls} = \frac{35}{60} \times 360^\circ = \frac{7}{12} \times 360^\circ = 7 \times 30^\circ = 210^\circ $$

Check: $150^\circ + 210^\circ = 360^\circ$.

Part 2: Bar Chart