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VCE 11 General 2023

1.02 Categorical data

Lesson

Introduction

In the world of data, often there is an interest in the number of times, or frequency, that something occurs. It could be the number of road accidents caused by drink driving, the number of hot days in a year, or the number of visits to a website in a month.

In situations like these, where the same data value can occur multiple times, the data can be organised into a frequency table.

Categorical data and frequency tables

As an example, say that the colour of every car that passed though a given intersection was recorded over a ten minute period:\text{green,\, white,\, yellow,\, white,\, black,\, green,\, black,\, blue,\, blue,\, silver,\, white,\, black,\, green,\, blue,\, blue,}\\ \text{white,\, black,\, silver,\, silver,\, red,\, red,\, red,\, black,\, white,\, blue,\, white,\, black,\, silver,\, silver,\, white,\, blue,}\\ \text{ white,\, black,\, yellow,\, blue,\, white,\, white,\, red,\, green,\, silver,\, black,\, white,\, black,\, white.}

The same colours are occurring multiple times, so it makes sense to organise the data using a frequency table.

A frequency table and a column graph both showing the data of vehicle colours. Ask your teacher for more information.

Notice that the frequency table has three columns:

  • The first column shows the subgroups within the data

  • The tally column (optional) uses tally marks to record the frequency of each subgroup

  • The final column sums the tally marks and records the frequency as a number

The sum of the frequencies is equal to the total number of data values. In this case, the colours of 44 vehicles were recorded.

The graph on the right is a column graph, which will be discussed further below. Column graphs are well suited for displaying frequencies for categorical data. The table and graph allow us to analyse the frequency distribution - how the frequency of outcomes is spread across the different categories.

Examples

Example 1

Mr. Rodriguez recorded the number of pets owned by each of the students in his class. He found that 15 people had no pets, 19 people had one pet, 3 people had two pets and 8 people had three pets.

Write Mr. Rodriguez's results in the frequency table below.

Number of petsFrequency
0
1
2
3
Worked Solution
Create a strategy

'Frequency' means the number of people. Use the given information to fill this column in.

Apply the idea
Number of petsFrequency
015
119
23
38
Idea summary

Frequency tables organise the same data value that occurs multiple times.

Column graphs

A column graph (or bar graph) is a graph used to display relative quantities of data in different categories. There are a wide range of topics where this type of graph can be useful.

A column graph showing the number of students with certain eye colours. Ask your teacher for more information.

For example, the number of tourists visiting different destinations in a month, the number of people with certain eye colours in a class or the popularity of the top ten YouTube channels.

Column graph features to remember:

  • Title the graph clearly

  • Label each axis

  • Label each category and give the vertical axis a clear scale

  • Make sure the columns are equal widths and equally spaced

  • Make sure that the height of the column matches the value it represents which is often the frequency of the category (how often that outcome occurred)

Examples

Example 2

Miss Yen recorded her students' favourite pets in the column graph below.

A column graph showing the data of students favourite pets. Ask your teacher for more information.
a

Which type of pet is most popular?

Worked Solution
Create a strategy

Choose the pet that has the tallest column in the graph.

Apply the idea

The most popular type of pet means that most people picked it. The bird has the most picked and tallest column in the graph.

The most popular type of pet is bird.

b

Which pet is least popular?

Worked Solution
Create a strategy

Choose the pet that has the shortest column in the graph.

Apply the idea

The bird has the shortest column in the graph. The least popular type of pet is cat.

c

How many students picked a fish as their favourite pet?

Worked Solution
Create a strategy

Look across to the number that matches the height of the bar for fish.

Apply the idea

The height of the bar for fish goes up to the line next to the number 8 on the left.

8 students picked fish as their favourite pet.

d

How many students were surveyed in total?

Worked Solution
Create a strategy

Add the heights of each column.

Apply the idea
\displaystyle \text{Total student}\displaystyle =\displaystyle 6+8+2+6+14Add the heights
\displaystyle =\displaystyle 36Evaluate the addition

Example 3

John had 7 blue marbles, 4 black marbles, 8 yellow marbles, 13 white marbles, and 6 red marbles. Use this information to complete the column graph.

A blank graph showing the relationship between number of marbles and its colour. Ask your teacher for more information.
Worked Solution
Create a strategy

Look across to the number and match the height of the bar of each colour.

Apply the idea

Here's the complete column graph.

A column graph showing the number of marbles and its colour. Ask your teacher for more information.
Idea summary

Column graph features to remember:

  • Title the graph clearly

  • Label each axis

  • Label each category and give the vertical axis a clear scale

  • Make sure the columns are equal widths and equally spaced

  • Make sure that the height of the column matches the value it represents which is often the frequency of the category (how often that outcome occurred)

Outcomes

U1.AoS1.2

the concept of a data distribution and its display using a statistical plot

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