Classifying and Displaying Categorical Data (VCE SSCE General Mathematics): Revision Notes

Classifying and Displaying Categorical Data

Introduction to variables and data

When we collect information through surveys or experiments, we are gathering data about different variables. A variable is any characteristic that can change or vary from one person to another.

Consider a student survey that asks about:

• Gender (male or female)
• Preferred coffee size (small, medium, or large)
• Number of brothers
• Hand span in centimetres

Each of these questions represents a different variable because the answers will vary from student to student. The actual values we collect are called data.

These variables, and the data they produce, can be sorted into two main types: categorical or numerical.

Understanding categorical data

Categorical data sorts people or items into groups or categories. When you collect categorical data, you are essentially placing each person into one of several possible groups.

However, not all categorical data works the same way. There are two subtypes:

Nominal data

Nominal data uses values that are simply names or labels. These labels identify which category something belongs to, but they don't tell us anything more than that.

Example: Gender recorded as 'M' or 'F'

The letters M and F identify whether someone is male or female, but they don't imply any ordering or ranking. We can't say that one is "more" or "less" than the other in any meaningful way. The values are just names.

Other examples of nominal data:

• Favourite colour (red, blue, green, yellow)
• Type of pet (dog, cat, bird, other)
• Country of birth

Ordinal data

Ordinal data does two things: it names categories AND allows us to order or rank them.

Example: Coffee size recorded as 'small', 'medium', or 'large'

These values not only tell us the size category, but we can also arrange them in order from smallest to largest. There is a natural ranking to the categories.

Other examples of ordinal data:

• Level of agreement (strongly disagree, disagree, neutral, agree, strongly agree)
• Education level (primary, secondary, tertiary)
• Satisfaction rating (very unsatisfied, unsatisfied, neutral, satisfied, very satisfied)

The key difference is that ordinal data has a meaningful order, while nominal data does not.

Understanding numerical data

Numerical data consists of values that represent quantities. These are numbers that you can perform arithmetic operations on, such as adding, subtracting, or finding an average. Like categorical data, numerical data has two subtypes:

Discrete data

Discrete data can only take specific, separate values. These values are typically whole numbers that come from counting.

Example: Number of brothers ($0$, $1$, $2$, $3$, ...)

You can have $0$, $1$, or $2$ brothers, but you cannot have $1.5$ brothers. The values are distinct and separate from each other.

Other examples of discrete data:

• Number of students in a class
• Number of cars in a car park
• Number of goals scored in a match

Continuous data

Continuous data can take any value within a range. These values typically come from measurement and are limited only by the precision of your measuring instrument.

Example: Hand span measured in centimetres

If a student records their hand span as $24.1$ cm, their actual hand span could be anywhere between $23.05$ cm and $24.15$ cm, depending on how they measured and rounded. The variable could theoretically take any value within a certain range.

Other examples of continuous data:

• Height in centimetres
• Weight in kilograms
• Time in seconds
• Temperature in degrees

A note on interval and ratio scales

There is another way to classify numerical data, though for most statistical purposes this distinction is not critical:

Interval scales have equal differences between successive values, but zero does not mean the complete absence of what is being measured.

Example: Temperature in degrees Celsius. The difference between $13°\text{C}$ and $14°\text{C}$ is the same as between $14°\text{C}$ and $15°\text{C}$. However, $0°\text{C}$ does not mean a complete absence of heat. Also, we cannot say that $40°\text{C}$ is twice as hot as $20°\text{C}$.

Ratio scales have all the properties of interval scales, plus zero means complete absence, and we can make meaningful ratio statements.

Example: Number of brothers or hand span in centimetres. Zero brothers means no brothers at all. Someone with $4$ brothers has exactly twice as many brothers as someone with $2$ brothers.

For statistical analysis in this course, we treat both interval and ratio data the same way, so you don't need to distinguish between them.

Types of variables summary

Variables that produce categorical data are called categorical variables. These can be further classified as:

• Nominal variables (e.g., gender, hair colour)
• Ordinal variables (e.g., size preferences, satisfaction levels)

Variables that produce numerical data are called numerical variables. These can be further classified as:

• Discrete variables (e.g., number of siblings, number of students)
• Continuous variables (e.g., height, weight, time)

The following diagram shows how different variable types relate to each other:

Classifying variables: worked examples

Let's practice identifying different types of variables with some examples.

Frequency tables

Once we have collected categorical data, we need to organize it in a way that makes it easy to understand and analyze. Frequency tables are the most common way to do this.

What is a frequency table?

A frequency table lists the possible values of a variable along with how often each value occurs in the data set.

Frequency can be recorded in two ways:

Frequency: The actual count of how many times each value appears.

Percentage frequency: The percentage of times each value appears, calculated using:

$\text{Percentage frequency} = \frac{\text{count}}{\text{total}} \times 100$

A frequency distribution is simply a complete listing of all values and their frequencies.

Bar charts

While frequency tables summarize data effectively, visual displays often make patterns easier to spot. For categorical data, the appropriate visual display is a bar chart.

Features of bar charts

A bar chart has these key features:

• The vertical axis shows frequency or percentage frequency
• The horizontal axis shows the categories of the variable
• The height of each bar represents the frequency (or percentage) for that category
• Gaps between bars indicate that each value is a separate category
• There is one bar for each category

The mode or modal category

One important feature that bar charts help us identify quickly is the mode or modal category.

The mode is the category that occurs most frequently - the most common or popular value. In a bar chart, this is the category with the tallest bar.

In Example 5, the modal category is 'pie' because it was chosen by more children ($13$) than any other lunch option.

When is the mode useful?

The mode is most useful when:

• A single category clearly occurs more often than others
• You want to identify the most popular or common choice
• You're conducting popularity polls or market research

Examples of questions answered by the mode:

• "Which TV station is most frequently watched between 6 p.m. and 8 p.m.?"
• "When is a supermarket busiest?"
• "What is the most common shoe size sold?"

The mode tells us what is typical or most common in categorical data, helping us understand what most people choose or prefer.