Lesson 1 ~25 min Unit 4 · Data & Chance +85 XP

Types of Data

Categorical vs numerical — understand what your data means before you do any maths with it.

Today's hook: Ever tried to find the “average” favourite colour in your class? You’d be stuck — because colours are categories, not numbers! Data comes in different types, and knowing which type you have decides EVERYTHING about what maths you can do with it.

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Think First

warm-up

Before you read on — look at these four pieces of information: eye colour, number of siblings, temperature in °C, favourite sport. Which ones are numbers? Which are just labels? Try sorting them, then read on to check.

Record your answer in your workbook.

The Big Idea

+5 XP

Data is information we collect. All data belongs to one of two main types: categorical (labels or groups) or numerical (actual numbers). Numerical data splits further into discrete (counted, whole numbers only) and continuous (measured, can take any value in a range). Knowing the type decides which graphs and statistics you can use.

Think of it as a tree: Data branches into Categorical and Numerical. Numerical then branches into Discrete (count) and Continuous (measure). You can’t find the mean of eye colours, but you can find the mean of heights — the type determines the maths.

Data → Categorical or Numerical → Discrete or Continuous

Can you do maths on it?

If adding or averaging makes sense, it’s numerical. If not, it’s categorical.

Count vs measure

Discrete = count (whole numbers). Continuous = measure (any decimal possible).

Type shapes your graph

Categorical → bar chart. Continuous → histogram or line graph.

What You'll Master

objectives

Know

The two main types of data: categorical and numerical
That numerical data is either discrete or continuous
Common examples of each type from everyday life

Understand

Why the type of data determines which maths tools you can use
The difference between counting and measuring
Why some numbers (like postcodes) are actually categorical

Can Do

Classify any variable as categorical, discrete, or continuous
Explain your reasoning using the count/measure test
Identify real-world examples of each data type

Words You Need

vocabulary

Categorical dataData that represents labels, names or groups. You cannot do arithmetic with it.

Numerical dataData that represents actual numbers you can add, subtract and average.

Discrete dataNumerical data that can only take specific (usually whole-number) values. You count it.

Continuous dataNumerical data that can take any value in a range (including decimals). You measure it.

VariableAny characteristic we collect data about (e.g. height, eye colour, number of pets).

Data setA collection of data values for one or more variables.

Spot the Trap

heads-up

Wrong: “Postcode 2060 is a number, so it’s numerical data.” You cannot meaningfully average postcodes — 2060 is just a label for a location.

Right: Ask “Does arithmetic make sense?” Averaging postcodes is meaningless, so postcodes are categorical even though they look like numbers.

Wrong: “Shoe size is continuous because it’s a number.” Shoe sizes come in fixed steps (6, 6.5, 7, 7.5 …) — you can’t have shoe size 6.372.

Right: Shoe size is discrete (countable steps). Height in cm is continuous (a person could be 163.47 cm).

Categorical Data

+5 XP

Categorical data places each observation into a category or group. The categories are labels — you can count how many fall in each group, but you cannot add, subtract or average the labels themselves.

Examples: Favourite subject (Maths, English, Science …), eye colour (blue, brown, green), type of pet (dog, cat, fish). You can say “12 students chose Maths” but you cannot say “the average favourite subject is 2.4.” That is meaningless. Categorical data is displayed with bar charts or pie charts — never a histogram.

Categorical = groups/labels, not arithmetic

Ask: label or number?

If you’re sorting into named groups, it’s categorical.

Count, don’t average

You can count frequencies of each category, but you can’t find a mean.

Best graph: bar chart

Use a bar chart or pie chart for categorical data.

Discrete vs Continuous Numerical Data

+5 XP

Both types are numerical, but they differ in what values are possible. Discrete data can only take specific values (usually whole numbers — you count it). Continuous data can take any value in a range (you measure it, and decimals are always possible).

Discrete examples: number of students in a class (27, 28, 29 — never 27.5), goals scored in a match (0, 1, 2, 3 …). Continuous examples: height in cm (163.47 cm is possible), time to run 100 m (12.83 s is possible). The key test: could the value be a decimal with unlimited precision?

Discrete: count it (whole steps). Continuous: measure it (any decimal).

The half-test

Can you have half a value? If yes, it’s continuous. “Half a sibling” makes no sense → discrete.

Instruments measure continuous

Rulers, scales, thermometers and stopwatches give continuous data.

Discrete can have gaps

A number line for discrete data has dots with spaces; continuous has a solid line.

Classifying Variables

+5 XP

To classify a variable, apply a three-step check: (1) Is it a label/group? → Categorical. (2) Is it a number you count? → Discrete. (3) Is it a number you measure? → Continuous. When in doubt, ask “Does averaging this make sense?” and “Could it be a decimal?”

Let’s classify some common variables: Blood type (A, B, AB, O) → labels → Categorical. Number of cars in a household → counted, whole number → Discrete. Mass of a student’s school bag in kg → measured, 4.73 kg is possible → Continuous. The three-step check never fails!

Label → Categorical | Count → Discrete | Measure → Continuous

Always justify

Don’t just name the type — say why (e.g. “you count it, so discrete”).

Numbers can be categorical

Jersey numbers, postcodes, phone numbers — all categorical despite looking numerical.

Use the flowchart

The three-step check works for every variable you’ll ever encounter.

Watch Me Solve It · 3 examples

Watch Me Solve It · Classify 5 variables

+15 XP per step

PROBLEM

Classify each variable: (a) hair colour, (b) number of siblings, (c) temperature in °C, (d) postcode, (e) time to run 400 m.

1

Apply the label test to (a) and (d)

(a) Hair colour: brown, blonde, red — these are labels → Categorical
(d) Postcode: 2060, 3000 — labels for locations, arithmetic is meaningless → Categorical

Even though postcodes are digits, averaging them makes no sense.
2

Apply the count test to (b)

(b) Number of siblings: 0, 1, 2, 3 — you count it, whole numbers only → Discrete

You can’t have 1.7 siblings. Gaps exist between values.
3

Apply the measure test to (c) and (e)

(c) Temperature: 22.5°C is possible → Continuous
(e) Time: 63.47 s is possible → Continuous

Thermometers and stopwatches give continuous readings.

Answers(a) Categorical (b) Discrete (c) Continuous (d) Categorical (e) Continuous

Watch Me Solve It · Discrete vs continuous

+15 XP per step

PROBLEM

Is “mass of a loaf of bread in grams” discrete or continuous? Justify your answer.

1

Apply the arithmetic test

Mass in grams is a number → it is numerical data (not categorical)

You can find the mean mass of many loaves — arithmetic makes sense.
2

Apply the half-test

Can the mass be 700.43 g? Yes! A scale can read to many decimal places.
3

Conclude

Mass is measured (not counted) and any decimal is theoretically possible → Continuous

Even if the label says “700 g”, the actual mass could be 698.7 g — measurement is continuous.

AnswerContinuous — mass is measured and can take any decimal value.

Watch Me Solve It · Why can’t you average eye colour?

+15 XP per step

PROBLEM

A student says: “I’ll find the average eye colour of my class.” Explain why this is impossible and what they could find instead.

1

Identify the data type

Eye colour = categorical (blue, brown, green, hazel — labels, not numbers)

You cannot assign a number to “blue” that has real mathematical meaning.
2

Explain why mean is impossible

Mean = (sum of values) ÷ n. You cannot add “blue + brown + blue” — addition only works on numbers.
3

Suggest what they CAN do

Find the mode (most common eye colour) and the frequency of each colour.

Mode and frequency work for categorical data. Mean and median do not.

AnswerEye colour is categorical — mean is impossible. Find the mode and frequency of each colour instead.

Common Pitfalls

heads-up

Treating categorical data as numerical

Students sometimes assign numbers to categories (1 = blue, 2 = brown) and try to find the mean. The mean of those assigned numbers is meaningless — it depends entirely on which number you assigned to which label.

Fix: Only use mean, median and mode for genuine numerical data. For categories, use mode and frequency tables.

Confusing discrete and continuous

If data is recorded as whole numbers, students assume it’s discrete. But age recorded as “14 years” is still continuous — your actual age is 14.37 years; we just round it.

Fix: Think about the underlying variable, not how it’s recorded. Ask “Could this theoretically be a decimal?”

Thinking all numbers are numerical data

Jersey numbers, phone numbers, postcodes and student ID numbers look numerical but are actually categorical. “Player 7 scored” — you wouldn’t add jersey numbers to find the average player.

Fix: Ask “Does adding or averaging this number make real-world sense?” If not, it’s categorical.

Copy Into Your Books

Data Type Tree

Data → Categorical or Numerical
Numerical → Discrete or Continuous
Categorical: labels/groups, no arithmetic

Classification Test

Label/group? → Categorical
Count it (whole numbers)? → Discrete
Measure it (decimals possible)? → Continuous

Discrete Examples

Number of siblings, goals scored
Number of students in class
Pages in a book

Continuous Examples

Height, mass, temperature
Time, distance, volume
Any measurement from an instrument

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Types of Data

4 problems

Four drill problems. Classify each variable and give a one-sentence reason.

1 Classify: age in years (e.g. 13, 14, 15).

Continuous — your actual age is e.g. 13.74 years (a decimal is always possible). We round it when we say “13 years old”, but age itself is measured continuously.Continuous numerical data
2 Classify: favourite music genre (pop, hip-hop, rock, classical).

Categorical — genres are labels/groups. You cannot add “pop + rock” and you cannot find a mean genre.Categorical data
3 Classify: temperature in °C recorded by a weather station.

Continuous — temperature is measured (not counted) and a value like 23.6°C is perfectly possible.Continuous numerical data
4 Classify: number of siblings a student has.

Discrete — you count siblings and you can only have whole numbers. Having 1.5 siblings is impossible.Discrete numerical data

Complete in your workbook.

Quick Check · 5 questions

What type of data is “eye colour”?

+10 XP

The number of goals scored in a soccer match is …

+10 XP

Height in centimetres is best classified as …

+10 XP

A student’s postcode (e.g. 2060) is …

+10 XP

Which measure of centre can be used for categorical data?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Classify each of the following as categorical, discrete or continuous: (i) blood type, (ii) number of books read this year, (iii) distance from school in km. Justify each answer.

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A researcher records the “jersey number” worn by each player in a sports team. Is this categorical or numerical? Explain.

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A student surveys her class and collects the following data: reaction time in milliseconds, number of languages spoken, favourite season, and whether they own a pet (yes/no). Classify each variable and explain which graph type would best display each one.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — Categorical. Eye colour is a label (blue, brown, green).

2. C — Discrete. Goals are counted in whole numbers only.

3. A — Continuous. Height is measured and can take any decimal value.

4. D — Categorical. Postcodes are location labels — arithmetic on them is meaningless.

5. B — Mode. Only mode works for categorical data.

Show Your Working Model Answers

Q6 (3 marks): (i) Blood type (A, B, AB, O) — categorical: it’s a label [1]. (ii) Number of books — discrete: you count whole books, can’t read 2.4 books [1]. (iii) Distance in km — continuous: measured, 3.47 km is possible [1].

Q7 (2 marks): Jersey numbers are categorical [1]. Although they are digits, averaging jersey numbers (e.g. (7 + 11 + 3) ÷ 3 = 7) tells us nothing meaningful about the players — the numbers are just labels [1].

Q8 (4 marks, 1 per variable): Reaction time in ms — continuous (measured); histogram or line graph. Languages spoken — discrete (counted, whole numbers); bar chart or dot plot. Favourite season — categorical (labels); bar chart or pie chart. Pet ownership (yes/no) — categorical (two categories); bar chart or pie chart.

Stretch Challenge · +25 XP, +10 coins

The Data Trap

A supermarket collects: (a) product barcode number, (b) price in dollars, (c) number of units sold per day, (d) product category (dairy, bakery, produce). Classify all four variables. Then explain: barcodes and prices are both numbers — why is one categorical and one numerical? Use the correct mathematical reasoning.

Reveal solution

(a) Barcode — categorical: it’s a label identifying the product; averaging barcodes is meaningless. (b) Price — continuous numerical: $4.73 is a valid price; you can find the average price of all products. (c) Units sold — discrete numerical: you count whole units; 47.3 units sold per day makes no sense. (d) Product category — categorical: dairy/bakery/produce are labels. The key distinction for barcode vs price: arithmetic on price (mean, total revenue) is meaningful; arithmetic on barcodes is not — the number is just an identifier.

Quick Review

Two main types

Categorical (labels) and Numerical (numbers)

Numerical splits

Discrete (count) and Continuous (measure)

Categorical test

Does arithmetic (mean, sum) make sense? No → categorical

Discrete test

Count it, whole numbers only, gaps between values

Continuous test

Measure it, any decimal possible, solid number line

Numbers can be categorical

Postcodes, jersey numbers, phone numbers are labels

Interactive: Data Type Sorter

Drag variables into the correct category (Categorical, Discrete, Continuous) and get instant feedback on your reasoning.

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