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Title: Mathspace VCE General Mathematics Units 1 & 2
Print run 1 - January 2026
ISBN: 978-1-963022-96-4
Editors and lead authors: Erin Gallagher, Jeremy Hunter, Elisha Templin, Nazeel Ali, Katrina Du, Neeti Meghani, Jessica Purton, Isha Tiwari
Writing and development team: Adam Humphreys, Ma Lourdes Ona, Sheramie Castillo, Regine Jara, Lesley Ann Lopez, Adriane Abunda, Jady Amarillas, Joy Ann Antonio, Macy Castillo, Christian Dinglasan, Adrian Doctolero, Dharent Fernandez, Dhave Fernandez, Rona Josue, Jessa Lambinicio, Eric Malabag, Julie Ann Manzano, Kyra Joy Manzano, Mary Joy Tacud, Albert Jara
Images and design team: Scott Nolan, Chastine Marquez, Keith Gimeno, Jasper Jumawan, Mary Matheu, Jemark Orevillo, Jessa Ortega
Number patterns and recurrence relations
Graphs and networks
Compare numerical variables
Shape and measurement
“Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.”
– Paul Lockhart
Big ideas
• Effective data analysis begins with classifying data by type (categorical or numerical) and organising it through appropriate visual displays to reveal initial patterns and frequencies.
• Key statistical measures are used to summarise and interpret the essential features of a dataset, including its central tendency, spread, and overall shape.
• Visual summaries, such as box plots, and comparative statistics enable the analysis of multiple datasets, allowing for clear interpretation of their similarities and differences in centre, spread, and distribution.
1 Data distributions
Chapter outline
Box plots help detect outliers! An outlier can be a surprising piece of data, like finding penguins in a grassy, rocky area–totally unexpected but real!
1.01 Types of data
After this lesson, you will be able to…
• distinguish between nominal, ordinal, discrete, and continuous data.
• classify real-world data sets into appropriate data types.
• compare and contrast the characteristics of nominal, ordinal, ratio, and interval scales.
Types of data
A statistical variable is a measurable or observable characteristic or attribute of an item or person. Statistical data are the actual values or observations collected for these variables. Data is collected to observe and analyse changes in these variables. For example:
Variable
Data
Height of students 162 cm, 168 cm, 173 cm, 183 cm
Type of animal Dog, Cat, Bird, Fish
Univariate data
Data relating to a single variable.
Statistical variables are classified as either numerical or categorical:
Numerical data
Data where each data point is represented by a number. Also known as quantitative data.
• Examples of numerical data include: the number of items sold each month, daily temperatures, heights of people, and ages of a population.
• The data can be further defined as either discrete (associated with counting) or continuous (associated with measuring).
Categorical data
Data where each data point is represented by a word or label. Also known as qualitative data.
• Examples of categorical data include: brand names, types of animals, favourite colours, and names of countries.
• The data can be further defined as either ordinal (it can be ordered) or nominal (unordered).
Discrete data
Numerical data with distinct, separate values. There is a definite ‘gap’ separating one data point from the next.
Discrete data usually, but not always, consists of whole numbers, and is often collected by some form of counting.
Examples of discrete data
Number of goals scored per match 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, . . .
Number of products sold each day 386, 397, 401, 410, 411, 422, 437, . . .
Shoe size 5, 5.5, 6, 7, 8.5, 9, 10, 10, 12.5
In each of these cases, there are no in-between values. For example, there is no way to score 2.5 goals. Discrete data can involve decimals as long as there is a definite gap between possible values. For example, a shoe will not ever come in size 8.145, so it is discrete.
Continuous data
Numerical data that can take any value within a range.
Continuous data often involves the use of decimal numbers, and is often collected using some form of measurement.
Examples of continuous data
Height of trees in a forest (in metres) 12.359, 14.022, 14.951, 18.276, 11.032, . . .
Times taken to run a 10 km race (minutes) 55.34, 58.03, 57.25, 61.49, 66.11, 59.87, . . .
Daily temperature (°C) 24.4, 23.0, 22.5, 21.6, 20.7, 20.2, 19.7, . . .
Since continuous data is typically measured, it is limited by the accuracy of the measuring device being used. So continuous data may appear discrete when it is rounded. For instance, it makes sense that time is continuous, even though it can only be measured to the nearest second (or millisecond, and so on).
Ordinal data
Categorical data with a natural order or rank.
The word ‘ordinal’ means ‘ordered’.
Examples of ordinal data
Product rating on a survey satisfactory, good, good, good, good, excellent, excellent, . . .
Exam grades C, C, C, C, B, B,
Size of fish in a lake small, small, small, medium, medium, medium, large, large, . . .
Nominal data
Categorical data with no inherent order.
The word ‘nominal’ basically means ‘name’.
Examples of nominal data
Nationalities in a team German, Austrian, Italian, Spanish, Dutch, Italian, . . .
Make of car driving through an intersection Toyota, Holden, Mazda, Toyota, Ford, Toyota, Mazda, . . .
Hair colour of students in a class blonde, red, brown, blonde, black, brown, black, red, . . .
Example 1
Which one of these variables is discrete?
A Height
C Daily temperature
Create a strategy
Choose the option that can be counted but is distinct and separate from one another.
B The time it takes to swim 200 metres
D The number of family pets
Apply the idea
The correct answer is option D: The number of family pets.
Classify this data into its correct category: Weights of kittens
A Numerical discrete
C Numerical continuous
Create a strategy
Determine if the data is represented by a number, or is separated into categories.
B Categorical nominal
D Categorical ordinal
Apply the idea
The weight of a kitten can be measured. So it is numerical or quantitative.
Weight is a measurement that can have any number of decimal places, so it is continuous. The correct answer is option C.
Example 2
Idea summary
A statistical variable is a measurable characteristic being studied. Data collected for one variable is called univariate data. Variables can be classified as either numerical or categorical.
• Numerical data (quantitative) consists of numbers. It is further divided into:
• Discrete data: Values are distinct and separate, often found by counting (e.g., shoe size).
• Continuous data: Can take any value within a range, often found by measuring (e.g., height).
• Categorical data (qualitative) consists of words or labels. It is further divided into:
• Ordinal data: Categories have a natural, meaningful order (e.g., product ratings).
• Nominal data: Categories have no inherent order (e.g., hair colour).
Level of measurement
To perform statistical analysis of data, it is important to understand what summary statistics can be meaningfully calculated, interpreted, and compared from a given set of data. Different levels of measurement can be applied depending on the properties of the given data.
The four widely applied levels of measurement are:
Nominal scale
A level of measurement with categories that do not have a natural order or ranking.
Example: Employment status, blood type, or eye colour
Ordinal scale
A level of measurement with categories together with an explicit ranking.
Example: Customer satisfaction ratings (very low, low, average, high, very high)
Other examples of interval scale include:
True zero
A point on a scale representing the complete absence of the measured quantity.
Interval scale
A numerical scale for numerical data with equal intervals but no true zero.
Example: Temperature in Celsius, as 30°C is not ‘double as warm’ as 10°C, but is 20°C warmer
• pH scale - 0 does not mean an absence of pH, it means as acidic as possible
• Dates - for the year 2025, 0 is not the ‘beginning of time’
• Time during the day using a 12-hour clock
Ratio scale
A numerical scale for numerical data with equal intervals and a true zero. Example: Mass, as 5 g is half as heavy as 10 g and 0 g means there is no mass
Other examples of ratio scale include:
• Temperature measured in Kelvin as 0 K has no molecular movement, so is a true zero
• Speed, because 40 km/h is indeed twice as fast as 20 km/h
• Time using a stopwatch
The significant difference between an interval scale and a ratio scale is the inclusion of a ‘true zero’ or ‘absolute zero’ Named variables
A < B < C < D
The properties of each scale can be compared in terms of how they lend themselves to being used to calculate and compare statistics:
the
say one value is _ units greater or smaller than the other
say one value is _ times greater or smaller than the other
Note: The mean of ordinal data can sometimes be calculated by assigning numbers to the outcomes. For example, for a survey asking for a customer satisfaction rating of very low, low, average, high, and very high, the numbers 1 to 5 can be assigned to the outcomes, and the mean level of satisfaction reported can be calculated. However, it is debatable whether this can be meaningfully interpreted, as the original categories are not separated into equal intervals.
Example 3
Classify this data into its correct category: Population of a town A Nominal B Ordinal C Interval D Ratio
Create a strategy
Use the table:
Categorises the values Y Y Y Y
Ranks values in order Y Y Y
Frequency distribution Y Y Y Y
Mode Y Y Y Y
Median Y Y Y
Mean Y Y
Can say one value is _ units greater or smaller than the other Y Y
Can say one value is _ times greater or smaller than the other Y
Apply the idea
The population can be compared through a ratio of values. For example, a population of 2000 is considered twice as large as a population of 1000. The correct answer is option D.
Idea summary
The four levels of measurement classify data based on their properties, which determines the types of statistical analysis that can be performed.
• Nominal scale: Used for categorical data where categories have no natural order or rank (e.g., blood type).
• Ordinal scale: Used for categorical data where categories have a meaningful, explicit ranking (e.g., satisfaction ratings).
• Interval scale: A numerical scale with equal intervals between values, allowing for meaningful differences. It lacks a true zero (e.g., temperature in Celsius).
• Ratio scale: A numerical scale that has all the properties of an interval scale but includes a true zero, allowing for meaningful ratios (e.g., height, mass).
The critical distinction between interval and ratio scales is the concept of a true zero (also known as an absolute zero), which represents the complete absence of the quantity being measured.
1.01 Practice questions
What do you remember?
1 Fill in the blanks with the terms from this list:
• Interval • Numerical • Categorical • Variable
a Univariate data can be numerical or ⬚ data.
b We can find the mean of ⬚ data.
c Univariate means one ⬚.
d There is no true zero on an ⬚ scale.
2 Determine whether each statement is true or false:
a A ratio scale has a true zero.
b An interval scale is a special type of ratio scale.
c Discrete data is a special type of numerical data.
d Categorical data can be broken down into ordinal and interval data.
e The mode of categorical data can be determined.
3 Identify whether these examples are numerical or categorical data:
a Favourite flavours
c Daily temperature
e Amount owed on lay-by
g Daily UV index
i Favourite colours
k Types of dogs at the park
Practice
b Maximum temperature
d Types of horses
f Types of vegetables
h Maximum rainfall
j Brands of phones
l Maximum snowfall
4 Identify whether these data variables are discrete or continuous:
a The number of classrooms in a school
b Daily humidity
c The time taken to run 200 metres
d Lengths of cats’ whiskers
e The distance from the Earth to nearby galaxies
f The number of people taller than you in each class
g Marks on the most recent Physics test
h Time it takes to fall asleep at night
i The amount of water consumed in a day
5 Identify whether these are nominal or ordinal categorical data:
a Eye colour
d Favourite song
b Birthdate
e State/territory of birth
c Temperature
f Year level
6 Classify these data examples as one of these types:
• Numerical discrete
• Numerical continuous
• Categorical ordinal
• Categorical nominal
a Types of trailers in a shopping centre car park
b Heights of people at an athletics carnival
c Weights of dogs
d The languages spoken in a class
e The number of languages students in a town speak
f The number of people at an athletics carnival
g The time spent playing games each day
h The number of sports people play
7 Classify these data examples as one of these types:
• Numerical continuous
• Numerical discrete
• Categorical
a The workplaces of people living in Sydney
b The body temperature of a hospital patient taken over a 21-hour period
c The brands of breakfast foods in a supermarket
d The weights of club members
e The number of people attending a netball game
8 Identify the level of measurement that can be applied to these data as nominal, ordinal, interval or ratio:
a Number of pets in a household
c Make of car
e Daily rainfall in a city
g Temperature measured in degrees Celsius
h Letter grades on an assignment (A to F)
i Year of birth
b pH of liquid
d Weights of eggs
f Military rank
9 Identify the level of measurement that can be applied to the data generated in each situation as nominal, ordinal, interval or ratio:
a The time taken for a team of swimmers to complete one lap of a swimming pool is recorded by their coach.
b The shoe sizes of shoe prints left at the scene of a crime are recorded by a detective.
c The religion practised by a person is recorded during a census.
d A driver is being rated by customers on a scale of 1 star to 5 stars.
10 Determine whether these statistics can be meaningfully computed, interpreted, and compared, from the data in each situation:
i Mode
iii Differences between outcomes
ii Median
iv Ratios between outcomes
a The temperature, in degrees Celsius, of the ocean at a particular location is monitored and fed back to a weather station.
b A group of people is surveyed and asked to indicate their attitude on a recent government policy by selecting a number from 1 to 5, where 1 = strongly disagree, 2 = disagree, 3 = neutral, 4 = agree and 5 = strongly agree.
c The flavour of ice cream selected by customers at a market stall is recorded over a day.
d The weight of puppies at a puppy school is recorded each week.
11 The graph shows the height of each student in Kate’s class:
a Is the variable ‘height of student’ continuous or discrete? Explain your answer.
b Is the variable ‘number of students in a given height range’ continuous or discrete? Explain your answer.
Extend your thinking
12 Patients at a hospital are asked to give their pain a rating from 1 to 10. Vera says: “The pain ratings are interval data because they are numerical and have a scale going up in equal amounts.”
Critique Vera’s reasoning and give a correct explanation for the level of measurement that can be applied to the pain rating data.
13 Recording the distance, in kilometres, between towns is supposed to produce continuous data. However, distances listed on road signs are always represented as whole numbers. Explain how this can be.
14 Explain in your own words why the amount of money spent on groceries is a discrete variable even though it involves decimal values.
15 Explain in your own words why postcode is a categorical variable while the number of houses in a postcode is a numerical variable.
