Association and causation: QCE General Mathematics Unit 3 Bivariate data analysis 2

Syllabus dot point

“Recognise and explain that an observed association between two variables does not necessarily mean there is a causal relationship, identify and communicate non-causal explanations including coincidence or the influence of another variable, and solve practical problems involving associations”

QCEGeneral MathematicsUnit 3: Bivariate data and time series…14 min readUpdated 23 Sep 2026

Quick answer

An association between two variables, categorical or numerical, does not necessarily mean one causes the other. Non-causal explanations include the influence of another variable that affects both, coincidence, and a causal link running in the reverse direction. Describe associations with "is associated with" or "tends to", explain any other variable by showing how it affects both, and solve practical problems by choosing the right tool (two-way table or scatterplot), describing the association and evaluating whether it could be causal.

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The answer
Practice questions

What this dot point is asking

Once you have found an association, the obvious next question is "does one variable cause the other?" The QCAA General Mathematics 2025 syllabus answers this directly: you must recognise and explain that an observed association between two variables (categorical, numerical, or one of each) does not necessarily mean that there is a causal relationship, and you must be able to identify and communicate possible non-causal explanations, including coincidence and the influence of another variable. The sub-topic finishes with solving practical problems by identifying, analysing and describing associations, which draws together two-way tables, scatterplots, $r$ , $R^2$ and the least-squares line.

Expect this in the external assessment and IA2 as short "explain" or "evaluate the reasonableness of the claim" questions, often attached to a regression question. The problem-solving and modelling task (IA1) also rewards discussing whether an association could be causal.

The answer

Association versus causation

Two variables are associated if their values are related: knowing one helps you predict the other. For categorical variables you see this as different percentages across groups in a two-way table; for numerical variables you see it as a pattern in a scatterplot, measured (if linear) by Pearson's correlation coefficient $r$ .

Causation means that a change in one variable produces a change in the other. Causation always produces an association, but an association can arise without any causation at all. So an association, however strong, cannot by itself prove causation.

Key fact

An observed association does not necessarily mean there is a causal relationship. Possible non-causal explanations include the influence of another variable (which affects both variables) and coincidence (a chance pattern). When describing an association, use "is associated with" or "tends to", and never "causes", "makes" or "leads to" unless causation has been established.

Non-causal explanations

The influence of another variable. A third variable, not included in the data, affects both variables and makes them rise or fall together. This is by far the most common explanation.

The number of pharmacies and the number of cafés in a town: both depend on the town's population.
Cold drink sales and lifeguard rescues: both depend on the temperature.
Children's shoe size and vocabulary: both depend on age.

Coincidence. With small data sets, or when many pairs of variables are examined, some will be associated purely by chance. The number of films an actor releases each year and the number of shark sightings at a beach might have $r = 0.71$ over ten years, but no mechanism links them. A coincidental association usually disappears when more data is collected.

Reversed direction. Sometimes there is a causal link, but it runs the other way from the one assumed. Suburbs with more police officers report more crime, not because police cause crime, but because more crime leads to more police being allocated. Deciding which variable is explanatory is a modelling choice, and it can be wrong.

Communicating a non-causal explanation

A full-mark answer usually has two parts:

Name the explanation: another variable (and which one), or coincidence.
Explain how it produces the association: how the other variable affects each of the two variables, or why chance is the only plausible reason.

For example: "The association between cold drink sales and rescues is likely due to another variable, daily temperature. On hotter days more people go to the beach, so more cold drinks are sold and more swimmers need rescuing." Naming "temperature" alone would usually earn only part of the credit.

When causation is plausible

Some associations have an obvious causal mechanism: hot days really do make people less likely to buy hot chocolate. The correct statistical position is still careful: the data is consistent with a causal link, but because it is observational (we simply recorded what happened), other variables could also be involved, such as school holidays or the number of customers. Establishing causation requires controlled experiments in which only the explanatory variable is changed. In "evaluate the reasonableness" questions, say both things: why the causal claim is plausible, and why the data alone does not prove it.

Solving practical problems with associations

The final syllabus point asks you to put the whole bivariate toolkit to work. A good approach follows the problem-solving and modelling steps:

Identify the variables and their types, and choose the explanatory and response variables.
Choose the tool. Two categorical variables: two-way table with percentages. Two numerical variables: scatterplot, $r$ , $R^2$ and, if linear, a least-squares line with a residual plot.
Analyse. Calculate the percentages or the statistics.
Describe the association (for numerical data: direction, form, strength; for categorical data: the pattern in the percentages).
Interpret in context, including predictions if asked, and whether they involve interpolation or extrapolation.
Evaluate: could the association be causal, or could another variable or coincidence explain it?

Worked examples: explaining associations in both kinds of data

Numerical data with another variable

Eight towns: pharmacies ( $x$ ) and cafés ( $y$ ): (2, 6), (3, 8), (3, 11), (5, 14), (6, 18), (8, 22), (9, 27), (12, 33).

Statistics (calculator): $r \approx 0.993$ , $R^2 \approx 0.986$ , least-squares line $y \approx 2.71x + 1.09$ .
Describe: Strong, positive, linear association: towns with more pharmacies tend to have more cafés. $98.6\%$ of the variation in the number of cafés can be explained by the variation in the number of pharmacies.
Explain: Town population influences both variables; larger towns have more of both businesses. The association is not causal.

Marker's note: " $98.6\%$ of the variation ... can be explained by the variation in ..." is a statement about the model's fit, not about cause. It is correct even though the association is not causal.

Categorical data with another variable

300 students: 144 of the 180 who eat breakfast daily passed a test; 72 of the 120 who do not passed.

Percentages: $80\%$ against $60\%$ .
Describe: Passing the test is associated with eating breakfast daily: the pass rate is 20 percentage points higher for daily breakfast eaters.
Explain: Another variable, such as a regular home routine or adequate sleep, could lead students both to eat breakfast and to perform better. The survey cannot show that breakfast causes better results.

Marker's note: the same reasoning applies whether the data is in a two-way table or a scatterplot.

Evaluating a causal claim

Hot chocolates sold ( $h$ ) against maximum temperature ( $t$ ) on eight days: least-squares line $h = -3.03t + 130.12$ , $r \approx -0.998$ . The owner says hot weather causes lower sales.

Interpret the slope. Each 1 °C increase is associated with about 3 fewer hot chocolates sold.

Evaluate. The claim is plausible (people prefer cold drinks when it is hot) and the association is very strong, but the data is observational. Other variables linked to temperature, such as school holidays or customer numbers, may also play a part, so the data supports but does not prove the claim.

Marker's note: in complex unfamiliar questions, a balanced evaluation (plausible, but not proven by these data) earns more than a flat yes or no.

Recognising coincidence

An actor's films per year and yearly shark sightings: $r = 0.71$ over ten years.

Look for a mechanism: None: films cannot affect sharks or vice versa.
Look for another variable: Nothing obvious influences both.
Conclude: Coincidence, made more likely by the small number of data points. More years of data would probably weaken the correlation.

Marker's note: say why other explanations fail before settling on coincidence.

Common traps

Treating a strong $r$ as proof of cause: Strength measures how closely the points follow a line, not why.
Naming another variable without explaining it: Show how it affects both variables.
Using causal words in descriptions: "Increases", "reduces", "improves" and "makes" all claim causation.
Assuming the explanatory variable must be the cause: The direction may be reversed.
Calling everything coincidence: Coincidence is the explanation of last resort, when there is no plausible mechanism and no third variable.
Forgetting categorical data: Associations in two-way tables are subject to exactly the same caution as correlations.

Exam technique

When a question asks whether an association shows causation, start with "No, an association does not necessarily mean causation", then give one specific non-causal explanation with a sentence showing how it works. For "evaluate the reasonableness" questions, comment on the strength of the association, the plausibility of the causal claim, and at least one other variable that could be involved.

Note

Two things can go up and down together without one causing the other. Towns with more pharmacies also have more cafés, but that is because bigger towns have more of everything, not because pharmacies attract cafés. Sometimes a link is just luck, especially when you only have a few numbers. So when you spot a pattern, say the two things are "linked" or "associated", and then think about whether something else could be pushing both of them.

Practice questions

Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.

foundation2 marks

Over one summer, a beach town records a strong positive association between the number of cold drinks sold each day and the number of lifeguard rescues each day. (a) Does this show that buying cold drinks causes people to need rescuing? (b) Give a non-causal explanation.

Show worked solution →

(a) No. An association between two variables does not necessarily mean one causes the other. (1 mark)

(b) Another variable, the daily temperature (or the number of people at the beach), influences both: on hot days more people visit the beach, so more cold drinks are sold and more swimmers get into difficulty. (1 mark for naming the variable and explaining how it affects both.)

foundation1 mark

Which one of the following is the best description of a pair of variables that are associated purely by coincidence? A. Two variables that are both affected by a third variable. B. Two variables whose association arose by chance and has no real connection. C. Two variables where one directly changes the other. D. Two variables with a correlation coefficient of zero.

Show worked solution →

B. Coincidence means the association happened by chance, typically in a small or selected data set, with no real link between the variables. Option A describes the influence of another variable, option C describes causation, and option D describes no linear association.

foundation2 marks

Rewrite each claim so that it accurately describes the association without claiming causation. (a) "Taller basketball players score more points because of their height." (b) "Studying at the library improves marks."

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(a) "Taller basketball players tend to score more points" or "There is a positive association between height and points scored." (1 mark)

(b) "Students who study at the library tend to have higher marks" or "Studying at the library is associated with higher marks." (1 mark)

core3 marks

For eight towns, the number of pharmacies (

x

) and the number of cafés (

y

) are: (2, 6), (3, 8), (3, 11), (5, 14), (6, 18), (8, 22), (9, 27), (12, 33). (a) Use your calculator to find Pearson's correlation coefficient

r

and the coefficient of determination

R^2

. (b) Describe the association. (c) Explain why opening a new pharmacy would not be expected to increase the number of cafés.

Show worked solution →

(a) $r \approx 0.993$ and $R^2 \approx 0.986$ . (1 mark)

(b) There is a strong, positive, linear association between the number of pharmacies and the number of cafés: towns with more pharmacies tend to have more cafés. (1 mark)

(c) Both are influenced by another variable, the population of the town. Bigger towns support more businesses of every kind, so both counts rise with population. The association does not mean pharmacies cause cafés. (1 mark)

core3 marks

A school surveyed 300 students. Of the 180 who eat breakfast every day, 144 passed a maths test. Of the 120 who do not, 72 passed. (a) Calculate the pass percentage for each group. (b) Is there an association? (c) Suggest a non-causal explanation for the association.

Show worked solution →

(a) Breakfast every day: $\frac{144}{180} \times 100\% = 80\%$ . Not every day: $\frac{72}{120} \times 100\% = 60\%$ . (1 mark)

(b) Yes. The pass rate is 20 percentage points higher for students who eat breakfast every day. (1 mark)

(c) Another variable could influence both, such as a regular home routine (or getting enough sleep): students with regular routines may be more likely both to eat breakfast and to study consistently. The data does not show that breakfast itself causes better results. (1 mark)

core2 marks

Across many suburbs, the number of police officers is positively associated with the number of reported crimes. A student concludes that police cause crime. Give a better explanation.

Show worked solution →

The direction of any link is more likely the reverse: suburbs with more crime are allocated more police officers. (1 mark) Also, larger suburbs (another variable, population) tend to have both more crime reports and more police. Either way, the association does not show that police cause crime. (1 mark)

exam4 marks

A café owner records the maximum temperature (

t

, °C) and the number of hot chocolates sold (

h

) on eight days: (14, 88), (17, 79), (19, 74), (22, 61), (24, 58), (27, 47), (30, 40), (33, 31). (a) Find the equation of the least-squares line in the form

h = mt + c

, with values to two decimal places. (b) Interpret the slope in context. (c) The owner claims "hot weather causes people to stop buying hot chocolate". Evaluate the reasonableness of this claim.

Show worked solution →

(a) From the calculator's linear regression: $m \approx -3.03$ and $c \approx 130.12$ , so $h = -3.03t + 130.12$ . (1 mark)

(b) For each 1 °C increase in maximum temperature, the number of hot chocolates sold decreases by about 3, on average. (1 mark)

(c) The association is very strong ( $r \approx -0.998$ ), and a causal explanation is plausible (people prefer cold drinks on hot days). (1 mark) However, the data is observational, so it only shows an association: other variables that change with temperature, such as the season, school holidays or the number of customers, could contribute. The claim is reasonable as a likely explanation but is not proven by this data alone. (1 mark)

exam3 marks

Over ten years, the number of films released by a particular actor each year and the number of shark sightings reported at a beach each year have a correlation coefficient of

r = 0.71

. (a) Describe the strength and direction of the association. (b) What is the most likely explanation for this association? (c) What would you expect to happen to the correlation if many more years of data were collected?

Show worked solution →

(a) Moderate (to strong), positive. (1 mark)

(b) Coincidence. There is no plausible way that either variable affects the other, and no obvious third variable that affects both; with only ten data points, a sizeable correlation can occur by chance. (1 mark)

(c) If the association is a coincidence, the correlation would be expected to weaken (move towards zero) as more data is collected, because a chance pattern is unlikely to continue. (1 mark)

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Flashcards General Mathematics deck

Sources & how we know this

General Mathematics General senior syllabus 2025 — QCAA