Correlation and causation, observation and experimentation for VCE General Mathematics Unit 3 Data analysis
“Distinguish between correlation and causation, identify possible non-causal explanations for an association, and explain the difference between observation and experimentation and the need for experimentation to determine cause and effect”
Correlation shows that two variables are associated, not that one causes the other. The association may be due to a confounding (common-response) variable, coincidence, or a causal link running the other way. Observational studies can only show association; a randomised, controlled experiment is needed to establish cause and effect. From observational data, write "is associated with" and "tends to", never "causes".
What this dot point is asking
Finding that two variables are associated is only the first step of a statistical investigation. VCAA wants you to know that association (or correlation) does not imply causation, to suggest non-causal explanations for an observed association (a third variable, coincidence), and to understand the difference between an observational study and an experiment: observation can reveal an association, but only an experiment can definitively establish cause and effect.
This appears in almost every Examination 1 as a "which conclusion is correct" question, and in Examination 2 as a request to describe an association or suggest a reason for it. The 2023 and 2025 multiple-choice questions below were answered correctly by only about of students, mostly because the incorrect options used the word "causes".
The answer
Association is not causation
Two numerical variables are associated (correlated) if knowing the value of one helps predict the value of the other: as one increases, the other tends to increase (positive association) or decrease (negative association). The correlation coefficient and the coefficient of determination measure the strength of a linear association.
Causation is a much stronger claim: that changing one variable produces a change in the other. A strong correlation is consistent with causation, but it is also consistent with several other explanations, so on its own it can never prove it.
A correlation, however strong, only shows that two variables are associated. To conclude that one causes the other you need a well-designed experiment, in which the researcher controls the explanatory variable and randomly allocates subjects to treatments. From observational data, write "is associated with", never "causes", "leads to" or "results in".
Non-causal explanations
When two variables are associated but you cannot conclude cause and effect, the association may be explained by:
- A confounding (lurking) variable, or common response. A third variable, not included in the analysis, affects both variables. Ice cream sales and sunburn cases rise together because both respond to hot sunny weather. Children's shoe size and reading age rise together because both increase with age.
- Coincidence. With enough variables and a small data set, some pairs will be associated purely by chance. A handful of years in which a sports team's wins track a country's rainfall is almost certainly coincidence.
- Reversed or unclear direction. Even when there is a causal link, it may run the opposite way to the one assumed. Towns with more police officers may have more crime not because police cause crime but because more crime leads to more police being employed.
Observation versus experimentation
The way the data was collected decides what you are allowed to conclude.
| Observational study | Experiment | |
|---|---|---|
| What the researcher does | Records the values of variables as they occur | Deliberately sets (manipulates) the explanatory variable |
| Who decides the groups | The subjects or circumstances | The researcher, ideally by random allocation |
| Other variables | Not controlled; may be confounding | Controlled or balanced by randomisation |
| What can be concluded | An association | Cause and effect, if well designed |
In a well-designed experiment the subjects are randomly allocated to a treatment group and a control group (which receives no treatment, or a placebo), and everything else is kept the same. Randomisation means the groups are alike, on average, in every respect except the treatment, including respects the researcher has not even thought of. So if the treatment group then differs from the control group, the difference can be attributed to the treatment.
Many important questions cannot be tested by experiment (you cannot randomly allocate people to smoke for 20 years), which is why careful observational studies, large samples and consistent evidence from many studies are used instead. For the purposes of VCE General Mathematics, the rule is simple: observational data shows association; experiments are needed to establish causation.
Writing conclusions VCAA accepts
When you describe an association in words, include direction, form and strength (and, if asked, the percentage of variation explained), and use association language:
"There is a strong, negative, linear association between weekly exercise and resting heart rate. Adults who exercise more tend to have lower resting heart rates. of the variation in resting heart rate can be explained by the variation in weekly exercise."
Words that are safe: is associated with, tends to, is related to, can be explained by the variation in. Words that claim causation: causes, leads to, results in, makes, increases, reduces, affects, improves. If the data is observational, avoid the second list entirely.
A careful point about : "64% of the variation in heart rate can be explained by the variation in exercise" is standard statistical language and is not a claim of causation. It describes how well the linear model fits.
Multiple-choice strategy
The standard VCAA question gives an association (often with or ) and four or five conclusions. Work through it in two passes:
- Cross out every option that says "causes" (unless the question describes a randomised experiment).
- Among the rest, check the direction against the sign of or the slope. A negative association means "more of one goes with less of the other".
Both the 2023 and 2025 questions below fall to exactly this method.
Naming a confounding variable
Across 50 suburbs, the number of cafés is strongly positively associated with the number of dog grooming businesses. Explain the association.
Look for something that drives both. Suburbs with larger populations (and higher incomes) support more businesses of every kind, including cafés and dog groomers.
Conclusion. Population (or income) is a confounding variable. The association is a common response; opening a café will not create a dog groomer.
Marker's note: name the third variable and explain how it affects each of the two variables. Naming it alone is usually worth only half the credit.
Correcting a causal claim
A report states: "Because between hours of homework and exam marks, doing more homework raises exam marks." Explain what is wrong and rewrite the conclusion.
What is wrong. The data is observational (students chose how much homework to do), so the strong correlation shows association only. Students who do more homework may also be more motivated or have more support at home, which could explain higher marks.
Rewrite. "There is a strong positive association between hours of homework and exam marks: students who do more homework tend to get higher marks."
Marker's note: the value is strong, but strength is irrelevant to the causation question.
Designing an experiment
A gym wants to know whether a new warm-up routine reduces injuries.
Design. Take the members who agree to take part and randomly allocate half to the new warm-up and half to the usual warm-up (the control group). Keep all other aspects of their training the same. After a set period, compare the injury rates in the two groups.
Why it works. Random allocation balances fitness, age, experience and every other factor across the groups, so a difference in injury rates can be put down to the warm-up.
Marker's note: the essential features are a comparison group, random allocation, and everything else held the same.
Converting to and interpreting
The coefficient of determination between the number of hours of sleep and reaction time in a sample of students is 0.49; the least squares slope is negative. Find and interpret .
. . The sign comes from the slope.
Interpretation. of the variation in reaction time can be explained by the variation in hours of sleep. The association is moderate and negative: students who sleep more tend to have shorter reaction times.
Marker's note: the 2025 report noted many students forgot the negative sign when converting to . Always check the slope.
- Choosing the "causes" option because the association is strong
- No value of or proves causation.
- Getting the direction backwards
- Negative association: more of one, less of the other. Read each option slowly.
- Naming a confounder without explaining it
- Say how the third variable affects both variables.
- Calling any study with two groups an experiment
- It is only an experiment if the researcher assigns the treatment. People who choose a diet are observed, not experimented on.
- Using causal verbs in a description
- "Affects", "increases" and "improves" all imply cause. Stick to "is associated with" and "tends to".
Dropping the sign when finding from . takes the sign of the slope.
In a multiple-choice question about conclusions, eliminate every causal option first, then check direction. In Examination 2, when asked for a possible reason for an association, write one sentence naming a plausible third variable and one sentence explaining how it drives both variables. When asked whether a conclusion of cause and effect is justified, answer "No, because the data comes from an observational study; an association does not imply causation, and a confounding variable such as ... could explain it."
Just because two things go up and down together does not mean one is making the other happen. Ice cream sales and sunburns both go up in summer, but ice cream does not burn you; the hot weather drives both. To prove that one thing really causes another, you have to run a fair test: split people into groups at random, change only one thing for one group, and see if that group ends up different. Without that fair test, the most you can say is that the two things are linked.
Exam-style questions
Questions in the style of VCAA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
2023 VCAA-style1 markA study of Year 10 students shows a negative association between topic test scores and the time spent on social media. The coefficient of determination is 0.72. From this information it can be concluded that: A. a decreased time spent on social media is associated with an increased topic test score B. less time spent on social media causes an increase in topic test performance C. an increased time spent on social media is associated with an increased topic test score D. too much time spent on social media causes a reduction in topic test performance E. a decreased time spent on social media is associated with a decreased topic test score
Show worked answer →
Rule out causation first. This is an observational study, so options B and D, which use "causes", cannot be concluded however strong the association.
Use the direction. A negative association means that as one variable goes up, the other tends to go down, so less social media time goes with higher test scores. Option C and option E describe a positive association.
The answer is A. ( of students chose it.) Note that the coefficient of determination of 0.72 tells us the association is strong (), but strength never turns association into causation.
Source: VCAA 2023 General Mathematics Examination 1, Question 10, and the 2023 examination report.
2025 VCAA-style1 markFor the 12 A-League men's teams after 27 games of the 2022 to 2023 season, the correlation coefficient between games won and goals scored against the team is . Based on the correlation coefficient, it can be concluded that: A. fewer goals scored against is associated with a smaller number of wins B. fewer goals scored against causes a smaller number of wins C. more goals scored against causes a smaller number of wins D. more goals scored against is associated with a smaller number of wins
Show worked answer →
A correlation coefficient describes association, not cause, so options B and C are rejected; the examiners' report says exactly this ("causation cannot be concluded").
is negative, so more goals against tends to go with fewer wins (and fewer goals against with more wins). Option A describes a positive association.
The answer is D. ( of students chose it.)
Source: VCAA 2025 General Mathematics Examination 1, Question 10, and the 2025 examination report.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation1 markAcross a group of towns, the number of ice creams sold each day is strongly positively associated with the number of people treated for sunburn. Give the most likely non-causal explanation.
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A confounding (common-response) variable: hot, sunny weather. On hot sunny days more people buy ice creams and more people spend time in the sun and get sunburnt. Both variables respond to the weather; neither causes the other. (1 mark for naming a plausible third variable and explaining how it affects both.)
foundation2 marksRewrite each statement so that it correctly describes an observed association without implying causation. (a) "Watching more TV makes students' marks lower." (b) "Owning more books raises a child's reading score."
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(a) "Students who watch more TV tend to have lower marks" or "Time spent watching TV is negatively associated with marks." (1 mark)
(b) "Children in homes with more books tend to have higher reading scores" or "The number of books in the home is positively associated with reading score." (1 mark)
The key words are "associated with" or "tend to", never "makes", "causes", "raises" or "leads to".
foundation1 markWhich one of the following study designs can establish that a new fertiliser causes an increase in crop yield? A. Recording the yields of farms that chose to use the fertiliser and farms that did not. B. Randomly assigning plots of the same field to receive the fertiliser or no fertiliser, keeping everything else the same. C. Measuring the correlation between fertiliser sales and crop yield across regions. D. Surveying farmers about whether they believe the fertiliser works.
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B. Only a randomised, controlled experiment can establish cause and effect: the researcher decides which plots get the fertiliser (at random), so the groups differ only in the treatment. In option A the farms chose for themselves, so better-resourced farms might both use the fertiliser and have higher yields (a confounding variable). Options C and D are observational or opinion data.
core2 marksData from 30 primary schools shows a strong positive association between students' shoe size and their reading age (). (a) Identify a likely confounding variable. (b) Explain how it produces the association.
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(a) The students' age (or year level). (1 mark)
(b) Older students tend to have bigger feet and, because they have had more years of schooling, also tend to read at a higher level. As age increases, both shoe size and reading age increase, producing a positive association between them even though bigger feet do not improve reading. (1 mark)
core2 marksA study of towns finds a strong positive association between the number of firefighters sent to a fire and the dollar value of the damage. (a) Explain why sending more firefighters does not cause more damage. (b) Which variable is more sensibly treated as the explanatory variable, and why?
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(a) Both depend on the size of the fire: a bigger fire causes more damage and also leads to more firefighters being sent. The association is a common response to fire size, not cause and effect between the two recorded variables. (1 mark)
(b) Neither variable causes the other, but if one must be chosen, the damage (or better, the fire size) explains the number of firefighters sent. This is an example of the direction being the reverse of the one suggested: large fires bring both more damage and more firefighters. (1 mark)
core2 marksThe coefficient of determination for the association between hours of weekly exercise and resting heart rate in a sample of adults is 0.64, and the association is negative. (a) Find . (b) Write a conclusion in words that VCAA would accept.
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(a) (negative because the association is negative). (1 mark)
(b) "There is a strong negative association between weekly exercise and resting heart rate: adults who exercise more tend to have lower resting heart rates. of the variation in resting heart rate can be explained by the variation in hours of exercise." (1 mark) It must not say that exercise causes the lower heart rate, because this is observational data.
exam3 marksA researcher wants to know whether listening to music while studying causes lower quiz scores. (a) Describe an experiment that could answer this question. (b) Explain why random allocation is important. (c) Explain why the conclusion from an observational study of students' own study habits would be weaker.
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(a) Take a group of students and randomly allocate half to study a passage with music playing and half to study the same passage in silence, for the same time, in the same conditions; then give all students the same quiz and compare the mean (or median) scores of the two groups. (1 mark)
(b) Random allocation makes the two groups similar, on average, in every other respect (ability, interest, sleep, and so on), so any difference in quiz scores can be attributed to the music rather than to differences between the groups. (1 mark)
(c) In an observational study students choose whether to listen to music. Those who choose music might differ in other ways (for example, they might be less focused or study for less time), so a difference in scores could be caused by those confounding variables rather than the music. Observation can show an association; only experimentation can definitively establish cause and effect. (1 mark)
exam1 markFor 40 countries, the correlation between the number of mobile phones per person and average life expectancy is . Which one of the following is the best explanation? A. Owning a mobile phone increases life expectancy. B. Longer life expectancy causes people to buy more mobile phones. C. A third variable, such as national wealth, is associated with both. D. The correlation must be due to an error in the data.
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C. Wealthier countries tend to have both more mobile phones per person and better health care, nutrition and sanitation, which lengthen life expectancy. National wealth is a confounding variable. Options A and B claim causation from observational data, and nothing suggests the data is wrong (option D).