Coaching Centre
Correlation and causation
Why a strong correlation does not prove that one variable causes the other: lurking variables, coincidence, reverse direction, and how to word conclusions about association.
Across a Sydney summer, the days with the most ice-creams sold are also the days with the most rescues at the beach. The correlation is strong and positive. Does eating ice-cream make people get into trouble in the surf? Of course not. Both go up on hot days. This is the most important idea in the chapter, and the HSC tests it almost every year.
Association is not causation
A strong $r$ tells you that two variables move together. It does not tell you why. There are four common explanations for a correlation, and only one of them is "$x$ causes $y$".
| Explanation | What's going on | Example |
|---|---|---|
| Causation | $x$ really does change $y$ | More hours of driving practice leads to fewer errors on the driving test |
| Lurking variable | A third variable drives both | Hot weather raises both ice-cream sales and beach rescues |
| Reverse direction | $y$ actually influences $x$ | Areas with more police have more crime, because police are sent where crime is high |
| Coincidence | The match is pure chance | Two unrelated trends that both happened to rise over the same ten years |
Data from NSW suburbs shows a strong positive correlation between the number of cafés in a suburb and the number of car accidents. A local councillor says "cafés cause accidents". Give a better explanation.
Wording your conclusion
When you only have observational data, a scatterplot and an $r$ value, write conclusions about association, never cause.
To show cause and effect, scientists run a controlled experiment: randomly split people into groups, change only $x$, and keep everything else the same. That's how medicines get tested. A survey or a set of observations can never do that on its own.
Start with All days: a strong correlation between ice-creams sold and beach rescues. Now switch to Colour by temperature. What do you notice about where the hot days sit?
Then choose Only days around 25°C. What happens to $r$ when the temperature is held roughly constant? Explain what this tells you about cause.
Your turn
Across Australian towns, there is a strong positive correlation between the number of doctors and the number of deaths each year. Should towns get rid of doctors? Explain.
Show the answer
No. Bigger towns have more people, so they have more doctors and also more deaths in total. Population is the lurking variable. The correlation shows an association, not that doctors cause deaths. Comparing death rates per 1000 people would be fairer.
A strong positive correlation between two variables means:
A positive correlation means larger $x$ values tend to go with larger $y$ values. It says nothing about cause.
Shoe size and reading ability are strongly correlated in primary school children. The most likely lurking variable is:
Older children have bigger feet and also read better. Age drives both, so it is the lurking variable.
Which study design could show that a new study app causes better marks?
Only a controlled experiment with random assignment removes lurking variables, such as more motivated students choosing to use the app.
Suburbs with more fire trucks at a fire tend to have more damage. The best explanation is:
The size of the fire is a lurking variable. Big fires bring more trucks and cause more damage.
A town finds that more umbrellas are sold on days with more car crashes. Which is the most sensible lurking variable?
Rainy days increase umbrella sales and make roads slippery, which increases crashes.
Which conclusion is correctly worded for observational data with $r = 0.78$ between exercise and mood score?
With observational data you can only describe association. $r$ is not a percentage.
Hospitals where more patients are treated tend to have more nurses. Thinking that "more nurses cause more patients" would be an example of:
The number of patients drives the number of nurses hired, not the other way round. The suggested cause is backwards.
Over 2010 to 2020, the number of streaming subscriptions in Australia and the number of solar panels installed both rose steadily, giving $r = 0.95$. The most reasonable comment is:
Two unrelated trends that both grew over the same years will always correlate strongly. Time, not either variable, is behind the pattern.
Exam style: Data from 40 Year 12 students shows a strong positive correlation between owning a laptop and HSC marks. Which is the best response to "laptops cause higher marks"?
Correlation shows association only. A plausible lurking variable, like family income or access to tutoring, could drive both laptop ownership and marks.
Exam style: A researcher keeps the temperature fixed by only looking at days between $24$°C and $26$°C. The correlation between ice-cream sales and beach rescues drops from $0.88$ to $0.07$. This suggests:
Once temperature is held roughly constant, the association almost disappears. So temperature was driving both variables.