Correlation vs Causation
Understand correlation and causation for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-06). A correlation means two variables tend to change together; a causal relationship means a change in one variable actually produces the change in the other.
You will learn to tell a correlation apart from a cause, see that even a strong correlation does not prove one, and look for other explanations such as a third (lurking) variable or coincidence — useful for judging claims like “ice-cream sales cause sunburn”.
Theory
A correlation means two variables tend to change together; causation means one variable actually brings about the change in the other. This Year 12 Standard 1 (NSW) guide shows why a correlation, however strong, does not prove a cause, how to spot a third (lurking) variable or a coincidence, and what makes better evidence.
A correlation means that as one variable changes, the other tends to change too — the variables are associated.
The direction of a correlation is positive if both variables increase together (points rise to the right) and negative if one increases as the other decreases (points fall to the right).
Causation means a change in one variable really causes the change in the other. A correlation, however strong, does not on its own prove a cause.
Before claiming a cause, look for another explanation: a third (lurking) variable that drives both, coincidence (both simply rose or fell over the same years), or a cause that runs the other way. Better evidence comes from a controlled experiment: change only one thing, keep everything else the same, and allocate people or plots at random.
A causal claim is reasonable when there is a clear way one variable affects the other (a heater warms a room) and other factors are kept the same.
There is no formula to calculate here. Keep the key rule in mind:
A correlation between \(A\) and \(B\) can come from any of these:
How to judge a claim that one thing causes another
- Check for a correlation first — with no trend, a cause is very unlikely.
- Look for a third variable that could drive both, and ask whether both simply grew over time.
- Reverse the claim: could \(B\) cause \(A\) instead?
- Conclude: observed data shows association only; a cause is reasonable when there is a clear mechanism and other factors are kept the same, as in a controlled experiment.
Ask what a cold week does to both variables.
| \(\text{cold weather}\) | \(\Rightarrow\) | \(\text{more hot chocolate sold}\) |
| \(\text{cold weather}\) | \(\Rightarrow\) | \(\text{people stay indoors together}\) |
| \(\text{indoors together}\) | \(\Rightarrow\) | \(\text{more colds spread}\) |
The weather (a third variable) drives both. No — the correlation is real, but hot chocolate does not cause colds; cold weather explains the link.
Check whether both variables simply grew over the same years.
| \(\text{online shopping}\) | \(\nearrow\) | \(\text{over time}\) |
| \(\text{solar panels}\) | \(\nearrow\) | \(\text{over time}\) |
| \(\text{both trend up together}\) | \(\Rightarrow\) | \(\text{coincidence}\) |
The correlation is coincidental; online shopping does not cause people to install solar panels.
Check what was changed and whether there is a clear mechanism.
| \(\text{only the time changed}\) | \(\Rightarrow\) | \(\text{all else the same}\) |
| \(\text{microwaves heat water}\) | \(\Rightarrow\) | \(\text{clear mechanism}\) |
Yes — this is a controlled experiment, so a causal relationship is reasonable.
Test the cause in both directions.
| \(\text{more sleep}\) | \(\Rightarrow\) | \(\text{less stress?}\) |
| \(\text{more stress}\) | \(\Rightarrow\) | \(\text{less sleep?}\) |
The cause could run either way, or another factor (such as work hours) may affect both. The correlation does not prove that sleep causes lower stress.
Look at how the groups were formed.
| \(\text{(i) app users differ in other ways}\) | \(\Rightarrow\) | \(\text{not proven}\) |
| \(\text{(ii) third variable}\) | \(=\) | \(\text{homework (effort)}\) |
| \(\text{(iii) random groups}\) | \(\Rightarrow\) | \(\text{fair test}\) |
(iii) Take one group of students, randomly give half the app and half no app, keep everything else the same, then compare their test marks — a randomised controlled experiment.
Common pitfalls
Frequently asked questions
What is the difference between correlation and causation?
Correlation means two variables tend to change together. Causation means a change in one variable actually brings about the change in the other. A correlation on its own does not prove causation.
Does correlation imply causation?
No. Even a strong correlation is only an association. The link may come from a third (lurking) variable, from coincidence, or the cause may run the other way.
What is a lurking variable?
A lurking or third variable is something that affects both variables at once, making them move together. For example, hot, sunny weather increases both ice-cream sales and sunburn cases.
How can you show that one variable causes another?
Use a controlled experiment: change only one thing, keep everything else the same, and put people or plots into groups at random. Data that is simply observed cannot prove a cause. A cause is reasonable when there is a clear way one variable affects the other and other factors are kept the same.
If there is no correlation, can there still be causation?
It is very unlikely. If the points form a random cloud with no trend, one variable is very unlikely to be causing changes in the other.