Mathematics

Correlation and Causation

Ice cream sales and drownings rise together every summer, and neither causes the other. The confounding variable, worked through, and what a causal claim needs.

Correlation and causation

Correlation is two quantities moving together in a pattern; causation is one of them making the other change, which correlation on its own can never establish.

Also called: Cause and effect
Where students meet it: Grade 8 statistics, in scatter-graph questions, and again in any exam question that ends by asking whether the data proves one thing causes another.

The short answer
Correlation means two quantities move together; causation means one of them actually makes the other change. Ice cream sales and drownings rise together every summer, but neither causes the other — hot weather causes both. A third variable like that is called a confounding variable.

An example

Ice cream sales ↑, drownings ↑, and the cause of both is the weather

Record weekly ice cream sales and weekly drownings in a coastal town across a year and you get a strong positive correlation. It is real: the two genuinely rise and fall together. Closing every ice cream shop would not save a single life, because hot weather is doing both jobs — it sends people to buy ice cream and it sends people into the sea, and more swimmers means more drownings.

Following the ice cream all the way

It is worth tracing this one properly rather than treating it as a slogan. The data are honest, the correlation is strong, and the conclusion is still nonsense.

The variable driving both is temperature, and it is called a confounding variable because it is tangled up with the two you plotted. Once you know how hot the week was, the connection between the two columns largely disappears — the sales tell you nothing further about the drownings.

That last sentence is also how you would test it. Group the weeks by temperature and look within each group: if hot weeks with heavy ice cream sales have no more drownings than hot weeks with light sales, ice cream is exonerated. Better still, count drownings per swimmer rather than per week, and the apparent summer surge shrinks.

Four reasons two things move together

Before deciding that A causes B, there are three other explanations to rule out. Naming them is usually what a question is looking for.

Coincidence matters more than people expect. Compare enough pairs of unrelated quantities and some will line up beautifully by chance alone, which is why a single striking correlation found by trawling data is weak evidence for anything.

A causes B — the claim being made.
B causes A — reverse causation. People who exercise less are more often ill; illness also stops people exercising.
C causes both — confounding, as with the weather.
Chance — the pattern is a coincidence, especially in a small sample or when many comparisons were tried.

What it takes to claim a cause

Three conditions have to hold before a causal claim is fair. The cause must come before the effect in time. The association must survive when the plausible confounders are held fixed. And there must be a mechanism that makes sense, ideally with the effect growing as the exposure grows and the same result appearing in different studies.

The cleanest way to satisfy the second condition is an experiment in which the researcher decides who gets the treatment, by a random allocation. Randomising evens out the confounders, known and unknown, between the groups. Observational data — data collected by watching rather than by intervening — can support a causal claim strongly, but it rarely settles one on its own.

Common questions

Does correlation ever imply causation?

Correlation is evidence for causation, never proof of it. Where the data come from a properly randomised experiment, a correlation between treatment and outcome does support a causal conclusion, because randomising removes the confounders. Where the data were merely observed, the same correlation leaves the alternative explanations standing.

What is a confounding variable?

A third quantity that influences both of the variables you plotted, creating a correlation between them without either one affecting the other. Temperature is the confounder behind ice cream and drownings. Where a confounder is suspected, the fix is to compare like with like — look within groups that share the same value of it.

How should I answer "does this graph prove that X causes Y?"

Say no, explain that the graph shows correlation only, and then name a specific alternative explanation for these variables. A generic "correlation is not causation" tends to score poorly. Suggesting a plausible third factor, or pointing out that the effect could run the other way, is what earns the mark.

Can there be causation without correlation?

Yes. If a drug helps one group and harms another equally, the overall correlation can be zero while real causal effects exist in both directions. It also happens when the relationship is curved, or when the data cover too narrow a range for the effect to show up.

Last updated August 21, 2026