Correlation Is Not Causation

A real pattern

A town records, month by month, how much ice cream is sold and how many people drown. Plotted against each other, the dots climb from lower left to upper right: in months when more ice cream is sold, more people drown. That is a positive correlation, and it is a real one. It shows up year after year.

It is tempting to read it as a cause: ice cream makes people drown, perhaps because swimming on a full stomach is dangerous. The scatter plot cannot say whether that is true. It shows only that the two measurements rise and fall together.

Each dot is one month. Months with more ice-cream sales have more drownings.

A third thing behind both

Ask what else changes from month to month. Hot weather does. On a hot day more people buy ice cream, and more people go swimming. More people in the water means more drownings.

So the weather drives both measurements, and neither one drives the other. Ice cream and drownings rise together because they share a cause. A hidden cause like this, which moves two measurements at once without appearing in either, is called a lurking variable.

The same thing happens elsewhere. Umbrella sales and traffic accidents rise together, because rain brings out umbrellas and makes roads slippery. Sales of electric fans and cases of sunburn rise together, because sunny weather brings both.

One cause with two effects. Hot weather sells ice cream and sends people swimming.

Holding the third thing still

Children with bigger feet tend to read better. Across a primary school, from the youngest class to the oldest, a scatter plot of shoe size against reading score climbs clearly.

Age is behind both. Older children have bigger feet, and older children have had more years of practice at reading. To see whether foot size itself matters, look only at children of the same age, so that age cannot change from one dot to the next.

For the nine-year-olds alone, the dots form a shapeless cloud. A nine-year-old with large feet reads no better than a nine-year-old with small feet. Once age is held still, the correlation is gone, which shows that age was carrying it.

The whole school, from the youngest children to the oldest: bigger shoe sizes go with higher reading scores.

The nine-year-olds only, on the same axes: a shapeless cloud, with no correlation.

Four ways a correlation can arise

When two measurements x and y are correlated, there are four possible explanations, and the scatter plot looks the same for all of them.

x causes y. More hours of practice on a piano lead to fewer mistakes in a performance.

y causes x. Towns with more crime tend to employ more police officers. The police do not cause the crime: the crime is why the towns hired them.

A third variable causes both, as hot weather does for ice cream and drownings, and age does for shoe size and reading.

The pattern is chance. With only a handful of dots, two unrelated measurements can line up by luck. A pattern that holds over many readings, and again in new data, is much less likely to be chance.

A question, not an answer

A correlation is evidence that two measurements are linked. It is a good reason to ask why. It is not, on its own, evidence of which one causes the other, or whether either does.

A correlation can still be used to predict. A lifeguard service that knows ice-cream sales are high this week can expect more swimmers and more emergencies, and can put more lifeguards on the beach. The prediction works whatever the cause.

What a correlation cannot support is a claim about changing one measurement to change the other. Banning ice cream would lower ice-cream sales, but it would not cool the weather or keep anyone out of the water, so the drownings would stay the same.

How a cause is shown

To show that x causes y, change x on purpose and keep everything else the same. That is an experiment. Split the volunteers into two groups by lot, for example 40 volunteers into two groups of 40 ÷ 2 = 20, give one group the treatment and the other none, and compare the results. Splitting by lot spreads every other difference, such as age, fitness or the weather on the day, evenly between the groups, so a difference in the results comes from the treatment.

Some experiments cannot be done: no one can be made to smoke for thirty years. Then a cause is argued from many separate studies that agree, a clear mechanism for how x could change y, and larger amounts of x going with larger effects on y.

The usual mistakes

Reading a correlation as a cause. A third variable driving both gives exactly the same scatter plot.

Calling the two measurements unrelated. The dots really do climb together, so the measurements are linked. What is in doubt is the explanation, not the pattern.

Naming a third variable that moves only one of the two. The price of ice cream changes how much is sold, but it has nothing to do with who goes swimming, so it cannot explain why drownings rise with sales. A lurking variable has to change both measurements.

Proposing a ban or a change from a correlation alone. That needs a cause, and the scatter plot never tested what the change would do.