Correlation vs causation: why linked doesn't mean caused

Correlation vs causation is the difference between two things moving together and one thing making the other happen. Ice cream sales and drownings rise and fall together, a correlation, but ice cream doesn't cause drowning: hot weather drives both. A correlation is a pattern in data. Causation is a claim about what would change if you intervened, and showing it takes more than the pattern.

In short

• A correlation measures how closely two quantities move together, from −1 to +1; it says nothing by itself about why.
• A correlation can appear without causation through chance, a hidden third factor, cause running the other way, or the way the data were selected.
• The surest test of a cause is a randomized experiment; when that's impossible, natural experiments and careful reasoning about the evidence can get close.

What is a correlation, exactly?

Two quantities are correlated when knowing one helps predict the other. They are positively correlated when they rise together, like height and weight, and negatively correlated when one rises as the other falls, like a car's age and its resale price. The usual measure is Pearson's correlation coefficient, written r. It runs from −1, a perfect falling straight line, through 0, no straight-line relationship, to +1, a perfect rising one.

Here is r worked out by hand for five summer days of temperature and ice cream sales.

Day	Temperature (°C)	Cones sold	Temperature minus its average	Sales minus their average	The two gaps multiplied
1	20	120	−5	−32	160
2	22	130	−3	−22	66
3	25	155	0	3	0
4	28	170	3	18	54
5	30	185	5	33	165

1. Find the averages: 25 °C and 152 cones.
2. For each day, find how far each value sits from its average, and multiply the two gaps. Hot days with high sales give a positive product; so would cold days with low sales. The products add up to 445.
3. Square the temperature gaps and add them: 25 + 9 + 0 + 9 + 25 = 68. Do the same for sales: 1,024 + 484 + 9 + 324 + 1,089 = 2,930.
4. Divide the sum of products by the square root of 68 × 2,930: 445 ÷ √199,240 = 445 ÷ 446.4, which gives r ≈ 0.997.

That is almost a perfect positive correlation, and in this case hot weather really does sell ice cream. But the same arithmetic would give an equally high r for two things with no causal link at all. The number measures how tightly two things move together, never why. It also only measures straight-line patterns: a relationship that rises and then falls can have an r near 0.

What does it take for one thing to cause another?

X causes Y if changing X, and nothing else, would change Y. The words "and nothing else" carry the weight. Causation is a claim about a world that wasn't observed: what would have happened to these same people, or this same city, if X had been different. Data from one world can show only that X and Y go together, which is why the step from correlation to cause always needs an argument about what else could explain the pattern. That argument is the core of weighing evidence.

Why can a correlation mislead?

There are four main ways a correlation can appear without the cause people assume.

Reason	What happens	Example
Chance	Two unrelated series happen to move together, especially when many pairs are searched	Tyler Vigen's Spurious Correlations collects thousands, built by trawling public data for matches
Confounding	A third factor drives both	Hot weather raises both ice cream sales and drownings, because more people swim
Reverse causation	Y causes X, not X causes Y	Patients who take more painkillers report more pain: the pain causes the painkillers
Selection	The way the data were gathered creates the link	Among hospital patients, two unrelated illnesses can look linked, because having either one gets a person admitted

Chance correlations are easy to manufacture. Compare enough pairs of yearly figures, and some will match closely by luck, which is the same trap as testing many hypotheses and reporting the one that gives a small p-value. Confounding is the most common real-world problem: children with bigger feet read better, because older children have both. The selection effect in the table is called Berkson's paradox, and its everyday cousin is survivorship bias, where only the cases that made it through are seen. A fifth trap is subtler: people picked because they were at an extreme often look better or worse next time for no reason but luck, which is regression to the mean.

How did correlation mislead doctors about hormone therapy?

For years, observational studies found that women who took hormone therapy after menopause had less heart disease than women who didn't. Many doctors took the correlation as a cause, and hormones were widely recommended partly to protect the heart.

The Women's Health Initiative tested it with a randomized trial. More than 68,000 women aged 50 to 79 took part in its clinical trials, and in the hormone trials women were assigned by chance to hormone pills or a placebo. Estrogen plus progestin raised the risk of heart disease, stroke, blood clots and breast cancer, and the National Heart, Lung, and Blood Institute now says hormone therapy should not be used to prevent heart disease. The early correlation was real, but a likely explanation is confounding: the women who chose hormones tended to be healthier and better off in ways that protected their hearts anyway. This explains a piece of research history; it isn't medical advice, and hormone therapy is still an option for some women's menopause symptoms.

How do researchers find real causes?

The strongest tool is the randomized controlled trial. Assigning people to groups by chance means that age, income, health habits and every other factor, measured or not, end up roughly balanced between the groups. Then a difference in outcomes can be put down to the treatment. Comparing against a dummy treatment also removes the effect of expectation, which the placebo effect explains.

Randomizing is often impossible or wrong: nobody can assign people to smoke for 30 years. Then researchers look for natural experiments, situations where chance or a rule sorted people into groups much as a trial would. In London's 1854 cholera epidemic, John Snow compared households on the same streets supplied by two water companies, one drawing from a sewage-polluted stretch of the Thames and one from cleaner water upriver, and found far more deaths among the first company's customers. The 2021 economics Nobel went to David Card, Joshua Angrist and Guido Imbens for showing what cause and effect natural experiments can reveal, as the Royal Swedish Academy of Sciences put it.

Where neither is possible, scientists weigh the whole pattern. In 1965, the statistician Austin Bradford Hill, who had helped show that smoking causes lung cancer, set out nine questions to ask of an association, in The Environment and Disease: Association or Causation? They include how strong it is, whether it is found again and again, whether the cause comes before the effect, and whether more of the cause brings more of the effect.

How can you tell whether a claim of cause holds up?

A short checklist covers most headlines that say one thing "is linked to" another:

• Was it an experiment or an observation? Only randomization balances the factors no one measured.
• What third factor could drive both? Age, income, weather and general health are the usual suspects.
• Could the arrow point the other way? Ask whether the outcome could lead to the supposed cause.
• Who is missing from the data? People who dropped out, died, or were never counted can create a link.
• How many things were tested? One striking result among hundreds of comparisons may be luck.
• Does more of X bring more of Y, and does X come first? A dose and a timeline make a cause more believable.

Check yourself

• Without looking back, explain why a correlation of 0.997 between two things doesn't show that one causes the other.
• Give one example each of confounding and reverse causation, in your own words.
• Why does assigning people to groups at random solve the confounding problem?

Sources

• Correlation does not imply causation — Wikipedia
• The Environment and Disease: Association or Causation? — Austin Bradford Hill, Proceedings of the Royal Society of Medicine, 1965
• Women's Health Initiative (WHI) — National Heart, Lung, and Blood Institute
• The Prize in Economic Sciences 2021, press release — Royal Swedish Academy of Sciences, 2021
• Spurious Correlations — Tyler Vigen
• Berkson's paradox — Wikipedia