Correlation Is Not a Cause
Two variables can move together because one drives the other, because the second drives the first, or because a lurking third variable moves both — and correlation alone cannot say which. · 12 min
You have surely noticed two measurements that rise and fall together. The busier the ice-cream stands, the more swimmers get into trouble at the beach. Taller children tend to read better. Towns that send more firefighters to a blaze tend to suffer costlier damage. In each case the numbers genuinely move together. The tempting next thought — that one of them must be causing the other — is exactly the mistake this lesson is built to catch.
Guess before you learn
Across many summer months, ice-cream sales and drowning deaths rise and fall almost in step. What is the most sensible reading of that?
Hot weather is the hidden mover: heat sends ice-cream sales up and also sends more people into the water, where more get into difficulty. If the first choice tempted you, keep that pencil mark. Jumping from moves together to causes is the single most common error in reading statistics, and the rest of this lesson is about spotting it.
9–12
3–5
When two measurements rise and fall together, we say they are correlated. But a correlation has three possible stories behind it. Maybe the first thing causes the second. Maybe the second causes the first. Or maybe a hidden third thing causes both. The correlation by itself cannot tell you which story is the true one.
6–8
A correlation is a tendency for two variables to move together. It is a starting question, not a finished answer. Behind any correlation between A and B lie at least three explanations: A causes B, B causes A, or a confounding variable C causes both while A and B share no direct link.
The ice-cream case is the third kind: hot weather (C) raises both ice-cream sales (A) and swimming (B). To move from A and B are correlated to A causes B, you need more than the correlation — you need a reason to rule the other two stories out.
9–12
Given a correlation between A and B, exactly which causal picture holds is not visible in the data alone. Four candidates compete: A causes B; B causes A (reverse causation); a confounding variable C drives both (common cause); or the match is chance in a small sample (coincidence). A correlation is consistent with all four.
This is why the slogan runs correlation does not imply causation. It does not forbid it either — real causes do produce correlations. The correlation simply cannot, on its own, distinguish a genuine cause from a shared one. The one dependable way to isolate cause is a controlled experiment, where you change A yourself and watch whether B follows.
K–2
Two things can happen together without one making the other happen. When the sun is out, we eat ice cream and we go swimming. The ice cream did not send us swimming. The sunny day did both.
Undergrad
Represent the variables as a directed graph. An observed association between A and B can arise from a directed path A → B, a path B → A, a common cause A ← C → B (a confounder opening a backdoor path), or, more subtly, from conditioning on a common effect A → S ← B (collider bias). Each produces correlation without A directly causing B.
Randomized assignment of A severs every incoming edge to A, closing all backdoor paths in expectation, so a post-randomization association identifies a causal effect. Observational work must instead assume the confounders are measured and adjusted for — an assumption no correlation can verify. Simpson's paradox is the extreme warning: an association can reverse sign once a lurking variable is accounted for.
Postgrad
In the potential-outcomes framework, the causal effect of A on B compares B(a) across counterfactual settings of A, whereas a correlation summarizes the observational joint distribution P(A, B). A nonzero Pearson correlation is compatible with A → B, B → A, a latent common cause L, selection on a collider, or sampling noise; these are observationally equivalent without further structure.
Identification of a causal estimand therefore requires assumptions the data cannot check — ignorability (no unmeasured confounding) via Pearl's back-door criterion, or an exogenous intervention formalized by the do-operator, P(B | do(A)). d-separation on a hypothesized DAG tells you which conditional independences distinguish the rival structures; only intervention, or an assumption playing its role, licenses the causal claim.
confounding variable
A third quantity that influences both variables under study, producing an association between them without either one causing the other. Also called a lurking variable. Hot weather is the confounder behind ice cream and drownings.
So how do you tell a real cause from a shared one? Not from the correlation alone — you have to reason about the world behind the numbers. Two habits help. First, hunt for a lurking variable: ask what else could be driving both quantities up together. Second, where you can, run an experiment — change one variable deliberately and watch whether the other follows. Watching two numbers move together only raises the question. Cause is the answer, and a correlation never supplies it by itself.
Why is this true?
Why can a very strong correlation — say r = 0.95 — still fail to prove causation?
Because strength measures only how tightly two numbers track each other, never why. A lurking variable can drive both just as tightly as a true cause would, so even a near-perfect correlation stays fully consistent with no direct link at all.
Practice — new ink and old, interleaved
1.A study reports a correlation of r = 0.8 between hours of sunlight and monthly ice-cream sales. What may you correctly conclude?
2.A histogram is clearly left-skewed. Where is the mean relative to the median?
3.Sketch a scatter of five points showing a strong positive association between study hours (x) and test score (y).
4.A regression line for predicting test score from study hours is score = 8 · (hours) + 50. Predict the score after 6 hours of study.
5.Explain in one sentence why r has no units.
6.In one sentence: what is a confounding variable, and how does it make a correlation misleading?
A confounding variable influences both measured quantities at once, so they move together even though neither one causes the other.
How close were you? Grade yourself honestly — it sets your review date.
7.Neighborhoods with more bars also record more traffic accidents. A reporter concludes that bars cause accidents. The most likely lurking variable is:
8.Match each reading to what it describes.
9.Order these three scatters from weakest to strongest relationship.
- a wide, loose band
- a moderate oval cloud
- points nearly on a line