Statistics · Research Methods
Correlation vs Causation
One of the most important distinctions in all of science — and one of the most commonly violated in journalism, business, and everyday reasoning.
Countries with more chocolate consumption per capita win more Nobel Prizes. Ice cream sales are correlated with drowning rates. Nicholas Cage film releases correlate with swimming pool drownings. These are all genuine statistical correlations — and all meaningless for understanding what causes what. The confusion between correlation and causation is arguably the most consequential error in empirical reasoning.
📘 Key Terms
Correlation: A statistical relationship between two variables — when one tends to increase, the other also tends to increase (positive correlation) or decrease (negative correlation). Measured by the Pearson correlation coefficient r, ranging from −1 to +1.
Causation: One variable directly produces or influences a change in another. Causal claims require ruling out alternative explanations — not merely demonstrating a statistical association.
The Pearson Correlation Coefficient (r)
r = Σ[(xᵢ−x̄)(yᵢ−ȳ)] / √[Σ(xᵢ−x̄)² × Σ(yᵢ−ȳ)²]
| Value of r | Interpretation |
|---|---|
| r = +1 | Perfect positive correlation |
| 0.7 to 0.9 | Strong positive correlation |
| 0.3 to 0.7 | Moderate positive correlation |
| 0 to 0.3 | Weak or no correlation |
| r = 0 | No linear relationship |
| r = −1 | Perfect negative correlation |
Why Correlation ≠ Causation: Three Reasons
1. Confounding variables: A third variable causes both. Ice cream and drowning are both caused by hot weather (the confounder) — not by each other.
2. Reverse causation: The direction is wrong. Hospitalisation correlates with death — but being admitted to hospital doesn’t cause death; being seriously ill causes both.
3. Spurious correlation: Pure coincidence over the time period studied. The Cage/drowning example — both happened to trend together over a specific decade for unrelated reasons.
💡 Key Insight
Establishing causation requires more than a correlation — it requires a credible causal mechanism, ruling out confounders, and ideally a controlled experiment or a natural experiment/instrumental variable approach. Randomised Controlled Trials (RCTs) remain the gold standard for establishing causality because random assignment ensures confounders are balanced between groups.
⚠️ Common Error
Students sometimes write ‘the correlation shows that X causes Y.’ This is never justified by a correlation alone, no matter how strong. r = 0.95 between two variables is consistent with X causing Y, Y causing X, a third variable causing both, or a chance relationship over the sample period. Always write ‘X is correlated with Y’ rather than ‘X causes Y’ unless causal evidence is provided.
Q1. A study finds a strong positive correlation (r = 0.82) between the number of police officers per capita in a city and the city’s crime rate. A politician concludes ‘more police cause more crime.’ Critically evaluate this conclusion. [6 marks]
Answer: The correlation is real but the causal interpretation is almost certainly reversed. The most plausible explanation is reverse causation: cities with higher crime rates deploy more police officers in response to crime — crime causes policing, not the reverse. There is also potential confounding: urban areas may have both higher crime and larger police forces due to population density. Establishing causation would require an exogenous shock to police numbers (e.g. a budget shock that randomly affected some cities) to use as a natural experiment, or a randomised deployment of officers across comparable areas. The correlation alone cannot support the politician’s claim.
References
1. Pearl, J. and Mackenzie, D. (2018) The Book of Why. Basic Books.
2. Angrist, J.D. and Pischke, J.S. (2009) Mostly Harmless Econometrics. Princeton University Press.
3. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. W.H. Freeman.
4. Vigen, T. (2015) Spurious Correlations. Hachette Books.
5. Field, A. (2018) Discovering Statistics Using IBM SPSS Statistics. SAGE.