Module 01 of 15

Correlation vs. Causation

Why two things moving together does not show that one causes the other.

AMathematical Foundation

Pearson's correlation coefficient r measures how closely paired observations (xᵢ, yᵢ) cluster around a straight line. It ranges from −1 (perfect negative linear relationship) through 0 (no linear relationship) to +1 (perfect positive linear relationship). r² is the share of the variance in y that a straight line in x accounts for. A causal claim is different in kind: it says that changing X, with everything else held fixed, would change Y. Correlation is a property of data you observe; causation is a claim about what would happen under an intervention you may never observe.

r=\frac{\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum_{i=1}^{n}(x_i-\bar x)^2}\,\sqrt{\sum_{i=1}^{n}(y_i-\bar y)^2}}

In words: r equals the sum of the products of each x deviation and y deviation, divided by the square root of the summed squared x deviations times the square root of the summed squared y deviations.

$x_i, y_i$: the i-th pair of observations
$\bar x, \bar y$: the sample means of x and y
$n$: number of pairs

Assumptions

r measures linear association only; a strong curved relationship can have r near 0.
r is sensitive to outliers: one extreme point can create or destroy a correlation.
r is undefined if either variable has no variation.
Nothing in the formula identifies direction of causation or rules out a third variable.

BIntuitive Explanation

Ice-cream sales and drowning incidents rise together in summer. Banning ice cream would not prevent drowning: hot weather drives both. Whenever you see two variables move together, there are at least four stories: X causes Y, Y causes X (reverse causation), something else causes both (confounding), or the pattern is chance or a by-product of how the data were collected (selection). Correlation tells you that one of these stories is needed; it does not tell you which.

CConnection to Debate

Many debate claims have the form “places or people with more X have worse Y, so reducing X would improve Y”. For example, a speaker might argue that heavier social-media use is associated with lower wellbeing, so authenticity-focused platforms that change how people use social media would help. The rebuttal is not “correlation is not causation” as a slogan, but a specific alternative: perhaps people who already feel worse use social media more (reverse causation), or loneliness drives both (confounding). Strong responses name the alternative mechanism and ask what evidence would distinguish them: a randomised experiment, a natural experiment, or a dose-response pattern within the same people over time.

DReal-World Application

Social media use and wellbeing
Competing interpretation
Evidence on authenticity and wellbeing mixes designs. In a correlational study of 10,560 Facebook users, more authentic self-presentation was associated with slightly higher life satisfaction (Bailey et al., 2020, Study 1). That association cannot by itself show that posting more authentically would raise satisfaction: happier people may simply present themselves more openly. The same paper's two-week within-person experiment found higher mood in the authentic-posting week but no change in life satisfaction, and a randomised experiment found that realistic “Instagram vs reality” images reduced body dissatisfaction compared with idealised images, immediately after viewing (Tiggemann et al., 2020). The experiments support short-run causal claims; the correlation is consistent with them but does not establish them.
Sources
- Authentic self-expression on social media is associated with greater subjective well-beingErica R. Bailey, Sandra C. Matz, Wu Youyou, Sheena S. Iyengar (Nature Communications 11, 4889), 2020-10-06
- Social media is not real: The effect of ‘Instagram vs reality’ images on women's social comparison and body imageMarika Tiggemann, Isabella Anderberg (New Media & Society 22(12), 2183–2199), 2019-11-16(abstract only)
Zoning reform and rents
Competing interpretation
Pew's comparison of Minneapolis (housing stock +12%, rents +1%, 2017–2022) with the rest of Minnesota (+4% and +14%) is descriptive: Minneapolis differs from the rest of the state in many ways besides its zoning, so the gap is a correlation, not a measured effect of reform. Studies that compare upzoned areas with similar non-upzoned areas before and after reform are more informative, and they disagree: Auckland's 2016 upzoning was followed by an estimated 34,614 additional permits over 2016–2021, while Chicago's targeted upzonings raised property values without measurably increasing construction within five years.
Sources
- Minneapolis Land Use Reforms Offer a Blueprint for Housing AffordabilityLinlin Liang, Adam Staveski, Alex Horowitz (The Pew Charitable Trusts), 2024-01-04
- The impact of upzoning on housing construction in AucklandRyan Greenaway-McGrevy, Peter C. B. Phillips (Journal of Urban Economics, 2023), 2023
- Upzoning Chicago: Impacts of a Zoning Reform on Property Values and Housing ConstructionYonah Freemark (Urban Affairs Review 56(3)), 2019-01-29(abstract only)

These examples describe real policies and research. The numbers in the interactive below and in the practice questions are hypothetical teaching data unless a question says otherwise.

EInteractive Visualization

Add, drag or delete points on a scatter plot and watch Pearson's r and the fitted line recompute.

Illustrative example — hypothetical data.Points are hypothetical teaching data. Click the plot to add a point; drag points to move them.

Keyboard: Tab to a point, arrow keys move it (Shift for larger steps), Delete removes it.

Start from a dataset

Pearson r

1.00

r² (share of variance in Y linearly explained)

0.99

Points

Fitted slope

0.92

These points show a very strong positive linear association.

Even a strong r says nothing about which variable causes which, or whether a third factor drives both.

r = \frac{\sum_i (x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum_i (x_i-\bar x)^2}\,\sqrt{\sum_i (y_i-\bar y)^2}}

In words: r equals the sum of products of deviations from the means, divided by the square root of the sum of squared x deviations times the square root of the sum of squared y deviations.

FDebate Practice

GLimitations

Correlation is still informative: it can reveal patterns worth investigating, and a lack of correlation where a theory predicts a strong one is evidence against that theory. But observational correlations cannot, on their own, establish the direction of causation, rule out confounders or tell you what a policy would do. Causal claims need designs that create or exploit variation in X that is unrelated to other causes of Y (randomised trials, natural experiments, regression discontinuities, difference-in-differences) plus a plausible mechanism.