Correlation vs. Causation & Simpson's Paradox

Critical Thinking

Chapter 4 · Correlation vs. Causation & Simpson's Paradox

Two things moving together doesn't mean one causes the other — and sometimes an aggregated trend can point in the exact opposite direction of what's actually happening underneath it. This chapter covers a real, absurd illustration of the first problem, and the single most famous real case of the second.

Correlation Is Not Causation: A Real, Absurd Illustration

A Real, Genuinely Strong Correlation — r = 0.947

Statistician Tyler Vigen's real "Spurious Correlations" project catalogs genuine statistical coincidences — including US per-capita cheese consumption and the number of people who died getting tangled in their bedsheets, which correlate at r = 0.947 (a near-perfect statistical relationship, on a scale where 1.0 is exact). Nobody seriously believes cheese causes bedsheet deaths — Vigen's own real point is that with enough variables compared against each other, some will correlate strongly by pure chance alone.

Three Real Reasons a Correlation Can Exist Without Causation

Coincidence: with enough variables compared, some will correlate by chance alone.
Reverse causation: B might actually be causing A, not the other way around.
A confounding variable: some third factor causes both A and B independently.

Simpson's Paradox: A Real, Famous Case

In 1973, the University of California, Berkeley's graduate admissions data showed a real, apparent gender gap: 44.2% of male applicants were admitted, against 34.6% of female applicants — a statistically significant difference that led to a real lawsuit alleging bias.

The Real Reversal, Once Broken Down by Department

Statisticians Peter Bickel, Eugene Hammel, and J. William O'Connell, publishing in Science in 1975, found that once admissions were examined department by department, no individual department showed significant bias against women — and in four of the six largest departments, women were actually admitted at a higher rate than men. The real explanation: women disproportionately applied to departments with lower overall acceptance rates (more competitive fields), while men applied more heavily to departments that accepted a larger share of all applicants. The aggregate gap reflected which departments people applied to, not how individual departments evaluated men versus women.

This real pattern — where a trend visible in combined data reverses or disappears once the data is properly broken into its real subgroups — is called Simpson's Paradox, and the Berkeley case remains its most famous real-world example.

Two Real Traps, Compared

Trap	What actually goes wrong
Spurious correlation	Two unrelated things move together, by chance or a hidden third cause
Simpson's Paradox	An aggregated trend hides or reverses a genuinely different pattern in the real underlying subgroups

Hands-On Exercises

Exercise 1

Using the three real reasons a correlation can exist without causation, propose a plausible confounding variable that could explain a real correlation between "ice cream sales" and "drowning deaths" without either one causing the other.

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Exercise 2

Explain, in your own words, exactly what "department choice" had to do with the Berkeley admissions gap — specifically, why did women's real pattern of applying to more competitive departments produce a lower aggregate admission rate even without any individual department being biased against them?

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Exercise 3

A colleague says, "Our company's overall promotion rate for men is higher than for women, so our promotion process must be biased." Using the Berkeley case as a model, explain what additional real data you'd want to see before accepting that conclusion.

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Chapter 4 Quick Reference

Tyler Vigen's real "Spurious Correlations" project shows unrelated variables (cheese consumption, bedsheet deaths) correlating at r = 0.947 purely by chance
Correlation can exist without causation via coincidence, reverse causation, or a shared confounding variable
The real 1973 UC Berkeley admissions data showed 44.2% male vs. 34.6% female admission rates in aggregate
Bickel, Hammel & O'Connell's real 1975 analysis found the reverse pattern within four of six departments — the founding real-world case of Simpson's Paradox
The real cause was which departments applicants chose, not how individual departments evaluated candidates