Every group in the data trends down. The total trends up.

By the 42Math team · 1 July 20265 min read

Nobody has made an error, nothing is missing, and the sample is not too small. Two correct calculations on identical data point in opposite directions.

Open the Causality Sandbox and press its Simpson preset. The panel reports a correlation between X and Y of 0.1702 and a fitted slope of 0.0928. Positive: as X rises, Y rises.

Directly beneath, the same panel fits the same points again after splitting them by a third column, G. Slope within G = 0: −0.5842. Slope within G = 1: −0.6109.

Every group goes down. The total goes up. No rows were added, removed, weighted or transformed between those two calculations, only grouped.

How both can be true

None of it turns on sample size or statistical significance.

The third column is doing two jobs at once. The panel prints a correlation between X and that variable of 0.8061, and between Y and it of 0.6884. So the groups do not sit on top of each other in the scatter: one occupies the low-X, low-Y region and the other the high-X, high-Y region. Each group is an internally downward-sloping cloud, and the two clouds are stacked along a rising diagonal.

Draw a single line through both clouds and it is dominated by the offset between them, because that offset spans a much wider range than the width of either cloud. The pooled line is measuring the gap between the groups. The within-group lines are measuring the relationship inside them. They disagree because they are measuring different things, and the arithmetic is impeccable in both cases.

It also explains why the effect cannot be fixed by more data. Collect ten times as many points and both estimates get tighter — around their existing, opposite values. Simpson's paradox is not sampling noise, and it is not a failure of the regression. It is what happens when you aggregate over a variable that matters.

The case that made it famous

Berkeley's graduate admissions, autumn 1973. Around 44% of male applicants were admitted against 35% of women, a nine-point gap across more than twelve thousand applications, far too large to be chance.

Bickel, Hammel and O'Connell then did the split, department by department. Few departments departed significantly from expectation in either direction, and about as many appeared to favour women as men; what survived proper pooling was a small bias in favour of women. The nine-point gap was produced almost entirely by which departments people applied to. Women applied in greater numbers to departments with low admission rates for everyone; men applied to departments admitting nearly half of all comers. Department was playing exactly the role that G plays in the preset, correlated with the input and with the outcome.

What makes the Berkeley study worth reading is the care of its conclusion. The authors did not announce that there was no bias. They pointed out that the bias had moved: whatever was steering applicants toward crowded, competitive fields sat upstream of the admissions committees, and their data could not see it. Disaggregating told them where not to look, which is a real result and a modest one.

Which number should you use?

The obvious lesson — always disaggregate — is wrong.

Consider a drug trial where treatment assignment correlates with patient severity: sicker patients were given the new drug more often. Pooled, the drug looks harmful. Split by severity, it helps in every stratum. You want the split, because severity influenced who got the drug and also influenced the outcome. It is a confounder, and adjusting for it removes a distortion.

Now consider a trial where the drug works by lowering blood pressure, and you split by blood pressure measured after treatment. Within each stratum the drug appears to do nothing: you have grouped by its own mechanism. Here the pooled number is the right one, and adjusting destroys the effect you were trying to measure. Blood pressure here is a mediator, and the sandbox carries a preset for its mirror image: a collider, a variable that both X and Y point into. Group by a mediator and you erase an effect that is real; group by a collider and you manufacture one that never existed.

These two cases can produce identical tables of numbers. Nothing inside the data distinguishes them. The choice between pooled and split is a claim about which variable came first — about causal structure — and causal structure is not in the columns. This is why the paradox is a permanent feature of observational work rather than a puzzle to be solved once: the statistics narrow the possibilities, and something outside the statistics has to pick.

Simpson himself made the point in 1951, in a paper more cautious than its reputation. He constructed the reversal in a contingency table and then argued that the "sensible interpretation" depends on the meaning of the categories, not on the table. The arithmetic is not ambiguous. The question is.

What to do about it

Three habits follow, and they are cheap.

Plot before you fit. A pooled slope through two offset clouds is visible instantly in a scatter and invisible in a correlation coefficient. The regression tool shows the residuals alongside the fit, and stacked groups leave an unmistakable signature there: residuals that are large, structured and grouped rather than scattered.

Write down what you think caused what, before you compute. If you cannot say why a variable belongs in the model, you also cannot say whether adjusting for it fixes a confounder or destroys a mechanism — and you will be able to justify either choice after the fact, which is the danger.

Be suspicious of a single headline number computed across heterogeneous groups. National averages, company-wide metrics, aggregate trends: each is a pooled slope across clouds that may be moving the other way. The reversal in the preset needed only two groups and one correlated variable. Real data offers dozens of both.

Published 1 July 2026 · corrections welcome via the corrections page.