Make the total contradict every group.

Read the chart inside the colours and the message is clear and sensible: among younger people, more exercise goes with lower cholesterol; among older people, the same. Two groups, one honest downward trend each. Now rub out the colours and fit a single line to the cloud, and — with the age gap turned up — that line slopes the other way, announcing that exercise raises cholesterol. Both pictures are drawn from the very same dots. Neither is a trick of the pen.

What changed is not the data but the question. The group lines ask “for people of the same age, does exercise track cholesterol?” The pooled line asks “across everybody, do exercise and cholesterol move together?” — and across everybody, the dominant fact is that the older crowd sits up and to the right. The pooled line dutifully traces that, the march of age, and mistakes it for a fact about exercise.

Slide the age gap back to zero and the paradox dissolves: the groups land on top of one another, the between-group pull vanishes, and the pooled line agrees with the group lines. That is the tell. The reversal was never about exercise and cholesterol at all — it was age, the third variable, masquerading as a relationship between the other two.

This is exactly confounding, the lurking-variable problem, in its most theatrical form. Ordinary confounding weakens or fakes a correlation; Simpson’s paradox is the special case where the lurking variable is strong enough to flip the sign, so the total doesn’t just mislead, it says the literal opposite of every group. The cure is the same in both: find the third variable and compare like with like — age group with age group — rather than throwing everyone in one pile.

It is tempting to declare the group lines “true” and the pooled line “false”, but that is too quick. Which number to trust depends entirely on what the third variable is. If age is a confounder — something that affects cholesterol for reasons that have nothing to do with exercise — then you should hold it fixed, and the within-age trend is the honest one. That is the usual case, and it is why “adjust for age” is a reflex in medicine.

But flip the story. Suppose a drug works by raising some intermediate that then helps; splitting the data on that intermediate would slice away the very effect you are trying to measure, and the pooled figure would be the right one. Same arithmetic, opposite conclusion. This is the deeper lesson the full Simpson’s paradox entry draws out: the data hands you two true numbers and stays silent on which question matters. Only knowledge of how the world actually works — not the spreadsheet — can break the tie.

It is not a toy. The most famous case is the 1973 UC Berkeley admissions data: overall, men were admitted at a higher rate than women, hinting at bias — yet within almost every individual department, women were admitted at an equal or higher rate. Women had simply applied more often to the most competitive departments. The lurking variable was which department, and pooling across departments reversed the story.

The same shape turns up in a real comparison of kidney-stone treatments, where one treatment won for small stones and for large stones yet lost overall, because the tougher cases were handled by the better treatment. So when a single headline number contradicts what you would expect from the parts, do not assume someone lied. Ask the one question that defuses it: what were the groups, and did anyone check the trend inside them?