Low-Birthweight Paradox

Here is a genuine statistical riddle from medicine: among underweight babies, those whose mothers smoked had better survival than those whose mothers did not. Smoking protects babies? No — and seeing why not teaches you to catch a whole species of statistical lies.

The trick is the selection. Babies can be underweight for many reasons: some harmless, some deadly. Smoking mostly adds harmless smallness. So the smoking group is a mix of fine-but-small babies and sick ones — while the non-smoking group's small babies are almost all sick. Compare only the small, and smoking looks protective. The filter created the illusion.

This pattern haunts every comparison between filtered groups: why hospitalized mild cases fare worse than non-hospitalized severe ones; why the hired candidates of a harsh interviewer outperform those of a lenient one.

The rule to keep: whenever two groups were selected by different gates, comparing them measures the gates as much as the groups. Ask what filtered them before believing what distinguishes them.