Science Puzzle
The Two Maternity Wards
A town has two hospitals. In the large one, about 45 babies are born each day; in the small one, about 15. Roughly half of all babies are boys, but the exact share changes from day to day: some days it's higher than 50%, some days lower.
For a whole year, each hospital records the days on which more than 60% of the babies born were boys. Which hospital records more of those days, or do they record about the same number?
The Answer
The small hospital, by more than two to one. Over a year it can expect about 55 such days, while the large hospital can expect about 25.
Each birth is a bit like a coin toss. With only a few tosses, lopsided results are common: three heads in a row happens one time in eight. With many tosses, the lucky streaks in one direction tend to be cancelled out by streaks in the other, and the result stays close to half. Thirty heads in a row would happen about once in a billion tries.
The same is true of births. On a day with 15 babies, getting 10 or more boys isn't unusual. On a day with 45, getting 28 or more boys is much rarer. Small samples swing about; big samples settle down.
When the psychologists Daniel Kahneman and Amos Tversky put a version of this question to university students in the early 1970s, more than half answered "about the same". The mistake has real consequences. In the United States, people noticed that many of the best-scoring schools were small, and large sums of money went into creating small schools. Many of the worst-scoring schools were small too, for exactly the same reason: with fewer pupils, average scores swing further from the middle in both directions.
The principle: Sample size and variability. Small samples vary much more than large ones, so the most extreme results, both high and low, tend to come from the smallest groups.