Simpson's Paradox

Simpson's Paradox Puzzles

Jane Slugger leads the league in batting average for three years straight, yet she does not have the highest three year average! How is this possible?

This sounds impossible, but in fact it can happen. This phenomenon of apparent reversal of averages is called Simpson's Paradox.

Invent batting records for Jane and another player to see how this works. Jane should have a higher average than the other player for three years in a row, but the other player should have a higher cumulative average.

The formula for batting averages is

       (# hits)
  ------------------  .
   (# times at bat)

Here's another version of the same problem. Salary data for factory workers at the X Company are divided into three categories: 1 to 3 years of employment (junior); 4 to 8 years of employment (intermediate); and 9 years and higher (senior). The average salary for women is higher in all three categories, but the overall men's average salary is higher! How can this happen?