name | delay% alice | 30% bob | 20%
So Bob's flights are delayed less often, he's the better pilot, right? Yet:
name | night | day alice | 7/25 | 1/5 bob | 3/10 | 3/20
So now Alice looks like the better pilot!
But wait, what if the pilots are responsible for scheduling their own flights? Bob's individual batting averages might be somewhat worse than Alice's, but he's making better decisions about when to fly.
But wait, what if Alice and Bob fly out of the same airport(s), and they've agreed to let Bob fly during the day... (this is not meant to be an accurate representation of air traffic control)
When faced with a Simpson-like situation, a correct analysis usually requires considering the chain of causation, in particular, whether any stratification of the data depends on the independent variable being tested. If the stratification is a result of the test variable, it usually isn't a good one. In the Berkeley example, this possibility is the highly unlikely situation that applicants were automatically assigned to departments with gender taken into consideration -- so the stratification was valid.