The Reversal
He is better every year, and he still loses.
Two baseball players, David Justice and Derek Jeter, and the three seasons where their careers overlapped. One of them hit for the higher batting average in 1995. He did it again in 1996, and again in 1997. Then you add the seasons together and he loses.
Nothing below is faked and no season is left out. The averages are the real ones, and you are shown all of them before you touch anything. The only number you will change is how the at-bats are split across the seasons. You will still flip the winner.
The record
Here is everything, both men, every season. Batting average is just hits divided by at-bats. The at-bats column is the one to keep an eye on: it is on the table, in plain sight, and it is also the number that is about to do all the work.
| Player | Season | At-bats | Hits | Average |
|---|
Every batting average in that table is real and stays exactly where it is. Nothing is hidden from you now and nothing is revealed later that was withheld.
The one thing you will change is how each man’s at-bats are distributed across his seasons. You are not editing the record; you are choosing how to weigh it.
Re-weight the seasons
Drag the slider. At one end every season counts equally; at the other, seasons count in proportion to how many times the man actually batted, which is the real record and is marked home. Watch the two diamonds, each man’s combined average, slide along the axis.
Move the slider. Both ways of combining the seasons are legitimate; that is the whole problem.
Two hitters on one axis – each diamond is a combined average
Name the better hitter
· unlocks once you move the sliderStop wherever you find defensible and say who was the better hitter over these seasons. Pick the one you would put your name to. There is no unpick button, here or in a published table.
–
The at-bats you weren’t weighing
· unlocks when you commitYou changed no batting average. You changed the weights, and the winner changed with them.
The figure has not changed its axis or its data. The at-bats behind each season are simply drawn to scale now, and the weighting has slid to the real record.
So which number is right
· unlocks after the revealThe lazy version of this lesson is “so always break the data down.” That is false, and it is the first thing a statistician will test you on. Whether to split or to pool is not a statistical question at all. It is a question about cause.
The rule is about cause. If the third variable decided who got which treatment, the split numbers are right. If the treatment caused the third variable, pooling is right. And if something nobody measured is still in play, neither table is right.
At-bats are none of those. No hidden bias handed one man his playing time, and batting more did not change how well either man hit. So neither number here is wrong. The per-season answer tells you who was the better hitter in a season; the pooled answer tells you who got more hits per at-bat across the span. They answer two different questions. The table only ever showed you one of them.
What just happened
· unlocks after the repairYou flipped a verdict about two men’s careers without altering one digit of their record. Everyone who has ever quoted the combined number and everyone who has ever quoted the per-season one was reading the same twelve numbers and answering a different question, mostly without noticing there was a choice to make.
Every batting average was in front of you. What was never made visible was how unevenly the two men’s at-bats were spread across their good and bad seasons. That single omission is enough to reverse the winner, with nothing hidden and nothing faked. It is the variable nobody split by.
Edward Simpson set the paradox out in 1951 (JRSS-B 13(2), 238). Colin Blyth gave it its name in 1972 (JASA 67(338), 364). Judea Pearl settled which number is correct, and when, in 2014 (The American Statistician 68(1), 8). The clinical version the argument leans on is Julious & Mullee (BMJ 309, 1480, 1994). The baseball line is from Baseball-Reference; the example is a favourite of Ken Ross’s A Mathematician at the Ballpark.
“Always disaggregate” is wrong: sometimes the correct answer is in neither table. Simpson made the point himself in 1951. He wrote out a table of playing cards, called the combined answer the sensible one, then relabelled the identical numbers as a medical trial and called the split answer sensible instead. Same arithmetic, opposite verdict. The numbers never decide; the story attached to them does. Pearl’s account of exactly when to trust which table is argued rather than closed, which is itself worth knowing.