A trend that flips
Simpsons paradox is the name for a situation where a comparison points one way inside every subgroup and the opposite way when the subgroups are combined. It is not a trick or an error in arithmetic. Every number is correct. The reversal comes from how unevenly the groups are sized, and it is a standard reason to be careful with averages.
An illustrative example
The numbers below are invented for illustration. Two players, Ana and Ben, have each played 110 matches across two game modes, an easy mode and a hard mode. Wins are against the same pool of opponents in each mode.
| Mode | Ana | Ben |
|---|---|---|
| Easy | 9 wins from 10 (90%) | 80 wins from 100 (80%) |
| Hard | 30 wins from 100 (30%) | 2 wins from 10 (20%) |
Ana has the better win rate in the easy mode, 90 percent against 80 percent. She also has the better win rate in the hard mode, 30 percent against 20 percent. In both modes, Ana is ahead.
Now combine the modes.
- Ana: 9 + 30 = 39 wins from 10 + 100 = 110 matches. 39 ÷ 110 is about 35.5 percent.
- Ben: 80 + 2 = 82 wins from 100 + 10 = 110 matches. 82 ÷ 110 is about 74.5 percent.
Overall, Ben’s record looks far better than Ana’s, even though Ana did better in each mode. The totals tell the opposite story to the parts.
Where the reversal comes from
The cause is the mix of matches. Ana played most of her matches in the hard mode, where almost everyone wins less often. Ben played most of his in the easy mode, where almost everyone wins more often. The overall rate is a weighted average of the mode rates, and the weights are very different for the two players.
Ben’s combined rate is dominated by the easy mode (100 of 110 matches), so it lands close to 80 percent. Ana’s is dominated by the hard mode (100 of 110 matches), so it lands close to 30 percent. The difference in difficulty swamps the difference in skill.
Which number should you trust?
There is no automatic answer. It depends on the question being asked and on what is driving the grouping.
If the question is “who plays better in the same conditions”, the within-mode comparison is the fairer one, because it compares like with like. Ana is better in both.
If the question is “whose overall record is better”, the combined figure answers that, but it measures both skill and choice of opponent or mode. Here, it says more about which mode each player entered than about how well either played.
Statisticians usually think about the cause behind the grouping. If the mode affects the result and also affects who plays it, the combined number mixes two effects. This is why the paradox connects to correlation versus causation: an overall association can be produced by a hidden third factor, here the difficulty of the mode.
Examples in sport and games
The same structure appears whenever players or teams face different mixes of conditions. A batter or shooter may do better than a rival against both strong and weak opposition, yet have a worse overall average because they faced more strong opposition. A character in a video game may win more in every rank tier but have a lower overall win rate if it is mostly played at higher ranks. A treatment in a medical study can show the same pattern when patients are not assigned evenly.
These are patterns that can occur, not claims about any specific real team, game or study. When you meet a real-world headline of this type, the right move is to ask what the groups are and how large each one is.
Related traps
Simpsons paradox sits alongside other ways that statistics mislead. Small sample sizes can make a subgroup’s rate unstable: Ana’s 9 from 10 is a thin record, and a different run of ten matches could easily look quite different. Cherry-picked statistics can show only the grouping that suits a story, and the combined figure and the split figure can each be used selectively. Even the averages versus medians question is related, since a single summary number hides the shape of the data underneath.
A checklist for combined numbers
When a comparison is based on totals, it helps to ask a few questions:
- Is the data split into groups that differ in difficulty, size or conditions?
- Are the groups roughly the same size for the two things being compared? If not, the weights differ.
- Would the comparison look the same inside each group?
- Is the grouping something people chose, like a game mode, rather than something assigned at random?
If the answers show unequal mixes, the totals should be read with care. The paradox is not that statistics lie. It is that one summary number cannot always carry the full picture, and the way the data is grouped is part of the story.