The pooled figure
Across 116,382 completed professional singles tennis matches, a player who loses the opening set goes on to win 16.52% of the time.
One rate, one population, and it looks like a finished answer.
The conditional rates
Now condition on the margin of that first set, which is recorded for every match:
| first set lost | recovery rate | matches |
|---|---|---|
| 7-6 | 21.92% | 14,265 |
| 6-4 | 19.41% | 25,546 |
| 7-5 | 18.89% | 10,252 |
| 6-3 | 17.39% | 25,970 |
| 6-2 | 13.30% | 19,739 |
| 6-1 | 11.06% | 14,269 |
| 6-0 | 7.18% | 5,833 |
The pooled 16.52% is a weighted average of seven rates spanning 7.18% to 21.92%. It is arithmetically correct and it describes no actual situation, because no player is ever in the pooled state. Every player is in exactly one row.
Two features worth pointing at in a classroom
The monotonicity is complete. Seven ordered categories, and the rate falls at every step without a single inversion. For an ordinal predictor that is about as clean a signal as observational data produces, and it is worth more than any single effect size.
The near-equal cells make the comparison assumption-free. The 7-6 row holds 14,265 matches and the 6-1 row holds 14,269. Four apart. A difference between two groups of that size cannot be attributed to unequal precision. That is usually the first objection raised, and here it cannot be raised at all.
Where the confidence intervals change the reading
The same data split by competition level comes with 95% intervals, and they do more work than the point estimates:
| tour | rate | 95% interval | matches |
|---|---|---|---|
| men's professional | 18.21% | plus or minus 0.72 | 10,945 |
| men's second tier | 17.32% | plus or minus 0.43 | 30,406 |
| women's second tier | 17.25% | plus or minus 1.08 | 4,731 |
| women's professional | 17.04% | plus or minus 0.63 | 13,543 |
| women's entry level | 15.97% | plus or minus 0.42 | 29,331 |
| men's entry level | 15.18% | plus or minus 0.42 | 27,415 |
Rows two and three sit at 17.32% and 17.25%, which invites a comparison. Their intervals are 0.43 and 1.08, because one rests on 30,406 matches and the other on 4,731. Those estimates are indistinguishable and the table says so only if you read the fourth column.
Rows one and six are a different matter. 18.21% against 15.18%, intervals of 0.72 and 0.42, no overlap worth arguing about. One comparison in that table is unsupportable and one is solid, and nothing in the presentation distinguishes them.
A validation worth copying
The same corpus gives a service hold rate of 78.62% for men's professional matches, across 230,715 service games. Women's reads 64.36% across 266,503. Both reproduce figures published elsewhere by other methods.
That is the useful kind of check. The count was built to answer a different question and tuned to neither target. Landing on both is evidence the pipeline is sound, not evidence it was fitted.
The exercise
Take the pooled 16.52%, the seven conditional rates and the cell counts, and ask a class what the pooled figure is for. The honest answer is that it is a summary of a population nobody belongs to, and that every useful statement here needs a condition attached.
The figures come from the Live Tennis API , a JSON service for developers. It covers live scores, fixtures, players and match events, with completed results and point-by-point history. The study file is public at blog.livetennisapi.com/studies.json under CC BY 4.0, free to reuse with attribution. The whole table can be reproduced from it.
Method: completed best-of-three professional singles matches, 1 January 2023 to 14 September 2026. Retirements and walkovers excluded. Counted from the point-by-point record.