In Bill James we trust: using the Pythagorean Method to estimate winning percentage in the National Hockey League
Bibliographic record
Abstract
Sabermetrician Bill James (1980) advanced a formula predicting a baseball team’s winning percentage based on runs scored and runs allowed called the Pythagorean method. Cochran and Blackstock (2009) revised this method to estimate a hockey team’s winning percentage using goals scored and goals allowed over the course of a season. While it is valuable to estimate the end of season winning percentage, it is equally important to know whether the Pythagorean method can be used to predict winning percentages at critical moments during the season. Thus, the purpose was to compare the estimated winning percentage at various points in the season using the Pythagorean method to the actual winning percentage at the end of the season. Using archival data from every NHL regular season game from the 2005-06 through 2010-11 seasons (N = 7,365), the results indicated the Pythagorean method estimated both home and away winning percentages with a high degree of precision. For home winning percentage, the actual end of season value was 54.95% while the estimated values were 54.96% at pre-Christmas, 55.01% at the All-Star break, 54.52% at the trade deadline, and 54.84% at the end of the season. As for visiting winning percentage, the actual end of season value was 45.09% while the estimated values were 45.04% at pre-Christmas, 45.09% at the All-Star break, 45.57% at the trade deadline, and 45.26% at the end of the season. The results are discussed in relation to player personnel decisions.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.010 | 0.063 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.004 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.005 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.013 | 0.008 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".