Pythagorean Formula
The Pythagorean Expectation refers to a family of formulae designed to estimate how many games a team "should" have won, based on the number of points/runs they scored and allowed. It was invented by Bill James in the 1980's, when he realized that there is indeed a stable relationship between a baseball team's run differential and the number of games they win. The original version looked like this:
- Win% = (Runs Scored^2)/((Runs Scored^2) + (Runs Allowed^2))
James named his creation after the Pythagorean Theorem, because its squared terms brought to mind the famous trigonometric formula. More recently, Clay Davenport of Baseball Prospectus refined the formula, first discovering that an exponent of 1.82 was a better predictor of winning percentage than 2, and then creating a formula for a floating exponent (one that changes for each team):
- Exponent = 1.5 * log((rs + ra)/g) + 0.45
Still, in general, 1.82 works as well as the Logarithmic-Exponent method...
Baseball:
- Win% = (Runs Scored^1.82)/((Runs Scored^1.82) + (Runs Allowed^1.82))
Statistician Daryl Morey (now the General Manager of the Houston Rockets) would later find that the key concept of James' Pythagorean Formula (that run differential can predict winning %) applies not just to baseball, but to just about any sport that keeps track of points scored and allowed.
In basketball, Morey found that an exponent of 16 worked best. However, Morey did his research during the NBA's run-and-gun Eighties; APBRmetrician Dean Oliver found that 14 works better for today's slower-paced NBA. In addition, mathematician Ken Pomeroy discovered 10 to be the ideal exponent for college basketball's even slower pace (and shorter games). So, to recap...
NBA Basketball:
- Win% = (Points Scored^14)/((Points Scored^14) + (Points Allowed^14))
NCAA Basketball:
- Win% = (Points Scored^10)/((Points Scored^10) + (Points Allowed^10))
Morey also experimented with NFL scores over a ten-year period to determine an exponent for football, which turned out to be 2.37. "The number would probably be slightly different for college," he says, but since he never actually did the study on the college game (point differential in college football can be notoriously unreliable, thanks to the penchant for BCS-conference teams to play weak, blowout-prone competition at the beginning of the season), we just have to assume that 2.37 works for all levels of the game. So, in equation form, we have:
Football:
- Win% = (Points Scored^2.37)/((Points Scored^2.37) + (Points Allowed^2.37))
Hockey's goals-per-game scores tend to be comparable to that of baseball, so the NHL's exponent is generally just 2. To derive "Points" from the hockey formula, simply multiply the expected winning percentage by 2 times the team's number of games played. In other words...
Hockey:
- Win% = (Goals For^2)/((Goals For^2) + (Goals Against^2))
And
- Points = Pyth % * (2 * Team GP)
The Pythagorean Expectation, at least in baseball, is a very accurate predictor of team records. In fact, Pythagorean Win % in the current season is often a better predictor of actual team Win % in the following season than actual Win % in the current season is. In other words, teams that win a lot of close games (largely thanks to external factors like luck) tend to have that luck stripped away by the formula (meaning they won more games than they "should" have). Furthermore, teams that significantly outperform their pythagorean record in one season almost always "regress to the mean" in the following season -- a phenomenon known as the "Plexiglas Principle".
Some people have posited that the difference between a team's actual and pythagorean record may be due to good coaching, which seems to make sense (good coaches would have an advantage in close games, when their strategies can directly impact the outcome), but there is little statistical evidence to back this up. If good coaches could consistently outperform their pythagorean records, there would a correlation between the number of games by which they outperformed the formula from season to season... but the correlation coefficient from season to season for coaches in all of the major sports is very close to zero -- meaning almost all variance from season to season can be explained by pure luck.
