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## Pythagorean 16.5 Method

### Definition

A method that gives an expected winning percentage using the fact that the ratio of a team's wins and losses is related to the number of points scored by the team raised to some exponent, which is usually taken to be 16.5. Other methods use 13, 16.1, or 17.

### Applicability

This method applies in a straightforward and similar manner to both individuals and teams.

### Calculating Expected Win% using the Pythagorean Method

Expected Winning %=(Pts scored)^(16.5)/[Pts scored^(16.5) + Pts allowed^(16.5)]

or, equivalently,

Expected Winning %=(Off. Rating)^(16.5)/[Off. Rating^(16.5) + Def. Rating^(16.5)]

### Discussion

The Pythagorean Method for relating points scored and allowed to wins and losses is an approximation to a more theoretically correct method, called the Correlated Gaussian Method. Both methods are used in JoBS, but it is hoped that the Correlated Gaussian Method will be used more in the future.

The Pythagorean 16.5 Method was derived from the corresponding method in baseball used by Bill James. 'Derived' may not be the proper word because I'm not sure if I knew what I was doing when the formula came out. You see, the corresponding baseball formula is identical to the basketball formula except that the exponents are 2's instead of 16.5's. What the derivation entailed was estimating average margins of victory for both sports and playing around with the logarithm button on a calculator. The number 16.76... came up on the first try. My expectations were for something between 13 and 20, so 16.76 was originally rounded up to 17 and tested as a valid possibility. It was then replaced by 16.5 after a more thorough empirical study. Martin Manley looked into this issue and came out with 16.1 as the exponent. I saw someone else use 13 as the exponent. I use 16.5 through most of JoBS, but that will be changing due to the development of the Correlated Gaussian Method for doing the same thing as this does.

The principle behind the method - that a team's won-loss record is closely related to the number of points it scores and allows - should be no surprise. It just makes sense that teams that win 60 games outscore their opponents by more than teams that win 50 do. However, one of the things that the Correlated Gaussian Method has added is that consistency also plays a role. Teams that win 60 games do not have to outscore their opponents by more on average than teams that win 50. They just need to be more consistent from game to game.

On the other hand, luck resulting from 'well-timed scoring' is a weak force in the NBA. It doesn't separate the good teams from the bad teams; it just separates two teams of similar quality. Taking the luckiest and unluckiest teams in the NBA, we usually find a total deviation of 10 to 13 wins. Luck has a place in basketball, just as the weather has a place in football and as Wrigley Field has a place in baseball. Each has an effect on the game, but, in the long run, the better teams win with or without the advantage or disadvantage of such factors. (In the short run, like the playoffs, luck can be pretty important. Witness the 1995 Houston Rockets.)

Occasionally luck plays a major part in a team's season. The '85-86 Clippers won 32 games, while their point totals led to an expectation of only 21 wins. A third of their victories (!) came out of the Twilight Zone. The '86-87 Clippers came back to reality, going through a pitiful 12-70 season in a daze. The '86-87 Warriors exceeded their Pythagorean projection by eight games, winning 42 instead of 34 games. They, too, crashed the following season, winning only 20. Both the Clippers and Warriors lost key personnel in their follow-up seasons, but neither ever showed any signs of life anyway. This sort of collapse can be seen throughout the history of basketball, but it's also seen in baseball (and probably other sports). The baseball people called this the Johnson Effect. It's the same effect in basketball so it gets the same name.

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