Out with Pythagorean, in with the Correlated Gaussian Method

NORMSDIST?

What the heck is NORMSDIST? Statistician Dean Oliver's Correlated Gaussian Method, which, like Pythagorean, estimates a team's winning percentage, requires you to find the NORMSDIST of a number, which he defines as taking "the percentile of a mean-zero variance-one normal distribution corresponding to a value given by that in the [equation]." Microsoft's website says that the NORMSDIST function "[r]eturns the standard normal cumulative distribution function."

I have no idea what that is. But I'm using it anyway.

Oliver described his Correlated Gaussian Method in detail here. (For an easier understanding, just think of offensive rating and defensive rating as points scored and allowed.)

What isn't stated, but should be, there is that you need game-by-game scores to calculate the win percentage; that's what the denominator is: the standard deviation of the game-by-game rating or point differences. The reason you need game-by-game data is because the formula works partly by determining the consistency of a team.

His method was intended for use in basketball, but it can be used in other sports, like football or baseball.

In the NBA, Gaussian yields a 3.0 RMSE, compared to the RMSEs produced by my pace and possession exponents that you can find here. I haven't heard of it being used in the MLB, though I'd expect it to be near the top of this list.

I ran the CGM for the NFL, from 1988 (the season after the strike) through last year, or 20 years of data. I got all of the game-by-game stats from the wonderful pro-football-reference.com, and compared the expected win/loss records to the common 2.37 exponent and the custom exponent for the 1988-2007 period (2.59).

Here are the results:

Exponent	RMSE	Avg_%_Error
Gaussian	1.130	13.9%
2.59	1.226	16.2%
2.37	1.231	16.9%

The CGM beat out the other two exponents by almost one-tenth of a game, which is very significant—it comes out to more than three games in a league season. Because the CGM includes consistency in its equation, teams like the Broncos (somewhat consistent until their 30-point loss to the Pats) do not fare well; see table below.

Lucky and Unlucky Teams

Team	CGM Wins	Act.Wins	Diff.
San Diego Chargers	4.5	3	-1.5
Philadelphia Eagles	4.2	3	-1.2
Chicago Bears	5.1	4	-1.1
New Orleans Saints	3.9	3	-0.9
Seattle Seahawks	1.8	1	-0.8
Miami Dolphins	2.7	2	-0.7
Cleveland Browns	2.6	2	-0.6
Green Bay Packers	4.6	4	-0.6
Tampa Bay Buccaneers	5.6	5	-0.6
Cincinnati Bengals	0.4	0	-0.4
Detroit Lions	0.3	0	-0.3
New York Jets	3.2	3	-0.2
San Francisco 49ers	2.2	2	-0.2
Baltimore Ravens	3.1	3	-0.1
Kansas City Chiefs	1.0	1	0.0
Dallas Cowboys	4.0	4	0.0
Indianapolis Colts	2.9	3	0.1
Tennessee Titans	5.8	6	0.2
Arizona Cardinals	3.8	4	0.2
Oakland Raiders	1.8	2	0.2
Minnesota Vikings	2.8	3	0.2
Jacksonville Jaguars	2.7	3	0.3
Houston Texans	1.6	2	0.4
Pittsburgh Steelers	4.6	5	0.4
Carolina Panthers	4.4	5	0.6
St. Louis Rams	1.4	2	0.6
Washington Redskins	4.4	5	0.6
New York Giants	4.4	5	0.6
Atlanta Falcons	3.3	4	0.7
New England Patriots	3.3	4	0.7
Denver Broncos	3.0	4	1.0
Buffalo Bills	3.8	5	1.2

A negative difference means the team is underperforming, a positive meaning they're doing better than expected.

One other thing I like about the CGM is that it doesn't hurt outliers—the Titans have a .974 winning percentage under CGM, compared to .873 with Pyth.; and last year, the Patriots had a .936 winning percentage under CGM, compared to .860 with Pyth. In other words, it isn't afraid to give very good or very bad teams the record they truly deserve.