Predicting Parity: Part 1 (Pythagorean Expectation)

Anyone who follows football knows how big a part of the game that parity is. One team can be good one year and bad the next and vice versa for seemingly no reason. This series, called Predicting Parity, seeks to discover why that is and figure out how to predict it. The first thing we will look at is knows as the Pythagorean Expectation.

Overview

Use of the Pythagorean Theorem in sports began with Bill James in baseball, which what he called the Pythagorean Expectation. James theorized that the amount of runs a team scored and allowed was a more accurate predictor of the quality of a team than their actual win loss record. Much like the what you’ll remember from Middle School math, the Pythagorean Expectation in baseball used the following formula.

mathrm{Win} = frac{text{runs scored}^{1.83}}{text{runs scored}^{1.83} + text{runs allowed}^{1.83}} = frac{1}{1+(text{runs allowed}/text{runs scored})^{1.83}}

This same method has been adapted for football, though with a different exponent.

                          PF^2.37
Expected record =~  -----------------
                    PF^2.37 + PA^2.37

Below are two charts, one sorted by Pythagorean % and one sorted by the differential of Pythagorean wins and Actual wins from the 2011 season. These charts allow us to see which teams were the best last season and which teams were actually better or worse than their record would suggest. This is important for making predictions into 2012.

Teams sorted by Pythagorean Wins

Team PF PA Pythagorean % Pythagorean Wins Pythagorean Losses Actual wins Differential
SF 380 229 0.76857543 12.29720688 3.702793123 13 -0.70279312
NO 547 339 0.756562223 12.10499557 3.895004431 13 -0.89500443
GB 560 359 0.741491691 11.86386705 4.136132952 15 -3.13613295
NE 513 342 0.723312432 11.57299891 4.427001085 13 -1.42700109
PIT 325 227 0.700679179 11.21086686 4.789133141 12 -0.78913314
BAL 378 266 0.696949393 11.15119029 4.848809705 12 -0.84880971
HOU 381 278 0.67851866 10.85629856 5.143701439 10 0.85629856
DET 474 387 0.617888325 9.886213206 6.113786794 10 -0.11378679
PHI 396 328 0.609809 9.756943993 6.243056007 8 1.75694399
ATL 402 350 0.581343219 9.3014915 6.6985085 10 -0.6985085
SD 406 377 0.543796445 8.700743126 7.299256874 8 0.70074313
CIN 344 323 0.537252023 8.596032371 7.403967629 9 -0.40396763
DAL 369 347 0.536357794 8.581724706 7.418275294 8 0.58172471
MIA 329 313 0.52950451 8.472072155 7.527927845 6 2.47207215
NYJ 377 363 0.522406577 8.358505233 7.641494767 8 0.35850523
CHI 353 341 0.52048049 8.327687845 7.672312155 8 0.32768785
TEN 325 317 0.514762827 8.236205238 7.763794762 9 -0.76379476
SEA 321 315 0.511177714 8.178843429 7.821156571 7 1.17884343
NYG 394 400 0.491046127 7.856738031 8.143261969 9 -1.14326197
CAR 406 429 0.467397347 7.478357548 8.521642452 6 1.47835755
ARZ 312 348 0.435658147 6.970530348 9.029469652 8 -1.02946965
BUF 372 434 0.409668219 6.554691497 9.445308503 6 0.5546915
OAK 359 433 0.390746755 6.251948077 9.748051923 8 -1.74805192
DEN 309 390 0.365458736 5.847339769 10.15266023 8 -2.15266023
WAS 288 367 0.360201141 5.763218262 10.23678174 5 0.76321826
MIN 340 449 0.340954592 5.455273469 10.54472653 3 2.45527347
JAC 243 329 0.327811238 5.244979806 10.75502019 5 0.24497981
CLE 218 307 0.307597385 4.921558163 11.07844184 4 0.92155816
KC 212 338 0.248709041 3.979344659 12.02065534 7 -3.02065534
TB 287 494 0.216354866 3.461677857 12.53832214 4 -0.53832214
IND 243 430 0.205443127 3.287090036 12.71290996 2 1.28709004
STL 193 407 0.145752436 2.332038974 13.66796103 2 0.33203897

 

Teams sorted by differential

Team PF PA Pythagorean % Pythagorean Wins Pythagorean Losses Actual wins Differential
MIA 329 313 0.52950451 8.472072155 7.527927845 6 2.47207215
MIN 340 449 0.340954592 5.455273469 10.54472653 3 2.45527347
PHI 396 328 0.609809 9.756943993 6.243056007 8 1.75694399
CAR 406 429 0.467397347 7.478357548 8.521642452 6 1.47835755
IND 243 430 0.205443127 3.287090036 12.71290996 2 1.28709004
SEA 321 315 0.511177714 8.178843429 7.821156571 7 1.17884343
CLE 218 307 0.307597385 4.921558163 11.07844184 4 0.92155816
HOU 381 278 0.67851866 10.85629856 5.143701439 10 0.85629856
WAS 288 367 0.360201141 5.763218262 10.23678174 5 0.76321826
SD 406 377 0.543796445 8.700743126 7.299256874 8 0.70074313
DAL 369 347 0.536357794 8.581724706 7.418275294 8 0.58172471
BUF 372 434 0.409668219 6.554691497 9.445308503 6 0.5546915
NYJ 377 363 0.522406577 8.358505233 7.641494767 8 0.35850523
STL 193 407 0.145752436 2.332038974 13.66796103 2 0.33203897
CHI 353 341 0.52048049 8.327687845 7.672312155 8 0.32768785
JAC 243 329 0.327811238 5.244979806 10.75502019 5 0.24497981
DET 474 387 0.617888325 9.886213206 6.113786794 10 -0.11378679
CIN 344 323 0.537252023 8.596032371 7.403967629 9 -0.40396763
TB 287 494 0.216354866 3.461677857 12.53832214 4 -0.53832214
ATL 402 350 0.581343219 9.3014915 6.6985085 10 -0.6985085
SF 380 229 0.76857543 12.29720688 3.702793123 13 -0.70279312
TEN 325 317 0.514762827 8.236205238 7.763794762 9 -0.76379476
PIT 325 227 0.700679179 11.21086686 4.789133141 12 -0.78913314
BAL 378 266 0.696949393 11.15119029 4.848809705 12 -0.84880971
NO 547 339 0.756562223 12.10499557 3.895004431 13 -0.89500443
ARZ 312 348 0.435658147 6.970530348 9.029469652 8 -1.02946965
NYG 394 400 0.491046127 7.856738031 8.143261969 9 -1.14326197
NE 513 342 0.723312432 11.57299891 4.427001085 13 -1.42700109
OAK 359 433 0.390746755 6.251948077 9.748051923 8 -1.74805192
DEN 309 390 0.365458736 5.847339769 10.15266023 8 -2.15266023
KC 212 338 0.248709041 3.979344659 12.02065534 7 -3.02065534
GB 560 359 0.741491691 11.86386705 4.136132952 15 -3.13613295

As you can see, Miami leads the way in differential. Though they were just a 6-10 team, they actually played as well as a 8.47 team. This was because they actually had a solid defense. As many questions as they have offensively, their strong defense should prevent them from bottoming out once again in 2012. Bad teams like Minnesota, Carolina, and Indianapolis are also high up on this list so they should be able to bounce back some in 2012, while teams with solid records like the Eagles and Seahawks should improve and possibly make the playoffs.

On the other end of the spectrum, teams like Green Bay and New England had #1 seeds last year, but they also had differentials higher than -1. That being said, that’s normally the case with really good teams like that and teams with elite quarterbacks tend to frequently exceed their Pythagorean Expectation. Peyton Manning did it over his final 9 healthy seasons in Indianapolis, while Tom Brady has done so in 8 of his 10 seasons. Aaron Rodgers is not nearly as experienced as those two, but he’s certainly just as talented. Meanwhile, Brady frequently exceeds his Pythagorean Expectation and should be able to do so again this year. The Packers and Patriots will remain among the best teams in the league this year, barring injury to Rodgers or Brady.

Meanwhile, Manning’s new team, the Denver Broncos, are near the bottom of this list, but given that he missed all of last year with injury, is joining a new and inferior supporting cast, now to has play his home games out doors, had 4 neck surgeries in less than 2 years, and turned 36 in March, I don’t know if we can still consider Manning on the same level of Brady or Rodgers or Drew Brees. Manning is certainly an upgrade at quarterback over Tim Tebow even if he’s only 70% of his old self, but he might only be barely enough of an upgrade to cancel out their differential. Anyone expecting them to make an 3 or 4 game jump to being a 11-12 team will be disappointed. 9-7 or so seems more appropriate.

Meanwhile, average teams like Kansas City and Oakland could also be much worse this year than last year, especially Oakland. Oakland has lost a lot in free agency over the past 2 offseasons, including their two starting cornerbacks from their 2010 team and their top pass rusher, Kamerion Wimbley. With minimal cap space and draft picks, they haven’t been able to make up for all of these losses and now have a pretty thin team, especially defensively.

Kansas City, on the other hand, will be getting guys back from injury, including Jamaal Charles, Eric Berry, and Tony Moeaki, and Matt Cassel. However, this says they had the talent of a mere 4 win team last year. The year before, they won 10 games, but they only went 2-5 against teams with winning percentages of .500 or better, with those two wins coming against a Jacksonville team starting its 3rd string quarterback and the early season Chargers, against whom Matt Cassel passed for just 68 yards. The Chargers avenged that loss with a 31 point win later that year. Over those 5 losses, 4 were by double digits and 3 were by 21+.

The other two teams with -1 or worse differentials were the Giants and Cardinals. The Cardinals figure to have a worse record this season and while the Giants may do the same, especially in an improved NFC East, there are some that believe they’ve turned the corner after winning the Super Bowl last year and are now an elite regular season team. I am not one of those people, but I can understand it.

[switch_ad_hub]

[switch_ad_hub]

[switch_ad_hub]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s