Pythagoras on the Gridiron

by Aaron Schatz

One of the longest-held theories in the baseball analysis community is the Pythagorean Win Theorem [1].  Worked out by Bill James, this formula notes that the record of a baseball team can be approximated by taking the square of team runs scored and dividing it by the square of team runs scored plus the square of team runs allowed.

(It actually turns out that the correct exponent is 1.82, not 2 -- squaring numbers is just easier to remember.) 

Take the 2002 NL East [2] as an illustration.  Atlanta scored 708 runs and gave up 565, which works out to a projection of 98 wins.  They actually finished with 101 wins.  The Mets scored 690 and gave up 703, a projection of 79 wins.  They actually had 75 wins.  Montreal scored 735 and gave up 718, a projection of 83-79, and that was their actual record as well. 

Occasionally, a team misses its expected record by a hefty margin.  The 2002 AL East [3] was a good example of this: both New York and Boston would be projected to finish 101-61 based on their runs scored and allowed.  The Yankees actually won 103 games while the Red Sox won only 93 due to exceedingly poor performance in one-run games.  An eight game difference is pretty rare, however.  In general, baseball teams come within .025 of their projection, which is four games over a 162-game season.

Statistician Daryl Morey, a former employee for STATS, Inc., took this concept even further [4].  His research proved that the Pythagorean Win Theorem carried over into all the major sports.  A team's won-loss record could be roughly estimated using points scored and allowed, although the exponent differed for each sport.  For the NFL, it is 2.37. 

Using this equation, we can estimate which teams in the NFL had records better than their points scored and allowed should indicate, and which had records that were worse.  Compare the predicted record to the actual record, and we can figure out which teams might be more likely to improve in 2003, and which ones may go downhill.

Of course, 16 games is a very small statistical sample, so there should be a lot more variation in the NFL.  It's nearly impossible for the Pythagorean equation to equal a team's exact record anyway; barring ties, there are only 17 possible winning percentages for an NFL team.

Using the great site pro-football-reference.com [5], I went back and took a look at every team from 1983 through 2002 (1987 records include strike games).  Using points scored and allowed, I determined each team's projected winning percentage, and compared both a team's real and projected winning percentage to its wins the following season.  For the most part, the results mirror those in baseball.  Teams that under-perform their point differential tend to get better the following year, and teams that over-perform tend to get worse.

For fun, let's start by looking at the real extreme cases.  Here are the 10 most underachieving teams in the NFL from 1983-2002:

The ten teams each had three wins fewer than should have been expected, and eight of the ten improved the following year.  The biggest improvement, of course, came from the 1997 Jets, but we should all be lucky enough to have Bill Parcells coaching our team.  (Then again, given how long he stays around, maybe we shouldn't be wishing for that.)

Here's the other side, the 10 most overachieving teams in the NFL from 1983-2002:

There are more exceptions on this list than on the list above, but the trend remains similar.  Teams that win despite giving up more points than they score tend to decline the following year.  Take, for example, the 1991 [6] and 1992 Detroit Lions [7].  The team stayed virtually the same.  Barry Sanders at tailback behind Pro Bowler Lomas Brown.  Erik Kramer and Rodney Peete sharing the QB duties and throwing to Brett Perriman and Herman Moore.  Bennie Blades and Chris Spielman anchoring the defense.  Do you think that Detroit fans came out of 1992 wondering what the heck happened?  Well, part of what happened was that the 1991 Lions weren't really a 12-4 team.  They shouldn't even have been in the playoffs.  Their 1992 collapse was unexpected -- after all, they had scored more than they allowed the year before -- but not completely out of the blue.

The top team on this list, the 1992 Indianapolis Colts, is a really good example of why points scored and allowed is a better indicator of team quality than wins and losses.  The 1991 Colts [8] were 1-15.  When you see a team go from 1-15 to 9-7, you expect they would be up there on that underachiever list, right?  Actually, the 1991 Colts scored 143 points and allowed 381, meaning that their projected record was, yes, 1-15.

So 1992 comes [9], the Colts go 9-7, and Indy fans think they're on the way to the upper echelon of the NFL.  Except the Colts played like a 5-11 team that year, not a playoff team.  Their nine wins came by an average of six points, and the rest of the time they were getting blown out with scores like 38-0 (to Buffalo) and 28-0 (to Miami, at home). The next year, 1993 [10], they were back to last place at 4-12.  In reality, the Colts were a bad team during the entire Jeff George era, and in the middle they managed to pull off the luckiest season of the past 20 years.

The teams on this list that did not decline only stayed successful because the next year they dramatically improved their scoring so that it now matched their records.  The Titans went from allowing 324 points in 1999 to allowing a ridiculous 191 in 2000.  The 1990 Steelers scored 27 more points than the 1989 team, and gave up 86 fewer.

Let's expand and look at more general data.  From 1983-2002, 33 different teams lost two or more games than the Pythagorean prediction would project.  One of those teams was last year; I'll reveal its identity in a moment.  Out of the other 32:

• 23 teams improved the following year.
• 5 declined.
• 4 had the same record.

"OK," perhaps you are thinking, "but losing teams usually get better the next year.  They get an easier schedule and they get better draft picks."  So we'll look at teams that underachieved despite winning records.  From 1983-2002, 34 teams lost one or more games than the Pythagorean prediction would project, despite at least eight wins.  Two of those teams were last year.  Of the other 32:

• 19 improved the following year, including 12 by more than two games.
• 4 of those -- 1984 49ers [11], 1986 Giants [12], 1994 49ers [13], 2000 Ravens [14] -- won the Super Bowl.
• 12 declined.
• 1 had the same record.

So, it looks like points scored and allowed are a good indicator of a team's success the following season.  26 of the 32 NFL teams came within one game of their Pythagorean predictions once the numbers get rounded, but we can still rank all 32 teams based on predicted winning percentage.  Which teams were 2002's biggest under- and overachievers?
 

Yes, Jacksonville wins the trophy as the only team in last year's NFL to lose more games than they won despite scoring more points than they allowed.  The odd thing is that teams of this kind normally win blowouts and lose close games -- that's what happened to the 2002 Chiefs.  But the Jags had only one blowout, week four against the pre-Pennington Jets.  They just played a ton of close games and lost most of them; they were 3-8 in games decided by a touchdown or less.  After they lost to Houston by two at home, there was probably a run on whisky at every liquor store in northern Florida.

So does this mean we should expect big seasons from the teams at the top of this list?  Well, no.  Even with average luck, the Bengals won't suddenly make the playoffs, unless Marvin Lewis is the second coming of Bill Parcells and Carson Palmer is the second coming of Dan Marino (and in the NFL, both those things are possible).  In addition, the Pythagorean projection doesn't take schedule difficulty into account.  Sorry Jaguar fans -- in 2003 your boys have to face the defensive juggernauts of the NFC South and best top-to-bottom division in sports, the AFC East, instead of the more middling NFC East and AFC North.

But remember the chart of teams above .500 who under-performed?  Believe it or not, the Tampa Bay Bucs were even better than they seemed, and there's good news for fans of Kansas City, Miami, and Atlanta.  Of course, Miami fans knew there was good news the day Ray Lucas was waived, but Kansas City fans have the biggest reason to celebrate.  Not only were the Chiefs about 1.5 wins worse than their projection, they get to face the NFC North and AFC North this year.  Lions and Bengals and Bears, oh my!  As in: Oh my, those teams suck.

And now, the 2002 overachievers:

What's interesting about this list is that none of these teams really had a season that was particularly unexpected.  Green Bay, San Francisco, and Pittsburgh had all been playoff teams in 2001, and Tennessee and Indianapolis were each coming off a bad year but had been good in 1999-2000.  But that's looking to the past; the future trend doesn't favor these clubs.  Of the six teams who overachieved by at least one game in 2001, five won fewer games last year -- most notably St. Louis -- and the other one had a new quarterback named Michael Vick.  Indianapolis and Tennessee should be particularly interesting to watch, because they may be moving down while Jacksonville moves up, and with the tough matchups that AFC South teams face this year I'll be shocked to see a wild card in the division. 

It sure looks like Parcells has his work cut out for him with Dallas, doesn't it?  The Cowboys were actually even worse than 5-11 in 2002.  It makes you wonder, if Parcells is such a good coach, shouldn't he be able to maneuver a team to a record better than the sum of its parts?  There may be an answer to that question. 

Baseball analysts have fought for twenty years against the idea that performance in one-run games, and thus the amount a team outperforms its Pythagorean prediction, is due to "clutch hitting ability."  Nonetheless, there is a theory that a team's distance from the prediction isn't all luck.  Two factors at least, contribute to good or bad performance in close games: bullpen quality, and managerial decisions.

Well, in the NFL we don't have a bullpen, but we do have "clutch situations," because in football the rules of the game change late in each half.  All of a sudden, time is limited, and 60 yards just don't cut it when you need to get 80 in three minutes.  So it strikes me that poor clock management is probably a frequent problem with teams that under-perform their predictions, and vice versa.

It turns out in fact, that some coaches have a clear tendency to lead teams to "lucky" seasons, while others have a penchant for teams that win fewer games than their scoring would otherwise indicate.  Which coaches are which?  Check FootballOutsiders regularly, because I'll answer that question in an upcoming sequel to this article.
 

Referenced links:

http://www.baseballprospectus.com/news/19990630davenport.html [1]
http://www.baseballprospectus.com/news/20010516aim.html [15]
http://www.morey.org/dmorey/modpyth.html [4]