Ranking every possible super bowl matchup (and then some)

For those of you paying attention to sports in any way whatsoever you may have noticed that the super bowl is coming up this weekend.  It's pretty easy to find a wide array of articles and analysis about it, and a week or two ago I came across an article at the bleacher report with the title:

Ranking Every Possible Super Bowl Matchup
(http://bleacherreport.com/articles/1483126-power-ranking-every-possible-super-bowl-matchup?hpt=hp_t3 )

I was excited by the title because I thought this was going to be a ranking of *EVERY* super bowl matchup between every team to figure out which team would actually be the strongest, and not just a simple rundown of what the situation was from this point onward.

Since that was a disappointment, I figured I'd just do it myself.  Right?

Well, it's easy enough (if not a touch tedious) to pull down the scores from every game of this season.  Luckily, the NFL plays a relatively small number of games so it's a fairly reasonable set of data.  At most a team will play another team twice, so we can produce a somewhat odd 32x64 partially filled matrix containing all the win information in one direction and the loss information in the other direction.

The important thing that this allows us to do is to calculate some means and standard deviations.  Specifically, we can check out the mean score of each team both from an offensive and defensive standpoint.  The offensive score is the score that team was able to produce, and a higher score should indicate a better offense.  The defensive score is the score that the team allowed the other team to produce, and a lower score should indicate a better defense.

Right off the start this gives us some good numbers to check out - what teams performed the best and worst throughout the season as well as how consistent given teams were.

The Patriots showed the best offense this year, coming in just over 34 points a game on average.   

The worst team?  Sorry, Kansas City Chiefs fans.  Do I have any readers who are Kansas City Chiefs fans?  Sorry, your offense only produced a little over 13 points on average.

The best defense goes to the Seattle Seahawks, only right around 15 and a quarter points per game allowed on average, and the worst defense goes to the New Orleans Saints, allowing on average just over 28 and a quarter points per game.

If we compare the average points of every team against every other team we can get a feel for what their records would have been if a) every team played every other team once and b) every team had the same defense.  Obviously one of those is a bit larger of a jump, but let's keep an open mind for the moment.  This is how things would work out:

Team (offense)	Count wins	Count losses
New England Patriots	31	0
Denver Broncos	30	1
New Orleans Saints	29	2
New York Giants	28	3
Washington Redskins	27	4
Green Bay Packers	26	5
Atlanta Falcons	25	6
Houston Texans	24	7
Seattle Seahawks	23	8
Cincinnati Bengals	22	9
Baltimore Ravens	21	10
San Francisco 49ers	20	11
Tampa Bay Buccaneers	19	12
Minnesota Vikings	18	13
Dallas Cowboys	17	14
Detroit Lions	16	15
Chicago Bears	15	16
Carolina Panthers	13	17
Indianapolis Colts	13	17
San Diego Chargers	12	19
Buffalo Bills	11	20
Pittsburgh Steelers	10	21
Tennessee Titans	9	22
Cleveland Browns	8	23
St. Louis Rams	7	24
Oakland Raiders	6	25
Miami Dolphins	5	26
New York Jets	4	27
Philadelphia Eagles	3	28
Jacksonville Jaguars	2	29
Arizona Cardinals	1	30
Kansas City Chiefs	0	31

Due to the way this works out through mean comparisons, this is actually a ranking of how every team would do in a super bowl against every other team.  The Patriots would beat anyone, the Broncos would beat everyone but the Patriots, etc.  

We can find the probabilities (roughly) associated with this actually being the outcome by taking into account the stability of those means via their standard deviations.  A proxy for this that I'm calling good enough for our immediate purposes is the probability associated with t-tests between these individual means.  The product of these reversed probabilities (due to the fact that a win or loss is more probable when the p-value is small; e.g. .02 should actually be .98) gives us something we can put in a table.  YES I KNOW I'M KIND OF BUTCHERING THE POINT OF P-VALUES. 

Some of these numbers are actually reasonably finite, and we can add to the above table as such:

Team (offense)	Count wins	Count losses	Probability of occurrence
New England Patriots	31	0	0.47089515
Denver Broncos	30	1	0.019777304
New Orleans Saints	29	2	0.000572806
New York Giants	28	3	6.88355E-09
Washington Redskins	27	4	1.22143E-07
Green Bay Packers	26	5	2.99026E-08
Atlanta Falcons	25	6	7.57896E-08
Houston Texans	24	7	8.88084E-10
Seattle Seahawks	23	8	4.09893E-11
Cincinnati Bengals	22	9	1.49184E-09
Baltimore Ravens	21	10	5.36586E-12
San Francisco 49ers	20	11	1.34842E-11
Tampa Bay Buccaneers	19	12	1.50215E-10
Minnesota Vikings	18	13	1.74783E-09
Dallas Cowboys	17	14	4.93673E-10
Detroit Lions	16	15	7.0116E-10
Chicago Bears	15	16	4.6336E-12
Carolina Panthers	13	17	0
Indianapolis Colts	13	17	0
San Diego Chargers	12	19	3.8835E-10
Buffalo Bills	11	20	3.40025E-09
Pittsburgh Steelers	10	21	6.20755E-07
Tennessee Titans	9	22	1.45213E-08
Cleveland Browns	8	23	1.73087E-06
St. Louis Rams	7	24	1.08623E-06
Oakland Raiders	6	25	2.8135E-07
Miami Dolphins	5	26	1.65687E-07
New York Jets	4	27	2.73281E-08
Philadelphia Eagles	3	28	1.04094E-06
Jacksonville Jaguars	2	29	0.000267479
Arizona Cardinals	1	30	0.000383025
Kansas City Chiefs	0	31	0.137665451

You can see that things sort of follow an upside down bell curve (let's call it a valley curve) - the most probable outcomes are those at the ends, while those in the middle have a bit more noise in them.  More of those middle games are likely to be close enough to drop the cumulative associated probabilities.

What we should keep in mind is that there are a lot of potential outcomes here.  There aren't just 31 (31-0 down to 0-31), but every possible combination of individual wins/losses that would get you to that point.  There's only one way to get 31-0 or 0-31, but there are 31 ways to go 30-1 or 1-30 (you could win or lose to any given team, and each of those has a probability associated with it).  If you'd like to kill a bit more time before you get back to work you can start working out the number of ways you can get to each potential outcome.  It also explains at least a little bit of the valley curve that we have going. 

Yes, the clever among you might have just realized that this table is excluding some potentially important information.  This probability isn't the cumulative probability of all situations that would produce a given outcome, but rather the probability associated with the most likely sequence that would produce that outcome.

For example, the Broncos going 30-1 is actually the probability of the Broncos going 30-1 while losing to the Patriots.  There's another probability that they'd go 30-1 while losing to the Giants, or the Saints, or even the Chiefs (the probability of them just losing to the Chiefs *at all* in this metric is 4.57E-07; fairly unlikely).

There's also a strange coincidence here that you might notice - the Panthers and Colts actually produced the same mean score throughout the season.  There's an interesting discussion to be had about how the way points are earned (in chunks) allows this, but it's for another day.  We'll see it happen a few more times when we get to defense.

Overall these probabilities don't really instill a lot of confidence (except for the Chiefs - sorry again Chiefs fans).  We have to keep in mind that this is simply offense, and doesn't consider how difficult any teams' defense might have been.  Now that we've seen how this works we can also produce the same table based on the idea that a) every team plays every other team once and b) every team has the same offense.

Such a situation would mean that a team's defense was the only way to stand out, and we can produce the same table based on how things would play out from there:

Team (defense)	Count wins	Count losses	Probability of Occurrence
Seattle Seahawks	31	0	0.033645703
San Francisco 49ers	30	1	5.72087E-06
Chicago Bears	29	2	3.03016E-05
Atlanta Falcons	28	3	7.53205E-07
Houston Texans	27	4	0
Miami Dolphins	26	5	5.97182E-10
Denver Broncos	25	6	4.65242E-06
Cincinnati Bengals	24	7	1.68938E-11
Pittsburgh Steelers	23	8	5.27437E-11
New England Patriots	22	9	0
Green Bay Packers	21	10	3.75177E-13
St. Louis Rams	20	11	1.21268E-13
Baltimore Ravens	19	12	0
Arizona Cardinals	18	13	5.77952E-13
Minnesota Vikings	17	14	1.17577E-13
Cleveland Browns	16	15	5.5051E-11
Carolina Panthers	15	16	2.73832E-11
San Diego Chargers	14	17	1.6091E-12
New York Giants	13	18	0
New York Jets	12	19	1.07632E-10
Indianapolis Colts	11	20	9.97422E-12
Washington Redskins	10	21	2.72878E-09
Tampa Bay Buccaneers	9	22	6.3485E-09
Dallas Cowboys	8	23	3.63465E-07
Kansas City Chiefs	7	24	9.21125E-08
Buffalo Bills	6	25	7.97968E-11
Tennessee Titans	5	26	2.93401E-05
Philadelphia Eagles	4	27	0
Jacksonville Jaguars	3	28	0
Detroit Lions	2	29	8.82456E-09
Oakland Raiders	1	30	5.65719E-11
New Orleans Saints	0	31	1.93842E-07

The same things about the other charts apply to this one, though it also gives us a picture of how strong different teams' defense was.  Unfortunately, this is also confounded with the fact that different defenses played different offenses.  We could simply look back at offenses, but those were already confounded by the fact that different offenses played different defenses.  You can see we're in a bit of a loop here.

While we're trying to think our way out of that one we can kill some time by taking a look at the average quality of defense that different teams faced throughout the season.  We can do this by averaging - for each team - the average points allowed by their specific list of opponents.   The more points that your list of opponents allowed, the easier it is to score points against them.

Team	Opponent Defense
Atlanta Falcons	24.5025641
Pittsburgh Steelers	23.89198718
Cleveland Browns	23.5650641
Tampa Bay Buccaneers	23.50737179
Cincinnati Bengals	23.5025641
Indianapolis Colts	23.46634615
San Diego Chargers	23.40865385
Philadelphia Eagles	23.07948718
Jacksonville Jaguars	23.04807692
Houston Texans	22.96634615
Kansas City Chiefs	22.93269231
Baltimore Ravens	22.91121795
Carolina Panthers	22.88717949
Miami Dolphins	22.88461538
New Orleans Saints	22.86794872
Denver Broncos	22.81730769
Washington Redskins	22.81025641
Chicago Bears	22.76217949
New York Giants	22.67083333
Green Bay Packers	22.60576923
Oakland Raiders	22.55288462
Minnesota Vikings	22.53846154
Tennessee Titans	22.49038462
Buffalo Bills	22.46153846
New England Patriots	22.19230769
New York Jets	22.17788462
Seattle Seahawks	22.07467949
San Francisco 49ers	22.06730769
Dallas Cowboys	21.94230769
Detroit Lions	21.85576923
St. Louis Rams	21.62980769
Arizona Cardinals	21.39903846

Turns out things are actually pretty close when it gets to this level.  The Falcons faced the easiest defenses, with their average opponent allowing 24 and a half points.  The Cardinals - perhaps not enough to account for their fairly weak season - faced the most difficult defenses.

We can look at the same concept in terms of how well defenses performed against their opponents' offenses:

Team	Opponent Offense
Arizona Cardinals	23.18269231
Atlanta Falcons	22.5599359
Baltimore Ravens	23.43974359
Buffalo Bills	20.9375
Carolina Panthers	23.58397436
Chicago Bears	22.27403846
Cincinnati Bengals	21.35801282
Cleveland Browns	22.59358974
Dallas Cowboys	23.9974359
Denver Broncos	23.49519231
Detroit Lions	22.05769231
Green Bay Packers	22.73301282
Houston Texans	23.25
Indianapolis Colts	21.77403846
Jacksonville Jaguars	23.12019231
Kansas City Chiefs	23.48557692
Miami Dolphins	22.0625
Minnesota Vikings	22.62980769
New England Patriots	21.67307692
New Orleans Saints	23.35801282
New York Giants	23.96634615
New York Jets	22.07211538
Oakland Raiders	22.34615385
Philadelphia Eagles	23.73301282
Pittsburgh Steelers	21.9974359
San Diego Chargers	22.38461538
San Francisco 49ers	23.44455128
Seattle Seahawks	22.56730769
St. Louis Rams	23.55288462
Tampa Bay Buccaneers	22.97339744
Tennessee Titans	22.73557692
Washington Redskins	23.25224359

At this level of aggregation we again seem to be washing out all useful variance.  

Overall, I'm not sure there's really enough variance here to warrant the meaningful inclusion of it unless things are really pretty close.

Speaking of close, we should at some point probably try to figure out who is going to win the *actual* super bowl.  One last combination before we get to that.  We might be able to get a little more out of offense and defense if we look at them in combination.  We can do this by combining the win/loss records for each team to produce a table like this:

Team (overall)	Wins overall	Losses overall
Denver Broncos	55	7
Seattle Seahawks	54	8
Atlanta Falcons	53	9
New England Patriots	53	9
Houston Texans	51	11
San Francisco 49ers	50	12
Green Bay Packers	47	15
Cincinnati Bengals	46	16
Chicago Bears	44	18
New York Giants	41	21
Baltimore Ravens	40	22
Washington Redskins	37	25
Minnesota Vikings	35	27
Pittsburgh Steelers	33	29
Miami Dolphins	31	31
New Orleans Saints	29	33
Carolina Panthers	28	33
Tampa Bay Buccaneers	28	34
St. Louis Rams	27	35
San Diego Chargers	26	36
Dallas Cowboys	25	37
Cleveland Browns	24	38
Indianapolis Colts	24	37
Arizona Cardinals	19	43
Detroit Lions	18	44
Buffalo Bills	17	45
New York Jets	16	46
Tennessee Titans	14	48
Kansas City Chiefs	7	55
Oakland Raiders	7	55
Philadelphia Eagles	7	55
Jacksonville Jaguars	5	57

      
Looks like that helps to put a bit more spread on things, though our apparent best teams aren't the ones in the super bowl.  Not shocking, as randomness can really play havoc with things when you play so few games and leave playoffs and finals up to single elimination matches.  While I'd be a bit more excited to watch a super bowl between the Broncos and the Seahawks (or the Bears and the Jaguars), that's not what we have this year.  
  
The Ravens and 49ers - going back to the earlier table - put up the 11th and 12th best offenses on average.  They're actually pretty close on that metric - the Ravens averaged 24.875 points per game, while the 49ers averaged 24.8125 points per game.  Given that their pooled standard deviation on those means is 11.70 points there's very little reason to believe that one of these teams has a substantially (or statistically) better offense.

The 49ers scored less than a tenth of a point less than the Ravens on average, though they also faced slightly more difficult opponents.  Their opponents allowed 22.0673 points on average, while the Ravens' opponents allowed 22.9112 points on average.  While this might allow us to tip things *a little* more in favor of the 49ers I'd still be hesitant to say that anything was even close to a sure thing.  I've thought about it a while and don't know if I have any meaningful way to combine points earned and points allowed by specific opponents.  

Let's take a look at defenses - the 49ers did hold up to some of the early promise of a good defense by coming up as the 2nd best, allowing only right around 17 points on average.  The Ravens were somewhat in the middle of the pack, coming up as 13th best defense with right around 21 and a half points on average.

Remember where we got caught in a loop a while ago?  One of the problems was that we had offense and defense to worry about, though at least for this pairing it seems the offenses are pretty close.  The small point difference is also offset by the difference in opponents.

If defense is where the difference is it's hardly enough to be impressed by - the difference in defensive strength is 4 points, while the pooled standard deviation is just under 11 points.

The slight advantage held by the 49ers is also shown in that last table, as they show up as 6th overall while the Ravens come in at 11th.  Even this spread isn't huge, as it's partly due to the fact that a lot of teams are actually incredibly close in terms of mean points scored or allowed.  Forcing things into wins/losses allows for sorting, but carries a lot of error in these close match-ups that could have gone either way.  Let us keep in mind that the teams that are coming up on the top of our charts didn't have perfect seasons, but the games they lost they may have lost by very slim margins. 

All in all I was hoping that one of these teams would have meaningfully distinguished themselves on something, but it seems that these two teams in the super bowl really are pretty close - at least by the numbers.  If pushed it would seem that the 49ers have a slight edge, but what that relates to in terms of a point spread is pretty tricky.  If the 49ers are able to put up a defense that's able to stop 4.5 more points than the Ravens, and both play basically the same offense (with perhaps a slight advantage to the 49ers), then we're talking about less than a one possession spread.  Four to five points is right in that range of being just covered by a touchdown but not covered by a field goal.

If I had to make some guesses, then, the best things to work from are the scores we've seen so far - offensively 24.8125 vs 24.875 points per game, defensively 17.0625 vs 21.5 (49ers and Ravens, respectively).  Opponents of each team also gave up 22.0673 vs 22.9112 points on average, scored 23.4446 vs 23.4397 on average.

So, if team A is trying to score x points and team B is trying to hold team A to y points, the relative importance of offense vs defense would dictate the weighted average that is most accurate.  Given no reason to assume anything else I'm just going to call it an even split and take a normal average.  What that would mean is that the most likely score of this super bowl (still probably pretty unlikely) would be 23.15625 to 20.96875, 49ers.  Okay, so that score is not just unlikely, but impossible.  Silly imprecise sports. 

You might be going straight to the comments to point out that you can't earn 0.00005 points in a game of *normal* football.  More important is the fact that even the rounded score of 23-21 might not be the most common.  If we head back over to my favorite historical archive of all football scores ever ( http://www.pro-football-reference.com/boxscores/game_scores.cgi ) we can see that there have only ever been 46 games with an outcome of 23-21.  Given the amount of error we're playing with here I'm willing to take this prior information into account to some degree, especially given the fact that the *very* similar score of 23-20 is over three times as likely as 23-21.

In all, my best guess would be that the scores are pretty close, and somewhere in the low 20s both.  The 49ers seem to have a slight edge, but it's football and they only get to play one game.  Repeat this super bowl 100 times and then we can talk.


More than anything, it seems that this super bowl might actually be a close game.  I say that's always what I want, sooooooo I guess I'd better watch it.

Maybe I'll record it so I can get rid of the stupid commercials. (<- flame baiting)



 >>>>Update:
Here are the raw numbers for points scored and points allowed as requested in the comments.

Team	Points scored	Points allowed
Arizona Cardinals	15.625	22.3125
Atlanta Falcons	26.1875	18.6875
Baltimore Ravens	24.875	21.5
Buffalo Bills	21.5	27.1875
Carolina Panthers	22.3125	22.6875
Chicago Bears	22.5625	17.3125
Cincinnati Bengals	25.0625	20
Cleveland Browns	18.875	23
Dallas Cowboys	23.5	25.53333333
Denver Broncos	30.0625	18.0625
Detroit Lions	23.25	27.3125
Green Bay Packers	27.0625	21
Houston Texans	26	20.6875
Indianapolis Colts	22.3125	24.1875
Jacksonville Jaguars	15.9375	27.75
Kansas City Chiefs	13.1875	26.5625
Miami Dolphins	18	19.8125
Minnesota Vikings	23.6875	20.875
New England Patriots	34.8125	20.6875
New Orleans Saints	28.8125	28.375
New York Giants	27.46666667	21.5
New York Jets	17.5625	23.4375
Oakland Raiders	18.125	27.6875
Philadelphia Eagles	17.5	27.75
Pittsburgh Steelers	21	20.25
San Diego Chargers	21.875	21.875
San Francisco 49ers	24.8125	17.0625
Seattle Seahawks	25.75	15.3125
St. Louis Rams	18.6875	21.75
Tampa Bay Buccaneers	24.3125	24.625
Tennessee Titans	20.625	29.4375
Washington Redskins	27.25	24.25