Responding to the SMQ

The Sunday Morning Quarterback (SMQ), on his excellent site, posted a question about what many regard as the surprising outcome of the BCS Title game. Actually, a series of questions, but the essence seems to be - can we ever really know the “best” team in college football?

Lacking the good sense not to attempt such an undertaking, here is my contribution to the discussion.

As an investment professional, and all around statistics geek, I am been interested in probabilities for decades. My interest ranges from the merely curious (What is the chance my daughter will have my blue eyes?), to the pecuniary (What is the chance I will roll a “7” before I make my mark at craps?).

As an avid college football fan, and occasional college football “investor”, my interests in statistical probabilities and the college game have crossed more than once. Also, at the onset, let me say this - I agree with the SMQ that many times the outcome of any one college game simply defies explanation.

Unlike activities with a known number of outcomes, like the casino game of craps, a college football game has theoretically infinite outcomes. In craps, for instance, there are only 36 ways the dice can fall. Of those 36, 6 are the number “7”. Thus we know each roll has a 1 in 6 chance of being a “7”.

In college football, you cannot assign a simple number of outcomes to any given game. Using Florida’s most overmatched opponent as an example (and an example they were), Western Carolina has some chance in theory of beating Florida. Now it may be a chance so remote as to be almost impossible, but the chance exists. And, like the monkey randomly typing for eternity will eventually produce the works of Shakespeare, if Florida and Western Carolina played enough times, Western Carolina would win.

The first question in this case then becomes - how many times would Western Carolina need to play Florida on average to get a single win?

I think the number would be rather large, but for our purposes, lets say that WCU would beat Florida 1 in every hundred times. But this only answers part of our overall question. When WCU got that one-in-a hundred-win, what would be the score?

Well, I submit, that if we ran the game say, 10,000 times, each of WCU’s 100 wins could be a different score. So, what we have here is a series of probabilities within probabilities. For the point spread of each of the wins WCU got 1 in 100 times, I believe a bell curve analysis would be appropriate. Using the “normal distribution” of the bell curve, I would estimate that WCU would beat Florida by 3 points or less 68.3% of the time, or one standard deviation. However, if you took the analysis out to, say, 7 standard deviations, or a probability of 0.00000000026%, that would be the time that WCU blows Florida out by some extraordinary score (once again, the probabilities here are mere estimates, used for illustrative purposes).

In Western Carolina’s visit to Florida on November 18th, they lost 62-0. Flipping the example above around, Florida had a 99% chance of winning that game. However, Florida’s point spread using a normal distribution on a bell curve would have been much higher than WCU’s would be for there win. I would hazard a guess that the same 1 standard deviation range for a Florida win would be about 40 points. In a very small percentage of times, Florida would win by 1 point.

Now, I am well aware that the entire range of possibilities (combining wins with score, or just merely coming up with a score, which would tell you who wins) can be placed on a single bell curve. In our example above, the chance of WCU winning would be placed to one side of the curve, with the greater win being pushed further out. However, I wanted to separate the win from the score for a reason.

And that reason is that what we are trying to do in making analyzing a single game - such as in the instance of Florida versus Ohio State - is to come up with a probability as to what a series of outcomes would be. In other words, to see who is, statistically, the “better team”.

In looking at the Florida - Western Carolina score, there is much evidence to be gathered that Florida would beat WCU the vast majority of times they play. Is there the same evidence from the likewise lopsided margin by which Florida beat Ohio State?

I think our fallacy here - the way we make inferences of overall worth based on a game like the BCS Title game - is that basing a conclusion on a single outcome is wrong. We know from a simple game like craps that there are 36 possible outcomes. If I roll a single time, and don’t roll a “7”, the most probable outcome, does that mean I will never roll a seven? Of course not.

Likewise, in college football, where the possible outcomes are likely infinite, does a single game mean all games would be so similar. No, it cannot.

However, we can most certainly gather inferences from a single game as to what other games might look like. In the case of Florida-Ohio State, conventional wisdom clearly underestimated the strength and speed of Florida’s defense (as did Ohio State). Prior to the game, based on the team’s records and statistics, I estimated that Ohio State might win this game “7 in 10 times” Afterward, I think almost the opposite.

However, the problem will always be that our sample sizes are far too small. Without talking about R-Squareds, just know that even a 14 game season is far too small a sample to really gather anything approximating real knowledge about the “goodness” of teams.

Take this example - there are about 120 NCAA Division I-A teams. If we assume they each have a 50% chance of winning any game (assume it, we know they do not), after one week 60 teams will be undefeated. After two weeks it should be 30, 3 weeks 15, 4 weeks 8 (round up), 5 weeks 4, 6 weeks 2, and by the 7th week 1 team. Now, can we say that the one undefeated team after 7 weeks is “the best”. Of course not. With the chance of winning being equal, they were the luckiest.

But the chance of winning is not equal. When you have 120 teams, and unequal chances of winning, the probability that after 12 games at least one or more teams are undefeated is very high. If that team had a fundamentally easy schedule (which I suggest Ohio State did), how much of that 12-0 record can be considered skill, and how much just the random probability of luck?

A problem I have had with all the post game analyses of Florida - Ohio State is that people seem perplexed Florida could do so well having played Vanderbilt and South Carolina so closely. I think this not only ignores my concerns about sample sizes and probabilities of outcome, it also ignores a very real part of the fundamental analysis people are trying to make in this type of comparison - in other words, what about Ohio State’s close games?

What about, in particular, Ohio State against Illinois. In that game, which Ohio State won by 7, Illinois out gained OSU 233 to 224 yards. In fact, there were many examples of games in which Ohio State gave up big yardage to poor Big Ten teams that went ignored (Mostly, I suppose, because the point totals were large - see this post).

My point is, if you were looking for the evidence that Ohio State could play badly, it was there. But, and this is getting away from this topic and towards another -“we”, collectively, did not want to see it. It did not play into the story of match-ups like “The Game of the Century” (OSU-Michigan).

And if we want to believe anything in college football, it is a good story.

Back to the analysis. Say the “true” probability of either team winning the BCS Title game was 50-50. When either team wins, then the probability of a point spread using the bell curve comes into play. Using this example, Ohio State could have gotten very, very unlucky. They lost the game, and the spread (27 points), fell somewhere outside a single, or likely even second, standard deviation. In other words, it was unlikely.

But, when you only get to play one game, the unlikely can happen.

However, know well this - if you are willing to believe that Ohio State was merely the victim of extraordinarily “bad luck” in this game, know that it also exists the possibility that their entire regular season and 12-0 record was the result of extraordinarily “good luck”. You can’t have it one way, without the other.

Finally, remember that college football is not a casino, where the house has a multitude of chances for the odds to come their way, no matter how long it takes. The sample size is very, very small, even minute, at 12 to 14 games. And determining who is really “best” in this small a sample size is unknown and unknowable.

In other words, no - we cannot ever know the “best” team.

Which, in turn, is what makes the game so much fun.