It's time for the annual update of my Superbowl Pool Stats. The most
popular Superbowl pool is the kind where you buy a square
on a 10X10 grid, each of whose cells corresponds to one of the 100
possible
pairs of final digits in each team's score. At the time you buy the
square, the digits are not yet assigned (after all, who would buy the
(5,5) square?), but are generally randomly
assigned *after*
all squares are bought. A
few years ago, I posted
information about the relative frequency
of occurrence of digits in the scores at the end of each quarter (many
pools pay out for the scores at the end of each quarter) in all
the previous Superbowls. Three years ago I updated
the graphs and included information about the relative frequency of all
the possible pairs in their actual and reverse orders (some pools pay a
small prize for having the correct reversed digits). And two years ago, I
promised to continue to update the information. It's a continuing tradition. Here is the updated
information:

- Quarter-by quarter histogram of the relative frequency of digits in the scores of the game winner.
- Quarter-by quarter histogram of the relative frequency of digits in the scores of the game loser.
- Quarter-by quarter histogram of the combined relative frequency of all final digits, regardless of whether they belonged to the winner or loser.

- First quarter winner-loser digit pair frequency - Combined correct and reverse order pair frequency
- Second quarter winner-loser digit pair frequency - Combined correct and reverse order pair frequency
- Third quarter winner-loser digit pair frequency - Combined correct and reverse order pair frequency
- End of game winner-loser digit pair frequency - Combined correct and reverse order pair frequency
- All-quarter winner-loser digit pair frequency - Combined correct and reverse order pair frequency

This year, the guy who runs my office pool is giving an immediate booby-prize / refund to the two people unlucky enough to get the squares (2,2) and (5,5). As you'd think, those two squares are pretty awful. No quarter of any Superbowl has ended with either of those pairs. Well, I did about that bad with the two squares I bought, without getting the booby prize. My squares are Colts - 2, Saints - 3 and Colts - 9, Saints 8. I'd win something with those scores reversed, too. But the (2,3) combo has never occurred in any Superbowl end-of-quarter score. And the (8,9) pair has only occurred once. I'm rooting for the Colts, and praying to St. Jude, patron of lost causes, to help my pool prospects.

I've been thinking about a way to populate this table via simulation. I think I'd need surprisingly little data to do so, which probably exists in some tabulated form somewhere: the total number of touchdowns scored in the NFL (by quarter, probably), percent of missed extra point kicks, the total number of 2-point conversions (and percent unsuccessful), total number of safeties, and the total number of field goals attempted/missed. From that, I could just generate a random sequence of "scores" weighted by their relative frequencies and run some monte carlo trials to fully populate that table of yours. I could get a little more advanced, but each advance would add a great deal of complexity to the data collection: e.g., only get data each year from the teams that went to the superbowl that year; bifurcate the data sets into "winning team" and "losing team" subsets, etc. The purpose is not, of course, to serve as a predictive measure, so getting the numbers in aggregate across all NFL teams and all years should be sufficient. The point is really to populate the very sparse table, particularly the "zero" (no data) squares. As evidenced by last night's game (wherein the most frequently populated square, 1/7, won again), it would obviously require a much greater number of games to get a feel for what's going on "down in the noise".

A question, since I am admittedly not a lifelong football watcher like some of you: what, intuitively, is unappealing about the 5/5 and 2/2 squares? After all, I can think of several scenarios which could render those scores; even though they all involve unlikely scoring opportunities (safeties, 2-pt conversions, or missed extra points), they're not unheard of.

Posted by: Ben | February 08, 2010 at 10:58 AM

With respect to your last question, Ben, I think it's the fact that they have unappealing numbers (they really are two of the worst in terms of digit frequency, though 8 actually ties 2 for second words), and their symmetry. It's like you're getting a double-whammy of badness. it really is hard to get 5 early in a game, and it takes an unlikely event (a safety). Ditto 2. Later in the game, these possibilities open up (12, 22, 35, etc), but they are still relatively unlikely scores, and the last requires a high scoring game.

If you're going to run your simulation, I think you'd need more than just frequency of occurrence data. Since my data is given for each quarter, you'd need the likelihood of each type of score in each quarter. For example, I think 2-point conversions are more likely late in a game than early. Ditto, six point scores (due to failed 2-point conversions, which are much more likely to fail than PATs).

I had a shot at winning a little money at the end of the first quarter and filling in one of the "zeros" in my pair chart. The Colts were up 3-0 with a few minutes left in the quarter. But the Saints had pinned them close to their own goal line with a great punt. Could have been a safety, and if that score had held for the rest of the quarter, I would've cashed in on the reverse of my Colts-2, Saints 3 combo. But the Colts had a great 11-yard run on the first play from scrimmage.

Posted by: Marty | February 08, 2010 at 11:17 AM