American Football Statistics

Recently I've been reading a book about how mathematical our everyday lives are (Towing Icebergs, falling dominoes and other adventures in applied mathematics by Robert B. Banks). In this book the author has an interesting chapter dedicated to the statistics of America football.

In this chapter he uses the data from the performances of the NFL teams from 1960 to 1992 to suggest that a first order linear discrete delay differential equation can be used to model the teams winning record for each season and that the performance is periodic with a specific time between each peak in the percentage of wins for that season.

The rationale is that (basically) the performance of the team from the previous season dictates the order (in reverse) of selection of new talent for the upcoming season.

The equation he derives is:

dU(t)/dt=a[U_m-U(t-τ)]

where U(t) is the proportion of wins in each season, U_m is the league wide average value of U and a is the growth coefficient.

Irritatingly he says that there are many ways to solve the equation but then uses an approximate, 'risky' method to produce a solution (Taylor series expansion in case you're wondering). He does however provide references so that you can follow up on the details.

The example he goes through in most detail is that of the Buffalo Bills. The graph of their performance for the years in question is below (thanks to this site for the data). It does indeed seem to be periodic between the years he mentioned.

This seemed odd to me. Obviously you wouldn't be able to determine the exact rank of each team in each year but you would know which teams were likely to be going down or up the rankings. You'd be able to tell, for example, that if your team did well one season, then they were likely to do less well the next.

However, when you look at the data for subsequent years the pattern was hard to determine. It seemed that the Bills were on a consistent downward trend after this. Although given the volatility of the data it's possible that a lot of frequencies would fit this chart.

Has something changed in the way selection is now carried out? I don't know. Is the pattern that he spotted (or at least went through) the same for other teams throughout this time period? Does it also change after 1992?

I'll follow it up by looking at all the teams that were around between 1960 and 1992 next time.