Pythagorean Theorem in Football

The NFL draft is quickly approaching, which means that I’m getting excited for the two weeks of draft talk before the topic completely removes itself from discussion throughout the summer. That’s a sad time given how much I love the sport, but I digress. Nevertheless, this article written by Bill Barnwell back in 2012 inspired me to show off what I learned about the Pythagorean Theorem – well how it applies particularly to football.

Here’s the formula.

Expected Win Percentage = Points For^2.37 / (Points For^2.37 + Points Against^2.37)

Let’s do a quick example. If we know that, in 2015, the Indianapolis Colts finished the season 8-8 while scoring 333 points and giving up 408. We’d use those inputs into our formula.

Pythag wins: 333^2.37 / (333^2.37 + 408^2.37)

If I did that right, that would give me a 38.2%-win percentage or would mean that given the point differential across games, we’d expect the Colts record to be 6-10 (if we round down). 38.2% * 16 games = 6.11. Since they finished 8-8, that would tell us they ‘over performed’ in that year, by two games.

That’s great and all if you want to tell how a team performed after the season, but I’m all about using numbers to predict the wins of a team’s next season. Thankfully, Pro-Football-Reference included a formula to predict year n+1 (or basically just the next season).

Predicted Year N+1 Wins = 4.07 + .12*(Year N Actual wins) + .38*(Year N Pythag wins)

Let’s apply this to the Indianapolis Colts given their 2015 results. We already calculated this year’s Pythag wins which is 6 and the actual wins, 8. Let’s plug it in.

4.07 + (0.12*8) + (0.38*6) = 7.31.

This would imply that the Colts will have a 7-9 record for the 2016 season. I didn’t stop there, though.  I wanted to see how good of a predictive model this is to forecasting future wins. Let’s try it.  Here are the Indianapolis Colts actual records, Pythag wins and Forecasted Pythag records.


Within the table, the pW and pL is calculated the way we’ve done above. Points For^2.37 / (Points For^2.37 + Points Against^2.37). In the N+1 pW and N+1 pL columns, we calculate them using the predicted wins formula above. For example, in 1996, the Colts had a 9-7 record and predicted to have an 8-win 1997 season. As you can see that was clearly off to what they actually did (3-13).

If you know some of the history around that special 1998 season, a man named Peyton Manning was at Tennessee getting ready to be drafted into the NFL. The Colts definitely thought that it was the end of Harbaugh’s time in Indy and wanted to ensure that they could pick up Peyton. I’m not saying that’s exactly what happened, but there are definitely other factors that will predict a team’s next season (who they draft, injuries, personnel changes, etc.). That’s why I took this in a different direction. Instead of putting weight on points for vs. points against directly on the next season, I tested a method that would account for a team’s consistency to over or under perform and by how much. Here are the results for the Colts.


If we take all of the seasons in the first table above. The Colts appear to ‘over perform’ 74% of the time, while exceeding their Pythag n+1 wins by an average of 2.9 games. Inversely, the Colts ‘under perform’ 26% of the time, and don’t reach their Pythag n+1 wins on average 3.9 games. If we take the results from 2015, the Colts are predicted to a 7 above. If they over perform, which we can expect 74% of the time, they’ll hit 10.3 wins given the table above. If they under perform, which we can expect 26% of the time, they’ll only get 3.4 wins. Then, I weighted the 74%*10.3 wins and averaged that with 26%*3.4 wins.

Finally, from averaging the two, I found the predicted wins from that gave me an 8 or 9 (if we round up) win season for 2016.

Photo by hyku

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