It's NFL playoff time again. I am in the process of redoing my playoff football model to more accurately reflect the probability of a given team to win a game based on the spread or the Sagarin rating difference.
Stern wrote a paper called "On the Probability of Winning a Football Game" (1991) in which he collected the final scores and the spreads from 1981, 1932, 1984 to determine the relationship between the two. He found that the final score difference between the favorite and the underdog, subtracting the spread could be modeled with a normal distribution with standard deviation of 13.89. The average was 0.07 which is effectively zero for the purposes of the analysis. The probability that a team will win a given game is then the cumulative normal distribution around the spread with a standard deviation of 13.86, or normsdist(spread/13.86) using Excel functions.
I wondered if the analysis had changed in 30 years so I pulled the data for this year through week 16. The plot is below:
The standard deviation is 13.92 with an average -0.17. Hardly any difference found from the earlier analysis for a lot of work to extract the data and get it into a format for the analysis, but at least we now know it hasn't changed. The normsdist function with the spread replaced with the Sagarin difference (home+home advantage-away) divided by 13.92 is what will be used in the game simulation for the playoff fantasy football.