The intent of this study is to see how much more likely strong teams are to convert third or fourth downs compared to weak teams.
In this study I plot various conversion probabilities of 3rd and 4th downs, as a function of team strength (and distance to go). The measure of team strength that I use is the offense’s end of season winning percentage minus the defense’s end of season winning percentage. For the purpose of this study offensive conversion percentage is the number of times a team coverts a third and fourth down divided by the number of attempts to make that conversion. And of course the defensive conversion percentage is the number of times the defense holds on third and fourth down divided by the number of attempts.
So my X and Y variables are:
X =(Offense end of season winning percentage) – (Defense end of season winning percentage)
Y = 1 (successful conversion or TD), 0 (failed conversion)
I know there are much better ways to measure how good an offense or defense is than the end of season winning percentage; but I think this is a good first step. I also think it’s accessible. If a coach wants to get a better idea of how likely they might be to convert in a certain situation the win-loss records might be the first thing they think of. It’s probably how they think of their team strength – “we are a 12 and 4 team”.
I then plot a linear fit from all the 1 and 0 y data points. This means I am assuming that the “true” probability of conversion is a linear function of team W-L record.
I use data from 2002 to 2011 (regular season plus playoffs). I use most of the “normal football” assumptions, but not all. I was slightly more aggressive with time, allowing the 2nd quarter to be counted all the way up to 7 minutes left. The 4th quarter is excluded; all of the 1st and 3rd quarters are included. The plays are only included if the line of scrimmage is between a team’s own 25 and the opponents 20. The score differential is limited to 10 points.
Below are the plots for 3rd and 4th down, for 1, 2, and 3 yards to go. (Note that the plots show win fraction instead of percentage).
4th and 1:
3rd and 1:
4th and 2:
3rd and 2:
4th and 3:
3rd and 3:
It appears that strong teams do end up converting at better rates…but most of the time not by that much. The most extreme is for 4th and 2, a difference of about 40% from the worst possible offensive record difference to the best possible offensive record difference (although no data has actually come from teams that extreme in record difference, it might be more appropriate to use a logistic fit when extrapolating that far out). 4th and 3 actually has a downward trend (good teams are actually less likely to convert), but lets just say that’s because of sample size. I haven’t tested the significance of that. There is a small bias, since teams that converted on any given conversion are more likely to have won the game, teams that failed are more likely to have lost. Correcting for this effect will only flatten the slope of these curves.