Something you always hear about is how home teams tend to have the advantage in a game. I checked Reddit to see if anyone had posted anything on this but the closest I got were analyses of nfl scores that have never ocurred and a bare distribution of scores , and so I decided to do some quantitative analysis of my own.
I looked at every professional football score from 1920 till
today and
plotted the distribution.
The gray diagonal indicates ties, and if home field advantage had no effect on the outcome we would expect the weighted average of the scores to lie on this line. However, as shown, the weighted average actually sits at (away points, home points) = (18.4, 21.5). From this graph I also define home field advantage to be the distance between the weighted average and the diagonal (I'd love to hear people's ideas on how else to quantify home field advantage). This distance comes out to be very close to 2 points. Below is an inset of the distribution which gives a better sense for the structure of the distribution
I also looked at the
point differential density function (right axis).
For a given point differential, this computes (home wins - away wins)/total games. Therefore, the range of values that this can take on fall between -1 and 1. If the home (away) team always wins then this graph would be constantly 1 (-1), whereas if home field advantage had no effect we would expect a flatline at 0. For all point differentials the home team manages to get the advantage. I also overlaid a plot of the total number of games that have ocurred at a given point differential (left axis), which gives an idea of the effect of normalization on each point (note that I normalize by the total games at that point differential, NOT total games for all point differentials). For this plot I ignore point differentials that ocurred less than 5 times (which conveniently end up being point differentials above 50).
Lastly I look at the
distribution of what I call the symmetric element ratios
.
For this plot I sum the elements that are symmetric about the diagonal (e.g. (17,20) + (20,17)), and then divide these elements by that sum. From this plot we get an idea of how frequent certain scores are relative to home vs away outcome. What's interesting about this plot is that it highlights how very lopsided games tend to very strongly favor the home team, for example how shutouts are almost always by home teams.
So overall, I would say that the home team has a definitive
quantitative advantage.
Take that Bill...
If you made it this far thanks for reading!
I used Pro-Football-Reference as my source. Last but not least, I would
be remiss if I didn't mention
Jon Bois' slightly
related Scorigami video. It is a must watch, along with
pretty much any other video he has ever made.