Referee bias is a hot topic in any sport, not just soccer. People often accuse referees of favoring the home team in matches. The accusation makes sense: with a stadium full of fans rooting for one team, you would think it would be hard not to favor the home team just a little bit.
But is there a bias evident in the data? To look at this, I looked at data from last season in the EPL. Referees have control over a number of parts of the game. The parts I looked at were fouls, yellow cards, red cards, and offsides. If refs exhibit a home bias in the EPL, they would call more fouls and offsides and give more yellow and red cards to the away team. Pretty simple logic. Let's look at the data piece by piece.
Clearly, the graph shows that average number of fouls is indeed higher for the away team. The away team is called for, on average, 13.04474 fouls, while the home team is called for only 12.09737. That's a difference of about a foul per game. I also ran a two sample t-test to test for significance. Basically what a t-test does is takes in to account the number of observations, mean, and standard deviation (measure of spread) and tests to see if they are equal. In the end, the test gives a p-value between 0 and 1. A p-value basically answers the question, if the two means were actually the same (fouls were the same for home and away), what is the probability that we there would be a difference in the means that we actually saw. A probability of 0 suggest that the means are different, and one of 1 suggests they are the same. Generally, a p-value of .05 or lower is statistically significant, meaning we can rightfully say the means are not the same.
Anyways, the p-value I came up with after running a t-test for home and away fouls was .0003. In other words, we can say that refs called more fouls on the away team at a statistically significant level.
Next I looked at yellow cards. Again, looking at the graph away teams received way more yellow cards on average than home team. Specifically, the home team averaged 1.413158 per game, while the away team averaged 1.955263 per game. That's a difference of about .5 per game. Again, I ran a t-test similar to the one above for fouls, this time for yellow cards. This time, the p-value was 0. This means there were definitely more yellow cards given to away teams than home teams at a statistically significant level.
Third, I looked at red cards. If there is a home referee bias present we would expect to see more red cards given to away teams. Like fouls and yellow cards above, the bias seems to continue. Looking at the graph, there were definitely more red cards given to away teams on average. Per game, home teams received .0605263 per game, and away teams received .1184211. In other words, away teams receive about twice the red cards than home teams. Again, are these numbers significant? Turns out, like fouls and yellow cards, they are. The p-value was .0042, again telling us that away teams received more red cards per game at a statistically significant level.
Finally, I looked at offsides. In this case, the home team was actually called more for offsides. Huh? What's going on here? Home teams, on average, were called for offsides 2.35 times per game, while away teams were only called 2.223684 times per game. Are referees not being biased for offsides, while they are for fouls, yellows and reds?
As always, we should check all possible scenarios. One explanation for the differences in these 4 differences in calls could come not from referee bias, but from the advantage that home teams have over away teams. Maybe teams that are losing naturally foul more, receive more yellow and red cards, and get called for offsides less. It's obvious that home teams have a big advantage over away teams in the EPL: To name just one statistic, home teams scored, on average, 1.63 goals per game, while away teams scored only 1.01 goals per game. This is a pretty wide margin.
If the apparent bias was actually due to the home team's advantage, then losing teams would follow the same pattern as away teams. In other words, losing teams would be called for more fouls and receive more yellow and red cards. Most importantly, losing teams would be called for offsides less. Well, let's look at the data for losing teams compared with winning teams side by side with the data for away teams compared with home teams.
Disregarding the draws column on the far left, the graphs look similar. Both away teams and losing teams are called for more fouls.
Again, if we look at the loss and win bars, they coincide closely with the away and home bars, respectively.
Three in a row. The bars look strikingly similar for away versus home and loss versus win.
Offsides: Finally, we should expect losing teams to be called for less offsides than winning teams, just like how away teams are called for less offsides than home teams...
Look at that! Winning teams are indeed called for more offsides than losing teams.
Conclusion: Based on the first half of the post, it truly appears that referees favor the home team with their calls. In fact, I convinced myself that was the case for a little bit. However, it really comes down to the advantage a home team has in a game instead of any referee bias. While this post doesn't show anything revolutionary about referee bias (admittedly, it would have been pretty cool to make a groundbreaking discovery proving refs favor home teams), it is a good reminder that data can often be deceptive in the way that you look at it. It's important to look at all angles to really understand what is going on beneath the surface before you jump to any conclusions.