Is Lionel Messi a freak event?

Probably not. Part of the research I do is on how crystals start to form, and this an example of what is called in the trade a “rare event”. So I have an interest in the statistics of rare, or extreme or freakish things. I googled “rare events” to see what others are up to, and one of the hits was a sports betting blog post by a guy called Zach Slaton, who analyses sports statistics. His basic conclusion was that the data suggest that Lionel Messi is a bit of a freak event. I am not sure I agree.

Above is a plot of data (red circles) for this season’s La Liga goal scorers (as of today, data taken from here). The y-axis is the fraction of the (242) La Liga footballers who have scored goals so far, and the x-axis is the number of goals they have scored. Messi is the red circle at the far right. There is one of him, so the fraction is 1/242, and he has scored 43 goals. The point next to him is Ronaldo, who has scored 31.

Slaton studied more data but he fitted an exponential function to the data. When I did that I got quite a poor fit. So I tried a power law. Above, the green line is my power-law fit, i.e., a fit of a function of the form y = b*xc, with c = -1.3 in this case. Power-law distributions like this are very broad, which is just another way of saying here that there is a very wide spread in the number of goals a player scores. The most common number of goals scored by a player is one, but some score more than 20, and Messi has hit the back of the next more than 40 times.

But from the plot above it looks like it is not just Messi, if he is excluded, there is still a wide range in the number of goals scored by a player. He is a very unusual player (statistically there are 241 players you’d rather have running at your team’s defence) but he is not some freakish outlier. As the distribution in the number of goals scored by players is so wide, you expect many players to score few goals but score to score many.