# Power Laws and Goals

I recently re-read a post by Soccer Statistically’s Ford Bohrmann, which showed how goals are distributed across players in a season. Although the ‘average’ player scores 1.83 goals in a season, this is highly deceptive because goals aren’t distributed normally across players.

In fact, this distribution can be modelled by power laws, whereby a lot of players score zero goals, fewer players one goal, even fewer players two goals and so on.

Taking inspiration from this, I’ve looked at how goals are distributed across players in a single game, as opposed to a whole season. We know that the vast majority of players in a match do not score, and that quantities of one, two and three goals are scored with decreasing likelihood.

This is shown by looking at over 40,000 Premier League appearances since 2009, whereby the vertical axis denotes how many times a player has scored 0, 1, 2 etc goals in a match:

We can model this data using an exponential function, and rescale the vertical axis such that it is a logarithmic scale (i.e. each increment is 10 times larger than the previous amount).

What is the application of this? Firstly it highlights what we already know – the rarity of goals in football. Only a few players get the glory of scoring even one goal, let alone two or three.

The regression line allows us to estimate how many players will score 0, 1, 2 etc goals on any given weekend. Say 277 players make an appearance in one round of matches (as happened last weekend); the line of best fit tells us about 239.4 players will not score, 32.5 will score one, 4.4 will score two and 0.7 will score three or more.

As it happens, the equation overestimates the number of players who will score one and underestimates the amount of players that don’t score; in the weekend just gone 22 players scored one and 251 did not score*. 4 players did score 2 goals, though.

We can also use the distribution to estimate the probability of very rare events – much like scientists forecast the rate of earthquakes. How often should we expect to see six goals in a match by one player? This has yet to happen in 21 seasons of Premier League football, but using the data from 2009-2013, we could say that we’d expect it to happen once in every 180,000 appearances – or roughly once every 18 seasons.

The estimated probabilities of such rare events are highly sensitive to small changes, however. If one of the two players in the sample with 5 goals in a match had scored 4 instead, the probability of a 6-goal appearance would jump to once every 29 seasons. Amazingly, we haven’t played enough Premier League football to get any kind of precise estimate (which would presumably drive David Mitchell mad).

I’m also reminded of offers by betting companies on first goalscorers and multiple goals; the power law distribution at least gives a base rate from which to adjust expectations on how many goals a player will score in a game.

A quick nod to Chris Anderson, too, who has neatly summarised the distribution of goals on a team basis.

*The raw numbers are much better in this respect, which estimate that 253 of 277 players don’t score, 21.3 score one, 2.3 score two and 0.4 three or more. Again, the reality was 251, 22, 4 and 0 respectively.

Data in this post provided by Prozone Sports.

Fascinating how neatly it fits, I’d always remain extremely cautious as the most interesting aspect of this (2+ goals) is the most prone to sample-size errors