As much as I hate football (for explanation – I’m a Blackburn Rovers fan; enough said) I’m enjoying my holiday read. It’s The Numbers Game by Chris Anderson and David Sally and is subtitled “Why everything you know about football is wrong”. It’s not actually the best thing to relax to as I keep jumping up and exclaiming its use/ful/less facts to my family across the sun lounger and then has me scrabbling for the calculator on my smartphone.

Hopefully, this blog will excise this particularly active demon and finally allow me to relax and read my novel, which is what I’d intended until I picked up The Numbers Game. I was struck by the comments made by Mssrs Anderson and Sally about the balance in football between skill and luck, which can be calculated in lots of different ways to be around 50/50.

### The basics: passing the ball

Their first figure contains data from English 1^{st} Division between 1953 and 1967 and shows the frequency of the number of passes in a move when an interception was made (Figure 1). The most common was zero at 39.4% (they must’ve been watching Rovers), i.e. they didn’t even manage one successful pass almost 2/5 of the time before interception by the opposition. Players managed a single pass before interception on the second 27.5% of the time and two successful passes before interception 16.5% of the time. Unsurprisingly, a sequence of 6 passes only occurred 1% of the time.

The data points towards the fact that football spends a lot of the time at equilibrium with attack and defence cancelling each other out so that goals become a rarity. The numbers were updated by Anderson and Sally using 2011-12 data for the Premier League (source: StatDNA) which showed that the distribution had shifted so that there were fewer sequences of two or less passes and more sequences of four or more passes (Figure 2). Perhaps this is not so surprising: the modern game is a lot about possession football and one might expect more trains of long sequences of passes compared to the players of the mid-20^{th} Century.

### An 1880s Model of football

What is surprising, though, is that the distribution is remarkably similar to a finding made by Simon Newcombe in the 1880s and named after Frank Benford for his work in the 1930s. Benford’s Law or the First Digit law predicts how many times the digits 1 to 9 would appear in a list of numbers. In this case, the law works if we change the thing we are counting from “number of passes” to “the number of the pass when an interception occurs” so that we are looking at the number of consecutive passes made. This makes the first pass a digit 1 rather than 0 (which possibly makes more sense anyway).

Figures 3 and 4 show that Benford’s Law predicts the number of passes of the 2011-12 Premier League pretty accurately. The message to coaches it gives is that they should expect around 30% of first passes to be intercepted, with 70% being successful. Around half of moves with one or two passes will be intercepted and the other half will get through. The % of interceptions increases as the number of passes increases so that there is only a 1 in 10 chance that any move with up to 7 passes will be successful.

### Why Benford’s Law?

This is where I admit that I don’t know why Benford’s Law should hold for the number of passes in football. It seems that, despite the best efforts of the players on the pitch, they more or less follow a theory of numbers which also applies to a wide range of real life situations (one of which is cricket). It seems that the Law describes human behaviour very well and soccer players are not as individual as they might like to think.

—

References:

Reep, C & Benjamin, B (1968) Skill and Chance in Association Football, Journal of the Royal Statistical Society, 131, 581-585.

The Numbers Game, Chris Anderson and David Sally, Penguin, (2013).

wiredchopSteve,

Great article. I have an observation about Benford’s law. The probability mass function is limited to the digits 0-9, but football passes theoretically have no upper limit (perhaps, how many passes can you make in 45 minutes?) Given that the recent data seems to end prematurely it would be interesting to see how many pass sequences extend beyond nine. I think an aggregation of data from a league would give this balanced perspective, I wonder what the graphs would like if you focused on teams from polar ends of a division?

Thanks again for the article,

Simon Choppin

stevehaakeSimon, it is also possible to calculate the probabilities of the 2nd digits too so that you can work out the probability of a 10, 11, 12, 13 etc. But it would be good to look at how far different teams deviate from Benford’s Law and see if the more successful ones are below in Figure 4 and the less successful ones above.