Is Premier League Defending Getting Worse?

In Radio 4`s More or Less, sports statistician Rob Mastrodomenico took a look at a BBC blog that discusses whether premier league defending is getting worse. He focusses on the fact that the number of goals per game has risen from 2.59 in 2010/11 to 2.97 in 2011/12.

After discussing the football and the pundits, he turns to the statistics. It is unarguable that 2.97 is bigger than 2.59, but he asks whether the difference is statistically significant. He concludes that it is not and calculates how great the difference would need to be for it to be statistically significant.

Now the question of statistical significance arises when we have the necessarily limited information that we gain from a sample and we want to know what that tells us about a population. Because our sample is an incomplete picture of the population, we are met with uncertainty -- and that`s where probability comes in. But if we have all of the data, there is no uncertainty and no role for statistical inference. If there is a difference between two things then we must conclude that, well, there is a difference between the two things.

Mastrodomenico`s analysis of the BBC`s football data treats the games played during the first half of each season as a sample of all the games that could theoretically be played. He sees each season as having an unknown, underlying goals-per-game average and the figures of 2.97 and 2.59 as estimates of their respective averages. It is thus reasonable to carry out standard statistical inference procedures.

It`s rather like tossing a coin. We can suppose that any given coin has a particular propensity to come down heads. If we were to toss it many, many times we might end up with a very good idea of what that propensity is. It might be 50:50 heads:tails or, in the case of a biased coin, it might be 60:40. In practice we wouldn`t toss the coin millions of times, say, but a couple of hundred times isn`t too arduous and the resulting proportion of heads would give us an insight into the coin`s underlying propensity towards heads. If we obtained 110 heads out of 200, we might reasonably argue that this is not inconsistent with a long-term propensity of 50:50 heads:tails although we would expect 100 heads out of 200 tosses, the process is a random one and a small deviation is not unlikely.

The problem for me in this approach is that I don`t believe football matches are like coin tosses. Every coin toss is essentially the same as every other coin toss: though the outcome varies, the circumstances surrounding it are the same. And you can, in principle, repeatedly toss the coin as often as you want. But football matches are subject to many different variables: no two matches are the same. And you cannot endlessly repeat matches in an attempt to discern an underlying goals-per-game average.

In some respects probability and statistics are not like other mathematical disciplines: there are differing schools of thought. There isn`t universal agreement, for example, on what a probability actually is and how you can calculate it. Some people believe that a probability is inherent to a situation so that all observers would agree on what that probability is. Others argue that if you have different information to someone else you can reasonably come to a different assessment of the probability of a particular event happening.

Mastrodomenico believes that any given football season has an underlying goals-per-game average and that we can gain insight into what that average is by looking at the actual games played. I look at it differently. I believe we can only go by those actual games. I think football matches are fundamentally different to coin tosses and cannot be treated in the same way. He concludes that we have insufficient evidence of a difference between last season and this season; I say there is a difference. Of course, whether that difference is meaningful and what caused it are completely different questions . . .