The table never lies, or so they say. So when Manchester City were crowned Premier League Champions last week everyone seemed to agree that they were the best team in the league. As Roberto Mancini said, they had scored more than United and conceded less and beaten them twice in the league. Although United finished on the same number of points it would be difficult to find a measure by which they deserved the title over City. Or would it?

While looking at discrete probability distributions I had asked my students to model whether Lionel Messi’s phenomenal goalscoring this season for Barcelona followed a Poisson distribution. At the time Messi had played 52 games for Barcelona in all competitions and scored a total of 70 goals, giving a mean number of goals per game of $1.35$.

So, for example, the probability of Messi scoring a hat-trick would be given by:

\[\frac{1.35^3 \cdot e^{ -1.35}}{3!}\]

In this case approximately $0.11$. Over 52 games we might therefore expect $52 \times 0.11$, say $5.5$ hat-tricks. In fact he scored 7 in this period.

My students were sceptical that the Poisson distribution would provide a good fit in this case. Their reasoning was that the likelihood of Messi scoring is heavily influenced by the team he is playing against.

A pleasantly self-referential question popular in textbooks refers to typos in textbooks. Now, perhaps in a 52 page book there are 70 typos; we would ordinarily use the Poisson distribution to find the probability of no mistakes, one mistake and so on, on a given page. What if there are 10 pages in our textbook responsible for half of the mistakes – perhaps they are heavy in equations. The mistakes are not randomly distributed, and by overestimating the mean number of errors on the remaining 42 pages, our calculations overestimate the likelihood of seeing an error on these pages.

This is how the Poisson distribution compared to Messi’s actual goalscoring antics for the season:

This predicted Messi scoring in a few more games than he did. Perhaps my students’ instincts were right! I started to wonder how things would look if I just looked at La Liga games. I left this question unsolved for the moment, and decided instead of looking at a single player to look at a team.

I went back to the Premier League to compare City’s attack with what I might expect from a Poisson distribution with a mean of $2.45$, as given by City’s $93$ goal tally over the $38$ game season. I found the following.

Again, I had to be honest: this didn’t look great, and I began to think my students had a point. City’s most common result was to score three goals; maybe having scored two, the likelihood of scoring a third increases as the other team – quite possibly losing – push forward. The Poisson distribution may not be a good fit for the rate at which teams score goals, but again my thoughts wandered.

Although I have not convinced myself that the Poisson distribution is a very good model to use here, if it were, would it work for defensive statistics as well? Using the mean number of goals conceded I could come up with a similar table to the one above. It turns out in the case of City, the Poisson model looks better for their defense, with predictions never further than a game from the actual data.

Now I had a model, albeit one I wasn’t completely convinced by, for City’s attack and their defence. I could obviously use this to find the probability of City winning (scoring more goals than they concede), drawing (scoring the same number of goals as they concede) or losing (scoring fewer goals than they concede). What would happen if I ran the season again, using the assumptions that goals are scored and conceded according to a Poisson distribution? How many points would City get?

According to me $90.8$.

Obviously I know City are going to top the table with this model – they scored more and conceded less than any other team – but how would everyone else fare underneath this system? Does the table lie?

Anything odd? Well, the predicted points using this model do not seem far from what actually happened. Spurs will feel unlucky to have lost out to Arsenal in the race for the Champions League. Blackburn, Bolton and Wolves can’t have many complaints about being relegated and everyone is roughly where you would expect, at least within a place or two.

Except that is, for Newcastle and Sunderland. The North East rivals finished eight places apart this season, in 5th and 13th respectively. My model would have them sitting next to each other in 8th and 9th. How did Newcastle translate 56 goals for and 51 against into 65 points, while Sunderland, with only a seemingly slightly inferior 45 goals for and 46 against, were 20 points behind? It might be down to the fact that of the the 51 goals conceded by Newcastle, 37 came during their 11 losses – an average of more than three a game – while in these games they only scored 10 goals in reply. Sunderland only bothered to score 7 goals in their 15 losses but they conceded 29, an average of just under two a game. In fact on 12 occasions they lost by a single goal, and if it weren’t for a couple of uncharacteristic 4-0s against Everton and West Brom, their goal difference could have matched Newcastle. In comparison, when Newcastle lost they lost more heavily, on only two occasions by a single goal.

In the 90’s there was a snooker based game show on British TV called Big Break. John Virgo was famous for advising players to ‘pot as many balls as you can ‘. It seems that football is simple too, you need to score as many goals as you can at one end, and not concede at the other. Broadly speaking, the better a team is at this, the more points they will get.

As for Newcastle manager Alan Pardew, well – it is points that count, and he seems to have got more points out of his team than you might expect.  After all if you’re going to lose it doesn’t really matter how many you lose by (obviously in the case of Manchester United losing 6-1 to City it mattered considerably – United’s eight goal deficit in goal difference would have been wiped out if this game had finished 2-1 to City, and both teams would then have identical records in terms of goals scored and conceded. Accordingly a play-off would be required at a neutral venue to decide the title – making the most exciting season ever even more exciting!) If Pardew has found a way to lose games by bigger margins when they lose, in return for losing less, then I won’t argue with his manager of the season award.

I have used the word prediction in several places throughout. Of course I am not really predicting anything since the season is over and it is interesting but not terribly useful to use a season’s worth of results as the data from which to model that same season’s worth of results. I tried applying the model at points throughout the season. Perhaps at the half way point, I thought, when Wigan were languishing in the relegation places, I would notice something odd – could their rise to safety have been foreseen? In fact half way through the season, Wigan had scored a miserable 17 goals and let in 37, and I along with most of the football watching public would have confidently predicted they would be relegated. If anyone thought their goal difference would be the same on the last day of the season as it was at this point, after both scoring and conceding 25 goals in the remaining 19 games, then I wish they had told me.