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:
Number of Goals per game, $x$ |
Probability of $x$ goals per game |
Expected number of games with $x$ goals |
Actual number of games with $x$ goals |
0 |
0.26 |
13.5 |
17 |
1 |
0.35 |
18.2 |
14 |
2 |
0.24 |
12.3 |
11 |
3 |
0.11 |
5.5 |
7 |
4 |
0.04 |
1.9 |
2 |
5 |
0.01 |
0.5 |
1 |
6 |
0.00 |
0.1 |
0 |
7 |
0.00 |
0.0 |
0 |
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.
Number of goals per game, $x$ |
Probability of $x$ goals per game |
Expected number of games with $x$ goals |
Actual number of games with $x$ goals |
0 |
0.09 |
3.3 |
5 |
1 |
0.21 |
8.1 |
7 |
2 |
0.26 |
9.9 |
6 |
3 |
0.21 |
8.0 |
12 |
4 |
0.13 |
4.9 |
4 |
5 |
0.06 |
2.4 |
2 |
6 |
0.03 |
1.0 |
2 |
7 |
0.01 |
0.3 |
0 |
8 |
0.00 |
0.1 |
0 |
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?
Position under my system |
Actual Position |
Team |
Actual Points |
Predicted Points |
1 | 1 | Man City | 89 | 90.8 |
2 | 2 | Man Utd | 89 | 86.7 |
3 | 4 | Tottenham | 69 | 69.2 |
4 | 3 | Arsenal | 70 | 68.5 |
5 | 6 | Chelsea | 64 | 65.0 |
6 | 7 | Everton | 56 | 59.1 |
7 | 8 | Liverpool | 52 | 56.8 |
8 | 5 | Newcastle | 65 | 55.6 |
9 | 13 | Sunderland | 45 | 51.0 |
10 | 9 | Fulham | 52 | 49.9 |
11 | 10 | West Brom | 47 | 47.0 |
12 | 11 | Swansea | 47 | 46.9 |
13 | 12 | Norwich | 47 | 43.5 |
14 | 16 | Aston Villa | 38 | 40.2 |
15 | 14 | Stoke | 45 | 39.4 |
16 | 15 | Wigan | 43 | 38.7 |
17 | 17 | QPR | 37 | 37.2 |
18 | 19 | Blackburn | 31 | 34.5 |
19 | 18 | Bolton | 36 | 33.7 |
20 | 20 | Wolves | 25 | 27.5 |
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.
From what I’ve read on the topic, the Poisson systematically underestimates the probability of a draw – but it’s a cool model to start from.
What was the (model’s) standard deviation for the number of points?
I’m currently going through some A level Statistics exams this summer, so found this article of particular relevance and interest.
I just wanted to mention that the Poisson probability of a hat-trick is given by \[\frac{e^{-1.35} \cdot 1.35^3}{3!} = 0.106\dots = 0.11 \textrm{(2sf)}\] The argument you used was \[\frac{e^{-1.35 \times 3}}{3!} = 0.003\] which is much smaller.
Null hypothesis: The data can be reasonably modelled by a Poisson distribution.
Alternative Hypothesis: The data cannot be reasonably modelled by a Poisson distribution.
Using the test statistic: Sum[(Observed – Expected)^2/Expected] for the Messi goals and the City goals give the test statistics 2.514 and 3.774 respectively.
Right tailed test: Comparing the test statistics with the Chi-Squared distribution (2 degrees of freedom) at the 10% and 5% critical values finds that $(2.514, 3.774) \lt (4.605, 5.991)$. So whilst I cannot comment on the power of these tests, there is insufficient evidence to reject the null hypothesis.
There is no reason to doubt that the data can be reasonably modelled by a Poisson distribution at these levels of significance.
Again, I must emphasise that I only have a rudimentary knowledge of goodness-of-fit tests and their power, so only offer my interpretation of the information given. Reading back through the article I realize you mention the possibility of more typographical errors in equations, so I wonder if the typo on $\operatorname{P}(X=3|X \sim \operatorname{Po}(1.35))$ is in jest.
Kind Regards,
EP
It was kind of you to give me the excuse, and I wish I had been clever enough to insert the typo in the equation deliberately. Tempted as I am to claim this was entirely deliberate unfortunately it is just a good to honest typo. I was so pleased at deciphering how to use LaTeX to set an equation, I didn’t even notice it wasn’t the right one.
I’ve fixed the Poisson formula, and Edward, I’ve TeXed your comment. I hope that’s OK.
Mr. Gregg: would the Spearman rank correlation coefficient be a good test of your prediction for the final ordering of the table?
Spearman’s Rank CC = 0.9684 and PMCC on the predicted vs. actual points = 0.9771 which indicate very strong positive correlation well beyond the 1% significance level, but this is to be expected since the ‘predictions’ weren’t really predictions at all, but derived from the data itself.
The question is whether a given distribution (e.g. Poisson) is a good model of the data at hand, and if so then it might be assumed that it would be a reasonable model for future data, which is where the predictions come in.
From what I’ve seen with the goodness-of-fit tests I’ve done using the Chi-Squared test statistic, I’d happily conclude that the Poisson distribution is a reasonable model for the number of goals Messi scores in a match, but that doesn’t mean I’ll be heading off to Ladbrokes any time soon!
I agree, not worth heading down to the bookies. How do you get the mean number of goals scored before the season begins or foresee any changes in this value. Messi for example has scored more goals in each season he has played for Barcelona (a trend that presumably will not continue indefinitely)
I have added a spreadsheet here with all the values for this season, and an additional one set up for next season, to see how the model works as a predictive tool.
https://docs.google.com/spreadsheet/ccc?key=0AssaMdmNkO_PdExMMzNmUURxRkczbjA2REVNWlF2SlE
(Hopefully the first weekend won’t end with ten goalless draws.)
You can’t calculate the mean number of goals of a season which has yet happened, but you can use historical data (last season’s results) to calculate a sample mean, then to create a statistical model based on the historical information.
You check to see if the model matches the historical data well enough. If the model fits the data and you assume the future is similar to the past, then you can use the model of the historical data to predict next season’s results (with varying levels of success).
Regarding an increasing mean over time: You can do statistical tests to see if there is evidence that the true mean has, in fact, changed at all (compared to the existing model) and it’s not just due to random variations prevalent in real life which are allowed by the probabilistic nature of the model.
If there was evidence of an increasing mean, the increase over time could be modelled in numerous ways: As you said a linear increase is unlikely to carry on indefinitely like x = A + Bt. However the number of Messi goals per season may increase to a limit like x = A + B(1 – e^-kt), that is to say the number of extra goals he scores in a season decreases to zero over time.
What about off-formula events – Perhaps Messi gets injured in the first match?
In short, I’d say that any formulae/statistical models are imperfect at estimating such complex systems because they can’t account for such eventualities. I think it’s important to acknowledge that these things always have limitations.