England stats part 2

Because there were just too many graphs for one post. This time, World Cups and managers.

Today’s papers are full to the brim with questions. Why didn’t England win the World Cup? Was it psychological? Were there too few youngsters on the team? Are young England players improperly trained? Did England play as a team? Is 4-4-2 dead? Should Fabio go?

Unsurprisingly, Capello is not popular in the papers. The Mirror demands “FabiGO” and The Sun‘s headline is simply “It’s time you went, Capello“. The Daily Mail goes further, and somehow suggests that the wealth of Fabio and the team is precisely why England did badly:*

In contrast to England’s only successful manager, Sir Alf Ramsey, who lived in a modest semi in Ipswich and, at the end of his time in charge, was earning just £7,000 a year, Capello has always enjoyed the trappings of the high life, including a vast art collection said to be worth £17million.

Indeed, his convoluted financial affairs brought him to the attention of the Italian tax authorities, though he was never charged with any offence.**

In case you hadn’t guessed, Alf Ramsey was England manager in 1966. Of course, he was also England manager for the 1970 World Cup (in which we fell in the quarter finals) and 1974 World Cup (which we failed to qualify for entirely). And “only successful manager”? Bobby Robson and Terry Venables both took England to the semifinals of a major tournament (1990 World Cup and Euro 1996) and both could have got to the finals if it hadn’t been for bad luck on penalty shootouts. Because that’s what it is. Luck.

Every football match is made up of a thousand random events – every kick of the ball, every split second choice, left or right, every bounce, every gust of wind, every referee’s decision – and these are what ultimately decide the game. Just look at Lampard’s ghost goal, and compare it to Geoff Hurst’s famous goal in 1966. Lampard’s ball crossed the line, but wasn’t counted, while Hurst’s ball didn’t cross the line and was counted. A different ref, a different linesman, a different position, and it all might have been reversed.

Managers and players can affect the probabilities of these random events, of course (the mathematical name for this interplay of chance and skill is stochasticity) – that’s why England scores, on average, 1.9 goals per game while conceding 0.89, while poor old San Marino scores 0.15 per game while letting in 4.15 – but there will always an average and a variance. As it happens, England goals are almost perfectly approximated by a function called the Poisson distribution. The idea behind the Poisson distribution is that if discrete events (goals) happen over discrete intervals (matches), you can work out the odds of a given number of goals being scored in a match. The better something fits a Poisson distribution, the more stochastic the process looks, and England goals fit the Poisson distribution almost uncannily well – there is a 99.8% correlation between the Poisson distribution and the number of goals scored against England, for instance. As a physicist, I’d kill for that level of accuracy/precision in my models.

Goals scored for and against England per match, mapped to the Poisson distributions μ=1.90 and μ=0.89 respectively, r=0.976 and r=0.998

Just look at the sheer degree of randomness in there. Sometimes you’ll get lucky, and get a result far better than average – the 1966 Cup win, in which England scored a statistically unlikely, but accurately modelled, 4 goals – while sometimes you’re unlucky, and you do worse than average – failure to qualify in 1974, say.

So what is the average?

Well, at the World Cup, we advance, on average, to the 2nd stage (μ = 2.06, σ = 1.43 to be exact). In other words, we get out of the group stage and fall in the first game after that – exactly what happened this year. England have only competed in 16 World Cups (before 1950, the British teams weren’t big on international tournaments), and the format and structure of the World Cup has varied over the years (from 16 teams, 3 stages in 1950 to 32 teams, 6 stages today) so there’s not enough data to build a really accurate Poisson distribution, but even comparing this sparse data to a scrappy Poisson distribution, there’s a tentative fit.

World Cup progression. Failing to qualify scores 0, failing to leave the group stage scores 1, and every stage reached beyond that scores another point. μ=2.06, r=0.785

World Cup progression marked with arbitrary units. Failing to qualify scores 0, failing to leave the group stage scores 1, and every stage reached beyond that scores another point. μ=2.06, r=0.785

Winning the World Cup final in 1966 meant passing the group stage plus 3 more matches; equivalent to 5 of these arbitrary units – to do it this year would have required 6 (since the 1966 World Cup did not have a last 16 stage). As you can see, this is not likely – the Poisson distribution puts the odds at around 1.3%. In other words, even in the next 100 years, the chance of England winning the World Cup even once is around 30% – though the odds of making it to the final at least once are a slightly more palatable 75%.

Being an England supporter will be painful for a long while yet.***

So what of Fabio? Well, one World Cup isn’t nearly enough of a sample to go on, but according to the data so far, Fabio Capello has the highest win ratio of any England manager in World Cup history, as well as a solidly average loss ratio – incidentally, even the loathed Sven-Göran Eriksson has a higher than average win percentage, and his results are extremely similar to the vaunted Alf Ramsey’s (60%, 25%, 15% vs. 61%, 24%, 15%). Sven was forced to leave in disgrace, and it looks like Fabio might go the same way, which really doesn’t seem fair to me. I’m not going to lie and say England’s performance this World Cup has been perfect, but it never is. Our luck was simply slightly poorer.

* Incidentally, whatever complaints you may have about footballers’ pay – heaven knows I have a few – I would like to point out that the England team are not getting a penny for playing in the World Cup. Instead, any fees they might have earned are going to charity.

** So in other words, Fabio Capello probably did not commit tax fraud. Thanks Daily Mail.

*** The England women’s team has been fairly poor in the past (winning 49% of games, losing 31%), but since they’ve recently made the extremely major step of moving from amateur to professional and since then made it to the Women’s Euro 2009 finals, there’s a genuine chance of them beating the statistics. Roll on 2011!

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