Soccer obeys Bessel-function statistics

The soccer World Cup gets under way in Germany on 9 June. For a month, 32 national teams will compete for the world title. Metin Tolan is betting on the home team.

Tolan, an experimental physicist at the University of Dortmund, bases his prediction on an analysis, conducted with three colleagues, of some 34 300 past games—2000 professional games played in Italy, 5300 in England, and 27 000 in Germany. “We approximated a soccer team by a radioactive source. A soccer team emits goals according to Poisson statistics,” he says.

Calculating the probability that a team will win or lose a game by a given number of goals leads to what Tolan calls the “Bessel-function theory of football”—as soccer is called in most places outside the US. A modified Bessel function results from summing over products of probabilities expressed as Poisson distributions.

Tolan’s calculations assume that goals are independent of one another, which, he says, “is reasonable for soccer, but not, for example, for basketball, because there the points are connected.” The calculations wouldn’t work for tennis, either, he adds, because too many points are involved, and not enough chance. “The probability for surprise in tennis is not very high.”

But for soccer the Bessel-function fits are good. “We have no idea why. I never would have guessed that you would find anything regular in a chaotic game like soccer,” says Tolan. Bessel functions would probably not approximate minor league teams well, he adds. “The professional teams, while not of equal strength, have a certain level, and you have a sort of restricted system where not everything can happen.”

For this year’s World Cup, Tolan and his colleagues carried out 100 000 simulations based on past performance to get the probability of each team’s winning the title. “Statistics cannot predict the results of a specific World Cup,” says Tolan. “So this is where the fun begins.” The simulations put Brazil’s chance at 15% and Germany’s at 10.5%, he says. But home teams tend to score an average of 0.6 to 1 additional goal per game. Incorporating that “home advantage,” says Tolan, boosts Germany’s chance of winning to 33%.