Football fans love statistics and can talk endlessly about the number of goals their team has scored or conceded, the number of points they need to ensure promotion or the number of seasons since their left back last scored with his right foot. Now physicists at Warwick University in the UK have taken this obsession one stage further, and have used the methods of statistical physics to analyse the number of goals scored in domestic football games in more than 150 countries. Fans from all over the world will be interested to learn that 'the probability density functions of goals scored cannot be fitted over their entire range by Poisson or negative binomial distributions' (arXiv.org: cond-mat/0110605).
John Greenhough and colleagues in the space and astrophysics group at Warwick kicked off by analysing the scores of 13 000 English top division games and 5000 FA Cup matches between the 1970/71 and 2000/01 seasons. They found that the scores were closely fitted by either Poisson or negative binomial distributions. Probability distributions can be characterized by their mean value and their standard deviation: for a Poisson distribution the mean is equal to the square of the standard deviation, while for a negative binomial distribution it is less.
The appearance of such distributions in football scores was first noted in the 1960s. However, when the Warwick team looked at the number of goals scored by the home and away teams in over 135 000 games in 169 countries since 1999, they found that both the Poisson and negative binomial distributions were inadequate when either team scored more than six goals, or when the total number of goals was more than nine.
Greenhough and colleagues then changed tactics and switched to extremal statistics, which are used to describe rare events in many different branches of science. They found that a so-called Fréchet distribution could describe the distribution of goals scored by the home and away teams, while a Gumbel distribution was better for the total number of goals. Football fans will not be surprised to learn that the home team tends to win, but the average margin of victory – 0.51 goals per game – may come as a surprise.
The Warwick team finished with a statistical nugget for football fans everywhere: ‘a total goal score above 10 occurs approximately once in every 10 000 English top division matches (once every 30 years) but in worldwide domestic matches such a score is seen once in 300 games which amounts to about once every day!’
