Football score distributions

Fit of a number of model distributions to the empirically observed histogram of the goals scored in the qualification stage of the FIFA worl cup.

Football fever

Together with Elmar Bittner (now University of Heidelberg), Andreas Nußbaumer (now University of Mainz) and Wolfhard Janke from the University of Leipzig I had a scientific look at the problem of scoring in football league and cup games. In particular, we wondered, "how scoring in football and the component of self-affirmation depend on cultural and political circumstances.".
Considering the score distributions of the home and away teams, we found that one has to take into account a component of positive feedback or self-affirmation of the teams upon scoring a goal to model the observed distribution of goals. Without this effect of the "football fever", in particular matches with outlandishly high results should occur less often than actually observed.

More details can be found in

  • E. Bittner, A. Nußbaumer, W. Janke, and M. Weigel, Self-affirmation model for football goal distributions, Europhys. Lett. 78, 58002 (2007) [PDF].
  • E. Bittner, A. Nußbaumer, W. Janke, and M. Weigel, Football fever: goal distributions and non-Gaussian statistics, Eur. Phys. J. B 67, 459 (2009) [PDF].

Modelling the EURO 2012

Using the modified Bernoulli models with feedback mentioned above, it is not only possible to model score distributions of whole leagues, but also to determine the degree of "football fever" pertinent to specific teams. Analyzing the qualification stages of all past UEFA Eurocups from 1968 to 2012, one can extract the following plot of all teams competing in the final tournament. (These data are for model B with multiplicative modification of the scoring probability.)


We can even simulate the whole tournament according to the feedback models introduced above, using the scoring and self-affirmation parameters determined for the individual teams from the qualification stage as shown above. Then,

  • matches of the group stage are treated like league games
  • there are some (unnecessarily?) complicated rules for breaking ties, see the UEFA rules
  • in case of a draw, matches in the knockout stage are augmented by 30 minutes overtime and, if necessary, penalties (which are treated as purely random)

The plots below show the resulting probabilities to win the EURO 2012 for each team as per the morning of the date given on the plot. At each date, the matches that have already taken place in reality are successively taken into account.