Scoring that breakthrough goal can give a team the impetus to win.© Getty

Football-minded mathematicians have proved one of soccer's classic clichés - the theory that once a team scores, the 'floodgates will open' and they will romp to victory with a flurry of goals.

Sometimes thought of as footballing myth, this goal fever can be seen in soccer results stretching back for decades, says mathematician Martin Weigel of Heriot-Watt University in Edinburgh, UK, who did the analysis with colleagues at the University of Leipzig in Germany, the nation hosting the World Cup.

Weigel's team examined results in the German mens' and womens' premier football leagues, and from all previous World Cup tournaments. They discovered that high-scoring games happened more often than would be expected if teams' final scores were randomly distributed.

This suggests that teams do not simply score a number of goals proportional to their skill, but rather are spurred on to greater heights after they score. The researchers call the effect 'self-affirmation'.

"Each time a team scores, it generally increases the probability of scoring during the rest of the game," Weigel says. He and his colleagues designed a mathematical model in which a team's chances of scoring are multiplied for every goal, and found that it fitted the skewed distribution perfectly¹.

Evenly matched

The effect is more noticeable in lower-quality leagues than in the upper echelons of football, Weigel adds. In the World Cup finals, he explains, teams are more evenly matched and therefore less likely to gain a psychological upper hand. That might explain why the highest-scoring contest ever seen in the final stages of the tournament was Austria's 7-5 victory over Switzerland in 1954, whereas in the qualifying stages, which feature a wider range of countries, Australia hammered hapless American Samoa 31-0 in the run-up to the 2002 tournament.

The same theory might explain why the East German league, before unification in 1990, witnessed more outlandishly high scores than the more professional West German Bundesliga, Wiegel says.

Goal rush

Footballers should prepare to cope with the mentally draining effects of conceding a goal if they want to avoid being the victims of a goal rush, says Tim Rees, a sports psychologist at the University of Exeter, UK. "When teams are doing well they become more confident" he says. "But what if you go 1-0 down in the first five minutes? By all means prepare for success, but you also have to get teams to prepare for when things go wrong."

Rees encourages teams to develop a strategy to deal with conceding a goal. "The important thing is just to have a firm plan B," he says. Typically, he recommends that players spend the five minutes immediately after conceding a goal concentrating on staying relaxed and focused on their style of play. "The idea is not to panic and run around like headless chickens," he says.

Rees also points out the folly of attempting to defend a one-goal lead rather than seeking to kill off a vulnerable opponent — something that several of this year's World Cup contestants have been accused of in the past.

Weigel says that his data also show which of this year's World Cup hopefuls are historically best at exploiting the self-affirmation effect (see football fever). One is England, who famously demolished Germany 5-1 in 2001; Italy meanwhile, has lived up to its reputation for doggedly defending a slender lead.

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Heriot-Watt University

• References

  1. Bittner E., Nussbaumer A., Janke W.& Weigel M. preprint available at http://arxiv.org/abs/physics/0606016 (2006).