They call it “the beautiful game,” but from a statistical point of view, soccer’s beauty is little more than skin deep. At least that’s how Gerald Skinner from the University of Maryland and his mate Guy Freeman see things. It’s hard not to agree when you look at the game from their point of view
Their approach is to think of a soccer game as an experiment to determine which of two teams is the best. The question then is this: What is the probability that the outcome of the experiment truly represents the relative abilities of the two teams? And the answer, unfortunately, is: not very probable.
They’ve carried out a statistical analysis of soccer scores in various types of competition and found them wanting as a way to determine, with a reasonable degree of certainty, the best team. “For typical scores, the probability of a misleading result is significant,” they say.
One way to increase the statistical significance of a result is to repeat the experiment. This, say Skinner and Freeman, is essentially what happens in tournaments that are organized so that the fate of a team does not rest solely on the result of one match. For example, in the last World Cup, teams were organized into mini-leagues, which should produce a more reliable result. Unfortunately, the winning teams then go into a series of knockout rounds, which are notoriously poor experiments.
In fact, Skinner and Freeman carried out an extensive statistical analysis of the scores at the last World Cup. They point out that if the outcomes of games were a true reflection of the teams’ abilities, then the situation “Team A beats Team B beats Team C, which beats Team A” should never occur. They call this an intransitive triplet.
And yet these paradoxical results occur all the time. In the 2006 World Cup, there were 355 triplets of which 17 percent were intransitive. That may not sound like much, but consider this: If the results were entirely random, you’d expect only 25 percent of the triplets to be intransitive. Is it really possible that the outcome of the World Cup is little better than random?
The mini-league stage leaves 16 teams battling it out for the trophy in a knockout stage. Skinner and Freeman calculate that the chance of the best team winning the World Cup in 2006 was merely 28 percent. “Even on very optimistic assumptions there is less than one chance in three that it was the best team that won the cup,” they conclude.
That’s a shame. But how to fix the situation? Skinner and Freeman suggest continuing the match with successive periods of extra time until the goal difference becomes large enough to yield a chosen level of confidence. That sounds ugly.
So the tradeoff is between a beautiful game and a statistically significant one. An easy decision if ever there was one.
Ref: arxiv.org/abs/0909.4555: Are Soccer Matches Badly Designed experiments?