Is there any evidence of skill?

The variability in the league tables in the previous simulation leads us to question whether placings at the end of the 1999/2000 English soccer season might be simply caused by chance results from evenly matched teams.

If teams have different skill levels, and therefore different probabilities of winning, then there will be more variability in the final points in the table than if all teams are evenly matched. (The difference between the points won by the best and worst teams will be greater.) In the diagram below, we therefore investigate measures of spread in the simulated league tables.

We again assume equally matched teams with P(draw) = 0.2.

Click Accumulate then click Run League several times to simulate a few seasons. The diagram shows a jittered dot plot of the range of points in the league table (maximum minus minimum). This jittered dot plot shows how large the range is likely to be if all teams are equally matched.

English Premier Soccer League in 1999/2000

In the 1999/2000 season, the top team got 91 point and the bottom team got 24 points, a range of 67 points. From the simulation, a range of 67 points seems extremely unlikely from evenly matched teams — the range is usually between 20 and 45 in our simulations — so we can conclude that some teams really are better than others. (More detail about testing for evenly matched teams is given later.)

The diagram also shows the distribution of the standard deviation of points earned by the teams in the simulation. In the real 1999/2000 season, the standard deviation of the points earned by the 20 teams was 16.1. From the simulation, we again conclude that such a high standard deviation would be unlikely if all teams were evenly matched, so skill does seem to play a part in the results.

Simetimes we only need a single categorical summary from each run of a simulation to answer the question of interest. On the previous page, we were only interested in whether Team A was top of the league at the end of the season.

More often, one or more numerical summaries must be recorded and analysed from each run.