Evidence of skill?

The simulation on the previous page showed that there is considerable variability in the league table at the end of a season even if all teams are equally matched — the top team often has considerably more points than the bottom team even when we have given all teams equal ability in our simulation.

This variability in the league tables leads us to question whether an actual league table might be explained simply by natural variability of teams with equal ability. A simulation can throw light on whether all teams might have equal abilities.

English Premier Soccer League in 2008/9

The table below shows the points gained by all teams in the English Premier Soccer League at the end of the 2008/9 season. Each team played all other teams twice (once at home and once away) — a total of 38 games — earning 1 point for each draw and 3 points for each win. The Premier League Cup is won by the team with the greatest number of points at the end of the season (Manchester United in the 2008/9 season).

  Team Pts
1 Manchester United 90
2 Liverpool 86
3 Chelsea 83
4 Arsenal 72
5 Everton 63
6 Aston Villa 62
7 Fulham 53
8 Tottenham Hotspur 51
9 West Ham United 51
10 Manchester City 50
11 Wigan Athletic 45
12 Stoke City 45
13 Bolton Wanderers 41
14 Portsmouth 41
15 Blackburn Rovers 41
16 Sunderland 36
17 Hull City 35
18 Newcastle United 34
19 Middlesburgh 32
20 West Bromwich Albion 32

Simulation

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.)

The simulation below assumes equally matched teams with P(draw) = 0.25, the proportion of draws in the actual league that year. We will use it to investigate measures of spread in the simulated league tables.

Click Accumulate then click Run League several times to simulate a few seasons. The diagram shows a 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.

In the actual 2008/9 season, the top team got 90 point and the bottom team got 32 points, a range of 58 points, and the standard deviation of the points was 18.2. From the simulation with equally matched teams, such high spread of results seems extremely likely — so we can conclude that some teams really are better than others.

 RangeStandard devn
Actual 2008/9 soccer league     58 18.2
From simulation   between 15 and 45     between 5 and 12