Football: Was points per game fair?

Due to the COVID-19 pandemic, many football leagues around the world had to finish their seasons early with fixtures unfulfilled. Some of those leagues abandoned their seasons altogether, ruling them null and void, resulting in no promotions nor relegations. Alternatively a number of curtailed leagues decided to award final placings based on results that had been achieved so far. In those cases, the average points per game (PPG) was used as the deciding measure.

However, in an earlier article, I suggested that PPG may not be a fair performance measure as it didn't take into account the following two important factors. Firstly the number of home and away games played should be considered. When included, this gives us a new measure which has since been called weighted points per game (WPPG) in many quarters. Secondly, crucially we should consider the difficulty of opponents that teams had played so far. I proposed an alternative measure, adjusted points per game (Adj PPG) which accounted for both these factors.

Some leagues, such as the Premier League, managed to complete their remaining fixtures. Hence in this article I use the Premier League results to compare what actually happened with what was predicted using PPG, WPPG and Adj PPG, and determine which of those three was the most accurate, or fairest.

Qualification for Europe and relegation

Below is the final Premier League table and the tables which show what the result would have been if Adj PPG, PPG, or WPPG had been used.

Final Premier League table, Adj PPG table, PPG table, and WPPG table.

Position 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Team Points Liverpool 99 Man City 81 Man Utd 66 Chelsea 66 Leicester 62 Tottenham 59 Wolves 59 Arsenal 56 Sheffield Utd 54 Burnley 54 Southampton 52 Everton 49 Newcastle 44 Crystal Palace 43 Brighton 41 West Ham 39 Aston Villa 35 Bournemouth 34 Watford 34 Norwich 21

Team Adj PPG Liverpool 2.740 Man City 2.091 Leicester 1.824 Chelsea 1.644 Man Utd 1.554 Wolves 1.548 Sheffield Utd 1.524 Tottenham 1.464 Arsenal 1.383 Burnley 1.330 Crystal Palace 1.299 Everton 1.246 Newcastle 1.201 Southampton 1.195 West Ham 0.979 Brighton 0.957 Watford 0.937 Bournemouth 0.918 Aston Villa 0.873 Norwich 0.769

Team PPG Liverpool 2.828 Man City 2.036 Leicester 1.828 Chelsea 1.655 Man Utd 1.552 Sheffield Utd 1.536 Wolves 1.483 Arsenal 1.429 Tottenham 1.414 Burnley 1.345 Crystal Palace 1.345 Everton 1.276 Newcastle 1.207 Southampton 1.172 Brighton 1.000 West Ham 0.931 Watford 0.931 Bournemouth 0.931 Aston Villa 0.893 Norwich 0.724

Team WPPG Liverpool 2.821 Man City 2.049 Leicester 1.821 Chelsea 1.657 Man Utd 1.538 Sheffield Utd 1.531 Wolves 1.483 Tottenham 1.429 Arsenal 1.405 Crystal Palace 1.340 Burnley 1.338 Everton 1.293 Newcastle 1.217 Southampton 1.181 Brighton 1.010 Watford 0.940 Bournemouth 0.940 West Ham 0.936 Aston Villa 0.921 Norwich 0.736

Undoubtedly the most important outcomes to examine are the top four positions (i.e. those who qualified for the Champions League), the 5th and 6th positions (i.e. those who qualified for the Europa League), and the bottom 3 (i.e. those who were relegated).

We can see from the table above that all three measures forecasted exactly the same top five positions. They were successful in predicting which teams would make up the top five, but placed Leicester (who finished 5th) and Man Utd (who finished 3rd) the wrong way around. The final Europa League place was taken by Tottenham, but this was not predicted by any of the measures.

Regarding the bottom three, Adj PPG and PPG both correctly predicted two out of the three teams that were relegated, namely Norwich and Bournemouth, whereas WPPG only predicted 1 correctly, erroneously projecting West Ham for the drop.

Based on these simple indicators, this suggests that Adj PPG and PPG would have been equally appropriate measures, with WPPG less so.

Points Comparison

More importantly, it would be informative to assess the overall predictive accuracy for the whole table for each of the three measures. One way to do this is to compare the actual final points for all the teams with the points predicted by each measure.

By multiplying the values from the table above by the number of games played, 38, we can obtain the predicted number of points for each team. These are shown in the table below along with their differences from the actual points.

Premier League teams’ final points and their predicted points according to Adj PPG, PPG, and WPPG.

Teams Actual points Predicted points Adj PPG PPG WPPG Arsenal 56 52.6 54.3 53.4 Aston Villa 35 33.2 33.9 35.0 Bournemouth 34 34.9 35.4 35.7 Brighton 41 36.4 38.0 38.4 Burnley 54 50.5 51.1 50.8 Chelsea 66 62.5 62.9 63.0 Crystal Palace 43 49.4 51.1 50.9 Everton 49 47.3 48.5 49.1 Leicester 62 69.3 69.5 69.2 Liverpool 99 104.1 107.5 107.2 Man City 81 79.5 77.4 77.9 Man Utd 66 59.1 59.0 58.4 Newcastle 44 45.6 45.9 46.2 Norwich 21 29.2 27.5 28.0 Sheffield Utd 54 57.9 58.4 58.2 Southampton 52 45.4 44.5 44.9 Tottenham 59 55.6 53.7 54.3 Watford 34 35.6 35.4 35.7 West Ham 39 37.2 35.4 35.6 Wolves 59 58.8 56.3 56.4

Difference from actual points Adj PPG PPG WPPG 3.4 1.7 2.6 1.8 1.1 0.0 -0.9 -1.4 -1.7 4.6 3.0 2.6 3.5 2.9 3.2 3.5 3.1 3.0 -6.4 -8.1 -7.9 1.7 0.5 -0.1 -7.3 -7.5 -7.2 -5.1 -8.5 -8.2 1.5 3.6 3.1 6.9 7.0 7.4 -1.6 -1.9 -2.2 -8.2 -6.5 -7.0 -3.9 -4.4 -4.2 6.6 7.5 7.1 3.4 5.3 4.7 -1.6 -1.4 -1.7 1.8 3.6 3.4 0.2 2.6 2.6

The averages of the absolute differences (i.e. ignoring the minus signs) are:

Average of absolute difference
Adj PPG	PPG	WPPG
3.7	4.1	4.0

PPG scored the worst (4.1), WPPG performed slightly better (4.0), and the best achieving was Adj PPG (3.7). This suggests that Adj PPG is the most accurate of the three at predicting final league points.

Conclusions

None of the three measures, Adj PPG, PPG, and WPPG, were by any means perfect at predicting what actually happened in the Premier League. However, based on the calculations above, it can be asserted that PPG, the method actually employed globally by many football leagues for deciding curtailed seasons, is not as fair as Adj PPG. If faced with the same decision again, and with the COVID threat still lingering over us this is a not altogether unlikely possibility, the football leagues should think carefully about whether PPG is most appropriate, or whether Adj PPG would be more suitable.

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About the author

Slim Khan is a Teaching Fellow at Warwick University, and is particularly interested in mathematics and statistics related to games and sports.