The Role of Luck in the Premier League

How many times have we watched the ball hit the inside of the post and stay out? How many times has the bounce of the ball just taken it inches out of the reach of the striker? How many times has an opposition's shot taken a deflection and flown into the back of the net?

With football being such a low-scoring sport, luck plays a crucial role. The Premier League averages 2.77 goals per game, which is slightly above the general world average. However, with so few scoring plays, the importance of a single goal is high. Many goals or scoring opportunities involve luck in some form or another.

Luck in football can play a huge role. 

However, is there a way that we can measure the impact that luck can have on how a team performs over a season compared with the general talent in a team?

I was reading 'The Book: Playing the Percentages in Baseball' by Tom Tango and came across a formula that he uses to answer such a question. He hypothesises that overall talent can be estimated from the overall variance in a league, where:

Total Variance = Variance due to talent + variance due to luck

To calculate total variance, we will look at win percentages and calculate the variance over the 20 teams in the league. To calculate the variance due to luck, Tango suggests that we can use the following formula:

Standard Deviation due to Luck = √(0.5*(0.5/Number of Matches Played)

However, the first issue to tackle is that of draws. Tango used his formula in baseball, where there are only two possible outcomes - win or lose. Obviously, in football, the draw is a common result, occurring in slightly over 25% of matches, so we cannot simply ignore them.

A commonly used alternative is to consider a draw as half a victory and half a loss. While this is not the worst method, it does ignore the fact that a team gains three points for a victory and just one for a draw. Therefore, it seems logical to consider a draw as a third of a victory and two thirds of a loss. As a result, we can then use this to scale up to actual points later, rather than just win percentages.

So, now we have our method set out, it is time to look at the results. In the current season, the average win percentage in the Premier League is 45.27%. If we multiply this by the 37 games that every team has played, it gives us a value for the average number of wins in the Premier League - 16.75. Remembering that each win is worth 3 points, we find that the average number of points in the Premier League is 50.25.

The variance of the winning percentages across the division is calculated to be 0.0247. We can also calculate the standard deviation due to luck, which comes out at 0.0822, which, when squared, gives an variance due to luck of 0.00676. In other words, the proportion of the total variance that can be explained by luck is 27.4%.

While it is encouraging that 72.6% of variance in the Premier League is determined by the talent of the players in each squad, the luck factor can still play a huge role. As we found earlier, the average number of points is 50.25. Given the variance due to luck, this would suggest that the average team would expect to gain points in the range from 37-64.

The difference between 37 points and 64 points is huge in the Premier League. A total of 37 points in the current season would put the team in 17th position, just two points clear of Wigan in the relegation zone going into the final day of the season. Conversely, 64 points would place the team in 6th position above Everton and, although not this season given the cup winners, would usually be good enough for a Europa League place.

We often see teams over-performing in one season before seemingly under-performing in the following season, raising the questions whether they were really as good as they were in one season, or as bad as they were in the other.

Newcastle are the prime example - last season, they had a wonderful season, finishing 5th and just missing out on the Champions League, while this season, they only secured their survival last weekend after a 2-1 win at QPR. They were fortunate to accrue as many points as they did last season, while arguably they have been unfortunate this season.

Interestingly, they collected 65 points last season, which puts them at the very top end of what we would expect from the average team, while their 41 points this season is only just above the bottom end of what we expect to see. Given many people would argue that Newcastle would be very close to the average, middle of the road Premier League team, their two seasons can be viewed as perfectly demonstrating the impact of luck and the natural variance of points around the expected mean.