How Much Does Luck Affect Football Results?
How much does luck impact the result of a football match and how much does the result come down to talent? Today on the blog Martin Eastwood introduces us to the work of Tom Tango and how it can be applied to football to determine over the course of a season how much of a team's performance is due to luck and how much is due to skill.
One of the more intriguing aspects of football is the role of luck, or random chance. With football being such a low scoring game it instinctively feels that luck should play a large role - it only takes one fortunate bounce or deflected shot to turn a tied match into a victory or loss.
Luck, Talent And Variance
But how can we measure how large an effect luck actually plays in football compared with a team or a player’s talent? Two great posts on the subject have recently been written by James Grayson on his blog, the first discussing how accurate a model for predicting the Premier League could be and the second considering the number of games it takes for talent to become the greater than random variation.
Tom Tango, author of ‘ The Book: Playing the Percentages in Baseball’, suggests that talent can be estimated from the total variance in a league, where variance is the spread of data around its average value.
Total variance = variance due to talent + variance due to luck
The total variance is pretty easy to work out; we can just pick a statistic such as win percentage and calculate the variance over a large enough sample size. Then all we need to do is work out how much of that variance is due to talent and how much is due to luck. As a starting point, Tom Tango suggests that we can estimate the standard deviation due to luck using:
Standard deviation due to luck = Sqrt(0.5 x 0.5 / number of games played)
Unfortunately, this is where we run into a slight problem as Tom is using a binomial model that presumes two outputs – win and loss. Unlike many North American sports, we have three outcomes in football to deal with – win, loss and draw – so the estimation does not work.
The easiest solution to this is to just ignore the draws and use wins and losses but this removes around a quarter of all football results so is not a particularly useful approach. Another option is to consider each draw to be worth half a win and half a loss. This seems plausible until you consider that a draw actually provides one third of the points a win does so instead we will consider a draw to be one third a win and two thirds a loss.
Since we can now calculate the total variance from the win percentage and estimate luck’s variance from Tom’s equation above, we can rearrange the original equation and calculate the role talent plays using:
Variance due to talent = total variance – variance due to luck
Looking at all the 38-match English Premier League seasons to date gives us an average win percentage of 46%, with a standard deviation of 14%. Since standard deviation is the square root of variance, we need to raise this value to the power of two, giving us a total variance of 0.01868.
Next we calculate the standard deviation due to luck:
Sqrt(0.5 * 0.5 / 38) = 0.08111
Again, we need to convert this from a standard deviation to variance by raising it to the power of two, giving us a variance due to luck of 0.0066.
Finally, we calculate the variance of talent:
Variance due to talent = 0.01881 – 0.0066
Variance due to talent = 0.0124
So What Does This Actually Mean?
The first point to note is that the variation due to luck accounts for around 35% of the total variance leaving around 65% attributed to talent. This is good news for the skilful footballers as talent has approximately double the effect on a team’s win percentage that luck has.
However, this still means that pretty much one third of a team’s win percentage is purely down to random chance. Since we know the average win percentage is 45%, an average team can expect to achieve 51 points per season but this could potentially vary anywhere between 34 – 68 points due to luck.
Taking last season’s league table as an example, the average team could have ended up anywhere between 19th and 5th place just because of the random variance of their results.
Another calculation we can do is to work out at what point in the season luck and talent have an equal effect. For the English Premier League this works out at twenty matches, meaning that at around the half way mark of the season an average team’s league position is as much due to luck as it is to talent.
This then leaves eighteen matches for talent to overcome luck. Obviously, the first twenty matches are not completely random with the next eighteen determined purely on talent. Rather it is a gradual effect over the course of the season as the more talented teams slowly exert their dominance over random chance as shown in Figure 1.
So next time a team goes on a winning streak or an attacking player fails to score for a few matches remember that football is one third luck and it may well be variance you are looking at.
Follow Martin on Twitter: @penaltyblog
And read more of his work on his blog pena.lt/y/