To answer the question we need to determine the size of the each component. So firstly the random deviation. An average Premiership team will score 52.075 points per season, by virtue of 14.075 wins, 9.85 draws, and 14.075 defeats. The standard deviation in the points they score per game is thus 1.76 points. To determine the amount of random variation after a given number of games we simply plug the number of games into the following equation
Var (random) = SQRT (1.76 * number of games)
(which gives a curve related to the square root of the number of games played).
If we then calculate var (observed) at each point during the season, then var (talent) can be easily calculated via the equation
Var(talent)^2 = Var(observed)^2 – Var(random)^2
To calculate var (observed) I’ve taken each team season and determined the number of points each team had after a given number of games. For each game number (i.e., 1-38 in a 38 game season) I’ve then determined the standard deviation in points that is observed, and used the above equation to determine how much of the variation is random, and how much is due to talent.
Given that we now know both the random variation and the variation due to talent at each point during the season we can compare their relative sizes and determine at which point talent becomes more important than luck.
The resulting plot looks as follows:
So the lines intersect after 16 games (which matches nicely the 18 games I’d determined a couple of days ago using a slightly different method), where Var(random) and Var(talent) are both ~5.30 points. From then on talent becomes increasingly important and by the end of the season the variation due to talent outweighs random variation almost 2:1, hence we see a final league table that makes a lot more sense than the one we look at after five games.
Finally, each team has 6-8 Premiership games left. Over that small of a sample the variation due to luck outweighs the variation due to skill by an average factor of ~2:1. That’s worth bearing in mind when you’re being told about all the incredible survival/collapse stories over the next two months. And, throughout the season, most ‘form’ tables are based on a teams last six games…