One of the perennial discussions we hear near the end of the Premier League season is “What is the point threshold that guarantees a team is safe from relegation”. Right now, as we already know two of the three teams to compete in the Championship next year (well, ok, Middlesbrough still have a mathematical chance), it is the battle for a CL spot that is way more interesting. Chelsea is assured a top four spot as practically are Spurs (97.5% chance even if they lose all remaining matches), leaving two places for the challengers Liverpool, Man City, Man United, and Arsenal.

Before we go on to answer the question “How many points secure a CL spot”, we need to make clear that, theoretically, the answer is different after every match day is played. In a more balanced season, where teams closer to the top lose often to teams lower, we expect a lower threshold. In a season where five or six teams win more often, the point threshold for CL qualification is expected to be higher. That is, if at the beginning of a season we can make the statement “70 points send you to next season’s CL with 70% probability” it doesn’t mean that the same is true after half of the matches are already played.

First of all we take a look at the number of points of the top four teams since 95/96 (first PL season with 20 participants).

We see that there is an upwards trend in the points of the top four teams over time. This suggests that the gap between stronger and weaker teams is growing. In order to have a more rigorous analysis we should take this trend into account.

Now to the main question: What is the probability of qualifying for the CL for a given amount of points earned?

We will use two different methods to estimate the probabilities in question. The first is based on the final tables of the past 21 PL seasons. We don’t want to go deep into technical details, but just briefly, for a given amount of points X, we look at how often a team with X points made it into the top four. If there is more than one team with X points and not all of those teams either succeeded or failed, we count the season as partially successful (for an X-point team). If there are no teams with X points, we look at where an X-point team would have finished in the final table. Here’s what we get.

So, it looks like, based on the past, with 66 points a team has about 48% chance to qualify for next season’s CL. If we take into account the trend seen in the first plot, we tend to believe that the chances are at most that much. 79 points almost certainly guarantee a CL spot. But even with as little as 61 points there seems to be a realistic chance (17%).

The other method to tackle this, the parametric method, is based on simulating PL seasons using a model which associates different strengths to different teams. The way to do this is to take the team strengths and, allowing for some fluctuation of them along the season, simulate full PL seasons of matches. Then we can treat the simulated seasons like we treated the past seasons in the previous method and estimate the conditional probabilities in question. This second method has some advantages and some disadvantages: One of the disadvantages is that it is parametric, i.e. it assumes that the PL match results are generated according the underlying model. This is always an assumption in statistics and it can deviate from the truth. A big advantage is that it allows us to simulate only a part of the season, i.e. we can take the PL table as it stands right now and simulate only the remaining matches. Additionally it allows us to avoid the trend problem of the other approach, if the team strength estimates are current. In the chart below you can see what were the CL qualification chances (given points earned at the end of season) before the season started, and what these chances are now (again, given points earned at the end of the season).

First of all we see that the CL qualification chances at the start of the season (blue line) look a bit different in this plot compared to the previous one. One of the main reasons that drives the differences is that the previous plot is a blend of the sets of team strengths at the start of the previous 20 seasons, whereas the last plot is based solely on the strengths of the 20 teams of the current PL season. Interesting is also the difference between the blue and red lines in the plot above. As the current season is nearly over, there is a much smaller range of final points that can take a team to next season’s CL, compared to the situation at the start of the season. We also learn that this season the bar is higher than we would have expected. At the start of the season 70 points would have got you more than a 50% chance of CL qualification. Now, 70 points only has a 5% chance of being enough.

So, how many points do you need to make it to the CL? It depends when you ask. At this moment 73 points gets you a 51% chance, and 76 points a more than 95% chance.