Points to avoid relegation in the Premier League?

I know 40 points is the number usually trotted out (and City’s almost there! :cool:), but there have been teams that went down with 40 points or even a few more. I’m curious if there’s an actual, mathematically determined safety value, though. It seems like there ought to be, but it also seems like it’s far too complicated a problem for me to figure out.

20 teams.
Each team plays each other team exactly twice.
Each match has three possible outcomes: Home 3 points, Away nil; Home 1 point, Away 1 point; Home nil points, Away 3.
Bottom three teams at the end of the season are relegated.

I suppose goal difference might complicate things, but perhaps we can set it aside for purposes of this exercise?

What do you all think?

I’m pretty sure a “guaranteed” number of points to stay up can be calculated.

Every match has a maximum of 3 points.
Each team plays 36 matches, so there are a maximum of 20 x 36 / 2 x 3 = 1080 points.

If a team has 61 points, then it is impossible for 17 other teams to have at least that many, as that would require 18 x 61 = 1098 points, so a team with 61 points or more is guaranteed to stay up.

If 18 teams have 60 points, that is 1080 points, but 1080 points is possible only if there are no draws (since a draw has only 2 total points and a win 3, the total number of points = 1080 - the number of draws). This is impossible as if the top 18 have all of the points, then the bottom two have zero - but either the winner of the first match between those two got 3 points or they each got 1 for a draw, so it is impossible for two teams to end with zero points. This means 60 is sufficient to stay up.

I am not sure about 59.

Missed the editing window…

OOPS - each team plays 38 matches, not 36, so that’s a maximum of 1140 points, not 1080.
I’ll have to work on those numbers again.

If a team gets 64, it is safe as it is impossible for 18 teams to have 64 points; this would require 1152 points.

It is possible for 18 teams to have 63; each one has one win and one loss against each of the other 17, and two wins each over the bottom two. One of the 18 is relegated on goal difference.

Thus, 64 is the minimum needed to guarantee staying up.

Excellent. Thanks!

That is indeed excellent. I realise this is a theoretical thread, and I’m cool with that. But as the question has been asked and answered, I’ll just comment that in a normal season, very few teams get 64 points, and by the time anyone does, they are already mathematically safe from relegation anyway, by virtue of being ahead of the third from bottom team by more than 3 x the number of games remaining. As such, I wonder what the typical cut-off point is, i.e. at what point in the season does, say, 60 points tend to absolutely guarantee you safety.

To illustrate what I mean, let’s look at last season (2013/14). I have just discovered the official Premier League website has a really useful historic tables feature where you can look at any point in the season: here, just change the year. Anyway, after every team had played at least 27 games (Chelsea had played 30), Chelsea had 66 points. Third bottom Sunderland had 25 points from 27 games, with 11 to play they could ‘only’ get to 58 and so Chelsea were safe already at that point, as were Man City, Liverpool, and Arsenal on 60, 59, and 59 respectively. So the latter did not need to get to 64 to guarantee their safety that year. And that was with over a quarter of the season left.

For info, as it turned out, that season 33 points could have kept you up if your GD was better than Norwich, a pretty low total. I believe the highest ever points total that resulted in relegation was 43 by West Brom. In fact:

13/14: 33
12/13: 36
11/12: 36
10/11: 39
09/10: 30
08/09: 34
07/08: 36
06/07: 38
05/06: 34
04/05: 33
03/04: 33
02/03: 42
01/02: 36
00/01: 34
99/00: 33
98/99: 36
97/98: 40
96/97: 40
95/96: 38

The above shows the points total at the end of the season of the third from bottom team (i.e. the highest points total to get relegated, which is the same as the lowest points total at which it was possible to not be relegated). Seems I misremembered about West Brom - it is actually West Ham that holds the unwanted record of 42 in 02/03. The data only go back to the 95/96 season as prior to that the league was 22 teams with a 42-game programme.

The mean of the above is 35.84, the mode is 36, and the median is 36. The standard deviation is just under 3, so the classic number of 40 puts you just above one SD away from the mean. That’s where my meagre stats knowledge runs out; perhaps someone can say if this tells us anything about the likelihood of a team with 40 points or more being relegated in any given season? Based on the figures we have it’s happened 15% of the time so far, which sounds about right. It’s probably wrong to say this gives us an 85% confidence that 40 points is safe but that seems about right to me. I wonder what number gives you 99% confidence - suspect it’s around 44.

I suppose this also explains why 64 isn’t used more often. Incidentally, those numbers show that you could theoretically be an invicible side and be relegated (by drawing the vast majority or all of your games).

I’m not 100% sure they do show that, because while I agree drawing all your games would leave you on 38 points (lower than some of the totals), in such a season you would also be denying all your opponents 4 extra points, so the overall number of points in circulation would be lower. The teams that have been relegated on 38 points or more will likely have been in seasons where there have been fewer draws, because in such seasons there will be more points in the system.

Having said that, clearly it is possible to have a season where if one team draws all its games, all the other teams could have won one and lost one of their remaining 36 games. This would give them all 54 points, so in fact you’re quite right. But really this is just a well-known facet of the 3 points for a win system - you’d rather win one and lose one than draw 2. Under the (very) old 2 points for a win system, it would be equal and you probably couldn’t be relegated on 38 points. It’s probably possible to prove that mathematically using the same method as supplied by That Don Guy. Wait - that’s not true because it would be possible for all 20 teams to have 38 points, in which case relegation would be decided on goal difference then goals scored. What we can say is that it would be extremely unlikely for a team to be relegated if they drew all their games under the 2 points for a win system, and quite unlikely under the 3 points for a win system.

If I’m working this out right, it’s theoretically possible to stay up with just 18 points. The bottom three all lose all their matches and the 4th from bottom only has wins against the bottom three.

It should be possible to stay up with 6 points, i think. The bottom four teams lose all their games except to each other, and they all tie each other. No idea what the tiebreak would be in that case!

The first two are goal difference (total goals scored minus total opponents’ goals scored), then total goals scored, then I think it’s points in matches against the other teams in the tie (i.e. “head-to-head”).

Agreed. After that I think they just draw lots (or otherwise determine it by random chance). Of course it’s never got near to that, it’s so unlikely in a 38-game season. I think goals scored is the furthest tie-break ever reached in practice.

In 1988-89, Arsenal and Liverpool both had 76 points and +37 goal difference (thanks to a last-minute Arsenal goal); Arsenal had 73 goals to 65 to win the title. However, they could not participate in the next year’s European Cup tournament because of the ban on English teams following the 1985 Heysel incident.

This raises a related question: What’s the fewest points to make it into the Champions League slots (1-4)? (I suppose there’s also the question of fewest points to win the league.)

Fewest points to win the league has to be 38 - all teams tie all games. Similarly, fewest points for fourth place would be 32 - 3 teams will all their games (except against each other), the other 17 all tie all their games.
Of course, scenarios with points penalties could make those totals lower.

Probably no-one cares, but I thought I may as well correct myself as after seeing this actually occur (Chelsea and Man City are presently tied on points, goal difference, and goals scored) I checked the Premier League rules and they confirm that if 2 or more teams finish the season tied in this way, and it is crucial for the championship, relegation, or qualifying for Europe, then a play-off will be organised. Still can’t see it happening but it’s good they have that provision which is far better than random chance (and, of course, is an extra money-spinner should it ever happen).