## Points above replacement: narrowing the range for the best and worst players

Disclaimer: A lot of this was written as it entered my head. My stream of consciousness is a isn’t necessarily logical. If you can follow then kudos.

Last time I looked at replacement level performance and determined that the Premiership teams are essentially fighting for a share of the 410 ‘non replacement level’ points that are scored each season (link). Here I’m going to have a stab of narrowing this down to determine just how many points the best and worst players are worth. This time I’m going to do it using points, at some point in the future I’ll use shots and see if I come up with a similar answer.

We know that each season there are a total of ~800,000 ‘player minutes’ played*. The ~400 ‘non replacement level’ points I mentioned in the previous paragraph must somehow be divided between these 800,000 minutes. Divide the points by minutes and we find that an average Premiership player is worth ~ 0.0005 PAR per minute, or 1.80 PAR per season**.

The best players

The best Premiership team ever, in terms of points, was Chelsea in 2004-05, scoring 95 points (~63 PAR). Let’s imagine the theoretical scenario whereby that for every minute of that season the eleven Chelsea players on the pitch were both a) equally talented and b) the best players in the league. In this scenario each player contributed equally to the PAR earned by Chelsea that season and so the PAR earned by Chelsea should be distributed equally amongst those players.

63 PAR / 11 players ~ 6 PAR / player

In reality the players were neither all equally talented nor all the best in the league and as such some players must have been worth more than 6 PAR. Thus this value serves as the lower bounds for the PAR for the best player in the league.

The step to determine the upper bound is not perfect but I think it’s justifiable nonetheless (in my head it works as a logical jump). Let’s assume that ten of the players on the ’04-05 Chelsea team were exactly league average and that the extra PAR they earned was entirely due to one player. That one player would thus be worth:

63 PAR – (10 average players x 1.80 PAR per player) ~ 45 PAR

The worst players

This time we’ll use the 11 points scored by Derby in 2007-08. They finished that season 21 points below replacement, a remarkable feat achieved through a combination of incompetence and bad luck. Following the same logic as earlier lets assume that the players on the pitch were equally talented and we can distribute the (negative) PAR between them.

-21 PAR / 11 players ~ -2 PAR / player

Thus the ‘best’ that the worst player can be is -2 PAR. In reality, as earlier, this isn’t the case and the worst players will actually be worse than this.

As before I’ll assume that ten of the players on the team were exactly league average and their (lack of) performance was due to one player to determine the outer bounds of their value.

In this case that player is worth

-21 PAR – (10 average players x 1.80 PAR per player) ~ -39 PAR

All of this leaves us with a pair of equations.

The value of the best player in the league is therefore summarised as: 6 < PAR < 45

The value of the worst player in the league is therefore summarised as: -39 < PAR < -2

Now the ranges are massive here but the point of this isn’t to give a specific value, more as a guide. Basically if this reasoning is taken to its logical conclusion and results in a ratings system then the values for the best and worst players should fall within these ranges. If they fall outside these bounds then there’s a good chance that, somewhere along the line, something’s been screwed up.

I think I could make a pretty good guess at where the best and worst players lie within these ranges. The question is how best to use our knowledge to narrow the range?

*22 players x 380 games per season x 95 minutes per game = 794,200 player minutes per season
**presuming a player plays every minute of the season

## 2 thoughts on “Points above replacement: narrowing the range for the best and worst players”

1. Interesting, and I like how you’ve gone about it. Two things instantly pop into my head …

1) Question: You use 95 minutes/game and I can see your reasoning for doing so, as actions happen in extra time … but wouldn’t using 90 mins. be better? Normally extra time at the end of the first half is 1 minute which would bias players coming on in the second half, no? as extra time is usually longer at the end of the second half (in your case an average of 4 minutes) … hmm, I’m not explaining this right and it doesn’t really affect your numbers … I can’t explain it but there’s something that doesn’t sit right with me … ah well, minor point. 🙂

2) Thought: You’ve used 11 players to calculate PPAR (Player PAR) values … it is going to be difficult to later see how it is with shots/passes etc. (whatever other stat we can find), as there is a goalkeeper to deal with. Is this a problem? You reference to ice hockey a lot, how do they solve this (if they have such a stat)?
If you look at the Calgary Flames (go flames go!) then you can see that they have two star players … Iginla (gotta love him) and their ‘keeper Kipper (what a star) … how do their “PAR” numbers relate for example?

I think your values are spot on but when individual players are added to the equation it would be interesting to see how to get over the hurdle of the goalkeeper who is so different when compared to the 10 outfield players.
Maybe I’m thinking too many steps ahead here, but I think your basic reasoning is good and have no real comments on it. 🙂

cheers.

2. been thinking about this a bit more … recently read an article on baseball stats with regards to share a player has in a win … kinda like PPAR.