# Applying Pythagorean Expectation To The Premier League

Jan 17th, 2013 - Posted by in Betting Theory, Football

Editor of www.pena.lt/y - a website about the statistical and tactical analysis of football

How can we apply Pythagorean Expectation to football? Today on the blog Martin Eastwood takes a look at the current Premier League campaign and which clubs are performing above or under expectations.

The Pythagorean Expectation (below) is an equation originating from baseball to estimate how many games a baseball team could be expected to win based on the number of runs they score and concede. It was originally created by Bill James and was named after Pythagoras' theorem relating the sides of a triangle together: a2 + b2 = c2

Expected wins = runs scored2 / runs scored2 + runs allowed2

For such a simple equation it works pretty well, with teams winning more games than predicted considered to have been luckier than expected while those winning fewer games are thought to have had luck against them.

## Issues With Applying Pythagorean Expectation To Football

Bill James’ Pythagorean Expectation has also been used successfully with NFL and basketball but has so far not worked well for football. The reason being that unlike many American sports, football does not just have wins and losses, it also has draws to contend with.

The original equation predicts that scoring zero runs will give zero points. While this may be true in baseball it is not the same for football. If you substitute goals for runs, then a football team can still get a point even if they don’t score by keeping a clean sheet and achieving a nil-nil draw.

We can refine the original equation to scale the expected number of points scored from a goal to account for this and produce an equation that works well for football. As I have described previously, this modified equation can accurately predict football league tables with an average error of less than four points per season (Figure 2) and can be used from around week ten of the season onwards to make predictions.

Figure 2: Predicted versus actual points for the refined football Pythagorean Equation

## Applying Pythagorean Expectation To The 2012/2013 Premier League Season

The table below shows how the English Premier League table would look based on my football Pythagorean Expectation equation. The stand out performance is from Manchester United who are currently out-performing the prediction by a gigantic 13 points. Based on their goals scored and conceded they would actually be expected to be two points behind Manchester City who have currently over-achieved by four points.

Team Actual Position Played Goals For Goals Conceded Actual Points Predicted Points Points Difference
Man City 2 22 43 19 48 44 +4
Man Utd 1 22 56 29 55 42 +13
Chelsea 3 21 43 19 41 42 -1
Arsenal 6 21 40 24 34 37 -3
Tottenham 4 22 39 27 40 36 +4
Everton 5 22 35 26 37 35 +2
Liverpool 8 22 35 28 31 33 -2
Swansea 9 22 31 26 30 32 -2
West Brom 7 22 31 30 33 30 +3
Stoke 10 22 21 24 29 26 +3
Fulham 13 22 33 38 25 26 -1
West Ham 11 21 24 27 26 25 +1
Sunderland 14 22 24 29 25 25 0
Norwich 12 22 24 34 26 22 +4
Southampton 15 21 28 38 21 22 -1
Newcastle 16 22 27 39 21 22 -1
Reading 19 22 26 42 16 20 -4
Wigan 17 22 23 40 19 19 0
QPR 20 22 17 36 14 16 -2
Aston 18 22 17 42 19 14 +5

Chelsea may consider themselves to be rather unlucky to be in third place having scored the same goals as Manchester City from one less match. However, they have also conceded the same number as well from these matches reducing their overall points total.

At the opposite end of the table Aston Villa are expected to be bottom of the league having scored as few goals as Queens Park Rangers and letting in six more. This is skewed slightly as Aston Villa have had a number of high scoring defeats so rather than conceding those 42 goals evenly across the season, a large proportion came in a small number of heavy losses. The predictions also suggest that Reading are unfortunate to be in a relegation spot as they have outscored many of the teams around them but have struggled due to a poor defence.

Interestingly, if you add up both the predicted points and actual points you get the same total – 590 – showing that at this stage of the season the overall number of points has also been predicted correctly.

As with any mathematical model, the Pythagorean Expectation is not perfect but it certainly has its uses and can provide an interesting estimation of how teams are performing.

And read more of his work on his blog pena.lt/y/

## Related posts

Thanks for the comments, if you are interested in finding
out more about the Pythagorean Expectation and football then you can take a
look at my website – www.pena.lt/y - where I
discuss the theory behind it in more detail. All the formulas used are available
on the site too.

I see the top of the article is edited with the extra information, how good!
Now, while you may not know enough maths to know about it, I think you (Joe) were refering to Taylor's Theorem, or Taylor series, it is the same.
I will explain, but if you want to know more...
http://en.wikipedia.org/wiki/Taylor_series
Basically, this is a numerical aproximation to a given function (drawn curve on X/Y axis). Depending on how close you want your approximation to be, you need more or less factors.
I assume that is what you meant, because with it you get a close guess of a function, but it is infinite in the way that it will never be perfect.
By the way, some terms may be wrong named, since I have never studied maths in english. :)

The predicted points in the model are actually 568 and not 590. Actual points are indeed 590.

oh my my, i did say i was no mathmatican after all and i did get them mixed up. Good Call Crator, you certainly do read the comments and good on you for picking this up. They are different and it is able trigonometry here so the only i get figure out of what football has to do with pythagorean, is the way the players should be positioned, in order to get that ball across and positioning themselves in offensive half. This can be seen in man u and barca players when a few players take part.

I'm not sure if I understood you Joe, but Pythagorean Theorem is not about any never ending number, it's the formula which gives the length of the longest side of a right-angle triangle (a triangle with a 90º angle).
That formula reads as H^2 = a^2+b^2, with H being the hypotenuse (long side) and "a" and "b" being the other sides.
From what I found out, Pythagorean expectation helds no relationship with Pythagoras the mathematician, and has this name just because it is similar. I must say I find this a bit strange though...

hmmm, i'm no mathematician but i thought pythagorean theory is an never ending number which also could translate to any number used at any point. So i wouldn't expect any formula to translate into football but never the less, an interesting theory.

I also found it strange that it wasn't included, but here you go:
http://en.wikipedia.org/wiki/Pythagorean_expectation

Excelent analysis, but what I am lacking here is the formula for predicted points for given time