# Fantasy Football Sabermetrics – Pythagorean Win Expectation

If we’re talking about fantasy football sabermetrics, then the Pythagorean Win Expectation has to be a part of the discussion. The PWE is a way to look at the points scored to points against differential and see how that correlate to wins. Bill James is credited for creating it as a way to show which teams were most influenced by luck in baseball.

Mr. James created the following formula to discern the expected win percentage for any given baseball team:

Wins = Runs Scored^2 / (Runs Scored^2 + Runs Against^2).

Using this formula, he was able to get a pretty accurate predictor of total wins. He proved it by applying the formula to past teams, and seeing how close it was to the actual result. It just so happened that the most accurate exponent was 2.

When I applied this formula to every single team we’ve ever had in my fantasy football league’s 10 year history, my predicted wins were off by around 2-3 on average. So, I did what other folks did for the National Football League and NBA – I played around with my exponent until I got the smallest possible average difference from actual results.

So the smallest possible number I could come with for average difference between real & predicted wins was 1.16353 wins. Now, Bill James and the rest of the Sports Numbers People aren’t too helpful to fantasy football sabermetrics amateurs, so I have no idea how much more accurate their numbers are. I think the more seasons (and bigger sample size), the better off we’ll be in getting this number accurate.

Anyway, after fiddling around with the exponent, the most accurate formula came out to:

Wins = Runs Scored^4.55 / (Runs Scored^4.55 + Runs Against^4.55).

Weird, right? Turns out the most accurate number for basketball is 13.91, as well.

So, how do you do this for your league? Now you can be a fantasy football sabermetrics master in seven easy steps!

1. You’ll need to copy and paste every team’s Points Scored and Points Against for each season. That’s easily done in a Yahoo! league, but I’m not so sure about ESPN.
2. Once those numbers are loaded into excel, create a formula in the spreadsheet for each team like the one I listed above.
3. That will give you the expected win percentage, so you’ll have to multiply that by the amount of games in each season to get expected wins.
4. Once you’ve got that, enter in the actual wins for each of these teams.
5. Subtract actual wins from expected wins to get your difference.
6. Now that you have a “difference” number for each team, enter the function (=STDEV(*enter selection of “difference” numbers*). That result is how many games off your formula averages in its predictions.
7. Play around with the exponents until you can get the lowest number possible!

With this method, I was able to create a spreadsheet that showed the luckiest and unluckiest teams of all time. Managers in my league found it interesting, and it gave us much to talk about. Likewise, if you’re able to find a smaller average difference from expected wins, let me know your sample size and if you did anything differently.

Much like the PWE for baseball, this can also tell us which fantasy teams are outperforming or under-performing expectations due to luck. With this knowledge we can see if our teams are truly “for real” or not. Although this is probably my least practical piece on fantasy football sabermetrics, it’s certainly lead to the most conversation and received the best reaction among the managers in my league!

## 8 thoughts on “Fantasy Football Sabermetrics – Pythagorean Win Expectation”

1. Did you ever find out what the accurate exponent was for a standard 10 or 12 team Yahoo league?

Also, you could use solver to find it precisely without the trial and error.

1. I have yet to take a look at that – in the future, I’ll try to find as many public leagues as I can and scrape data to get better results.

Thanks for the suggestion and your comment!

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3. Jim says:

This doesn’t entirely account for luck. Points against is something you have no control over. If your points against is obscenely high, your PWE will be small. I would argue that having a very high PA is extremely unlucky, but it won’t show up in PWE. I’m using the average points of the league instead of PA. I don’t know how accurate this is, but I think it accounts for the luck factor a little more.

Also, I have been playing around with the exponent, and I think somewhere from 3.4-3.5 is pretty close. 3.48 seems to be the magic number right now, but I’m still accumulating data.

Yes, I’m a math nerd with too much time on my hands.

1. Hi Jim, thanks for reading! Really interesting to see that your league’s exponent is lower. And I wouldn’t say you have too much time on your hands – certain fantasy football leagues can be a matter of life and death, this stuff is important :)

Your method definitely removes more luck – I think for this article, I just wanted to start and see the results of some proven ways of looking at luck in sports.

I think your instinct is correct. You can only completely remove luck if you ignore the other team in the matchup!

1. Jim says:

Actually, I think the exponent will depend completely on the scoring system of the league. Higher scoring leagues will probably have a higher exponent, and vice versa. Our league is a bit lower scoring than most leagues, so that could explain it.

4. John says:

Yeah just to reiterate the flaw between this idea and Bill James’ is that points scored against in fantasy football are independent of your own teams ability. In James’ formula you are rewarded for holding a team to minimal runs against. In fantasy football that is just a matter of fortune. A player in my league who is 1-6 finds himself with the third least points scored and lowest pts against. He’s 1-6 with a pyth w-l of 3-4 and an extrapolated 5-7. Suggesting he’ll only lose one more game all season?

I’m sure there’s strange exceptions to mess with any formula but this simple fact made me hesitant to present this to my league.

5. Rick Mc says:

I wanted to find some formula to account for what I perceived as wide variance in PF and PA versus wins in fantasy football. I searched for “fantasy football pythagorean record” and found this post. After reading it and the comments, I think my question may have been incorrect. I am trying to find out retrospectively how unlucky some teams have been or more generally how ‘flukey’ wins are in fantasy football. Does finding a pythag for your league tell you that? Or is the exponent some indication of the luck factor?