Mike Friere - So, What Is The Pythagorean Win Theorum?


In an effort to make "all that is old, new again", I have resurrected an old series of articles that I put together in a previous "Mack's Mets" lifetime that focus on the new wave of statistical analysis that has shaped baseball scouting and player rankings.

Some would refer to them as "Sabermetrics" and others would argue that it isn't necessarily new anymore.  Both of those statements are true, as we are all getting older by the minute, right?

However, to keep things somewhat fresh, I will go over a new statistic each week and then I will attempt to relate that measure to our favorite team and one or more of our current players to see how we rate, so to speak.

In the previous installment of this series, we took a look at wOBA and how it is a more specific measurement then "normal on base percentage".  For this article, let's focus on the "team aspect" and take a quick look at what is called the Pythagorean Win Theorum or "expected win formula" for those of you who prefer English (kidding).

The formula in question (see below) uses a team's runs scored and runs allowed to generate an "expected winning percentage" that can then be used to calculate a win/loss record based on the number of games the team has played.   So, what does the formula look like?

                                                                          (Runs Scored)2
   Expected Win Percentage   =          ----------------------------------------------
                                                            (Runs Scored)2 + (Runs Allowed)2

Considering all of the different calculations that have been thrown about during this series of articles, the formula listed above is actually pretty easy to digest.  Take a team's total runs scored (squared) and divide that number by the sum of a team's runs scored (squared) and a team's runs allowed (squared).  If you remember your basic order of operations (Parenthesis, Exponents, Multiplication, Division, Addition and Subtraction), then you shouldn't have any problems with this particular assignment.  The end result of this computation will be expressed as a percentage between 0.000 and 1.000, which will be used in the next phase of the overall exercise.

Expected Win Total = (Expected Win Percentage) x (Total Number of Games Played)

This figure will be expressed as a number between zero and the maximum number of games played, which can be subtracted from the total number of games to generate an expected win/loss record.

Here is an example, for your reference.  

Our mythical team (The Crushers) played in a total of 150 games for their season, where they scored 620 runs while allowing 456 runs to their opponents.  We have enough information here to use our listed formula and to generate an expected win total.

                    (620)2                      384,400                     384,400
EWP =       ---------------        =      -------------------             =   --------------   =   0.648%
            (620)2 + (456)2      (384,400) + (207936)         592,336

EWT =  (0.648) x (150) =  97.53 Wins (round up to 98 Wins)

Expected Record98-52, so The Crushers should be pretty successful, based on these figures.

As with most formulas, the more data that you have available to you, the more reliable the results.  To further illustrate this point, consider the following
two teams;

Team A has 1 win and 4 losses, while Team B has 4 wins and 1 loss.....you would expect the team with a better record to have a better differential and thus, a better
expected win percentage.   But, what if Team A's losses were all by one run and their only win was by four runs?   Furthermore, what if the exact opposite were true for Team B?  Both teams would have a similar expected win percentage (likely 0.500%), but Team B (0.800) would be three games ahead of Team A (0.200) in the standings and their win percentages would be WAY OFF, right?  Excessive results have a way of "normalizing" over the course of a full season, so this calculation is more valuable as you "take a look backward" OR to estimate how a team might perform in an ensuing season, provided the basic roster stays consistent.

There are slightly different versions of this formula that tweak the exponent, but the listed versions does a pretty good job of getting you in the "ballpark" so to speak. Oh and they also have versions of this formula that work for other professional sports, in case you are curious.

Lastly, taking this with a HUGE grain of salt, despite an UGLY loss last night the Mets are 12-3 on the young season with 71 runs scored and 51 runs surrendered.   If you use our fomula(s), that differential would equate to an expected win percentage of 0.660 or a record of 10-5, so the team may be overachieving a bit but their successful start is probably not a fluke.


Thomas Brennan said...

Is Pythagorus a righty or lefty, and can he throw 100?

Hopefully, whatever the Mets' expected wins will be based on 2018 runs scored and allowed, they will beat the formula and win more.

last night's game was no theorem, it was an exclamation point!

Mack Ade said...

did he say "runs screwed (2)" ???

Reese Kaplan said...

Using Mike's provided formula, it got me wondering if the man I love to bash was mathmatically a bad manager.


His Pythagorean win total should have been 559. The actual total was 551. So he is objectively below average. But I could have told you that :)

Mike Freire said...

I agree that TC could have done more with what he was given......early returns on Mickey Callaway are positive.

It's all about runs scores and runs prevented.

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