# Error in Pythagorean Expectation

Error in Pythagorean Expectation is the difference between its calculated winning percentage to the winning percentage that actually occurred.  Many sites like to claim Pythagorean Expectation is correct and reality is wrong; that a team “should have” won what the Pythagorean expectation formula states.  Mathematics can only estimate.  If there is a difference between reality and math, the error is in the estimation, the math, not reality.

The normal Pythagorean Expectation formula was posted in the previous two posts.  Here is a variation that claims to be more accurate.  We’ll see.

The above formula merely replaces the exponent of 2 with 1.83.  There is another version called the Pythagenpat which has a formula for the exponent states as folllows:

Number of games played for a full season is 162.  The following table lists all 3 of the above variations from the year 1990 to 2013.

 Year Pythag Pythag1.83 PyPat AVG 1990 173.11 168.55 168.26 384 1991 105.72 109.49 109.04 384 1992 173.73 173.48 173.14 452 1993 191.06 190.38 190.54 552 1994 150.40 144.80 149.12 374 1995 211.14 210.20 211.40 440 1996 139.26 142.73 140.42 436 1997 203.34 196.08 198.33 434 1998 179.06 182.30 180.60 640 1999 139.90 140.38 140.63 644 2000 154.86 149.55 154.19 512 2001 198.33 196.63 194.98 630 2002 210.68 212.34 211.39 738 2003 197.64 203.77 201.97 650 2004 182.86 191.84 185.99 688 2005 223.62 222.74 221.49 516 2006 179.09 179.98 180.23 490 2007 199.49 196.29 196.82 488 2008 200.32 194.59 197.39 554 2009 227.94 232.22 227.06 570 2010 163.49 149.12 150.44 532 2011 209.69 200.12 201.86 566 2012 161.24 167.61 166.40 608 2013 194.33 187.61 188.95 624 TOTAL 4370.3 4342.8 4340.64

How error is calculated was shown in the previous post.  The error numbers in the above are a summation for the entire league for each year.  The last column is:

AVG = Sum(abs(Team WAA)) across all teams in the league.

The above shows that PyPat version has the least error but not by much.  Overall error seems to hover around 6-7 WAA/team/year where WAA is measured as (W-L).  This means that if Pythagorean Expectation says a team should be at 81-81, its margin of error is WAA/2 or between 84-78 and 78-84.

Pythagorean Expectation only uses runs as inputs.  Although the sheer number of runs scored and runs scored against should be good estimators, the timing of those runs also matter.  A team that under performs its PE has problem scoring in clutch situations and scores a lot in non-clutch situations, and vice versa.  Some players excel when their team is way ahead and others excel when the game is on the line.   The modeling of this timing and differentiating clutch and non clutch players has to be done by a different model.    PE can only make a rough  estimation based upon the above brute force analysis.