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| '''Pythagorean expectation''' is a formula invented by [[Bill James]] to estimate how many games a [[baseball]] team "should" have won based on the number of [[run (baseball)|runs]] they scored and allowed. Comparing a team's actual and Pythagorean winning percentage can be used to evaluate how lucky that team was (by examining the variation between the two winning percentages). The name comes from the formula's resemblance to the [[Pythagorean theorem]].<ref>[http://thegamedesigner.blogspot.com/2012/05/pythagoras-explained.html The Game Designer|title=Pythagoras Explained]</ref> | | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
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| The basic formula is:
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
| | * Only registered users will be able to execute this rendering mode. |
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| :<math>\mathrm{Win} = \frac{\text{runs scored}^2}{\text{runs scored}^2 + \text{runs allowed}^2} = \frac{1}{1+(\text{runs allowed}/\text{runs scored})^2}</math> | | Registered users will be able to choose between the following three rendering modes: |
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| where Win is the winning ratio generated by the formula. The expected number of wins would be the expected winning ratio multiplied by the number of games played.
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| ==Empirical origin==
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| Empirically, this formula correlates fairly well with how baseball teams actually perform. However, statisticians since the invention of this formula found it to have a fairly routine error, generally about 3 games off. For example, in 2002, the New York Yankees scored 897 runs, allowing 697 runs. According to James' original formula, the Yankees should have won 62.35% of their games.
| | :<math forcemathmode="png">E=mc^2</math> |
| :<math>\mathrm{Win} = \frac{\text{897}^{2}}{\text{897}^{2} + \text{697}^{2}} = 0.623525865</math> | |
| Based on a 162 game season, the Yankees should have won 101.07 games. The 2002 Yankees actually went 103-58.<ref>[http://www.baseball-reference.com/teams/NYY/2002.shtml Baseball-reference.com | 2002 NY Yankees Statistics<!-- Bot generated title -->]</ref>
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| In efforts to fix this error, statisticians have performed numerous searches to find the ideal exponent.
| | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| If using a single number exponent, 1.83 is the most accurate, and the one used by baseball-reference.com, the premier website for baseball statistics across teams and time.<ref>[http://www.baseball-reference.com/about/faq.shtml#pyth Baseball-reference.com | What is pythagorean winning percentage?<!-- Bot generated title -->]</ref> The updated formula therefore reads as follows:
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| :<math>\mathrm{Win} = \frac{\text{runs scored}^{1.83}}{\text{runs scored}^{1.83} + \text{runs allowed}^{1.83}} = \frac{1}{1+(\text{runs allowed}/\text{runs scored})^{1.83}}</math>
| | ==Demos== |
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| The most widely known is the Pythagenport formula<ref>[http://www.baseballprospectus.com/article.php?articleid=342 Baseball Prospectus | Articles | Revisiting the Pythagorean Theorem<!-- Bot generated title -->]</ref> developed by [[Clay Davenport]] of [[Baseball Prospectus]]:
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| :<math>\mathrm{Exponent} = 1.50 * log(\frac{R+RA}G) +0.45</math>
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| He concluded that the exponent should be calculated from a given team based on the team's runs scored (R), runs allowed (RA), and games (G). By not reducing the exponent to a single number for teams in any season, Davenport was able to report a 3.9911 root-mean-square error as opposed to a 4.126 root-mean-square error for an exponent of 2.<ref>[http://www.baseballprospectus.com/article.php?articleid=342 Baseball Prospectus | Articles | Revisiting the Pythagorean Theorem<!-- Bot generated title -->]</ref>
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| Less well known but equally (if not more) effective is the [[Pythagenpat]] formula, developed by David Smyth.<ref>[http://gosu02.tripod.com/id69.html W% Estimators<!-- Bot generated title -->]</ref>
| | * accessibility: |
| :<math>\mathrm{Exponent} = (\frac{R+RA}G)^{.287} </math> | | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| Davenport expressed his support for this formula, saying: <blockquote>
| | ==Test pages == |
| After further review, I (Clay) have come to the conclusion that the so-called Smyth/Patriot method, aka Pythagenpat, is a better fit. In that, ''X'' = ((''rs'' + ''ra'')/''g'')<sup>0.285</sup>, although there is some wiggle room for disagreement in the exponent. Anyway, that equation is simpler, more elegant, and gets the better answer over a wider range of runs scored than Pythagenport, including the mandatory value of 1 at 1 rpg.<ref>[http://baseballprospectus.com/glossary/index.php?mode=viewstat&stat=136 Baseball Prospectus | Glossary<!-- Bot generated title -->]</ref>
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| </blockquote>
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| These formulas are only necessary when dealing with extreme situations in which the average number of runs scored per game is either very high or very low. For most situations, simply squaring each variable yields accurate results.
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
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| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
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| There are some systematic statistical deviations between actual winning percentage and expected winning percentage, which include [[bullpen]] quality and luck. In addition, the formula tends to [[Regression toward the mean|regress toward the mean]], as teams that win a lot of games tend to be underrepresented by the formula (meaning they "should" have won fewer games), and teams that lose a lot of games tend to be overrepresented (they "should" have won more).
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
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| =="Second-order" and "third-order" wins==
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| In their Adjusted Standings Report,<ref>[http://www.baseballprospectus.com/statistics/standings.php Adjusted Standings Report]</ref> [[Baseball Prospectus]] refers to different "orders" of wins for a team. The basic order of wins is simply the number of games they have won. However, because a team's record may not reflect its true talent due to luck, different measures of a team's talent were developed.
| | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
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| First-order wins, based on pure run differential, are the number of expected wins generated by the "pythagenport" formula (see above). In addition, to further filter out the distortions of luck, [[sabermetrician]]s can also calculate a team's ''expected'' runs scored and allowed via a [[runs created]]-type equation (the most accurate at the team level being [[Base Runs]]). These formulas result in the team's expected number of runs given their offensive and defensive stats (total singles, doubles, walks, etc.), which helps to eliminate the luck factor of the order in which the team's hits and walks came within an inning. Using these stats, sabermetricians can calculate how many runs a team "should" have scored or allowed.
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| By plugging these expected runs scored and allowed into the pythagorean formula, one can generate second-order wins, the number of wins a team deserves based on the number of runs they should have scored and allowed given their component offensive and defensive statistics. Third-order wins are second-order wins that have been adjusted for strength of schedule (the quality of the opponent's pitching and hitting). Second- and third-order winning percentage has been shown to predict future actual team winning percentage better than both actual winning percentage and first-order winning percentage.
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| ==Theoretical explanation== | |
| Initially the correlation between the formula and actual winning percentage was simply an experimental observation. In 2003, Hein Hundal provided an inexact derivation of the formula and showed that the Pythagorean exponent was approximately 2/(''σ''√''π'') where ''σ'' was the standard deviation of runs scored by all teams divided by the average number of runs scored.<ref>{{cite news|first=Hein|last=Hundal|title=Derivation of James Pythagorean Formula (Long)|url=http://groups.google.com/group/rec.puzzles/browse_thread/thread/3be0e6ad49631ddb/bfb52d16b12955ac?q=hein+hundal+pythagorean&fwc=1}}</ref> In 2006, Professor Steven J. Miller provided a statistical derivation of the formula<ref>{{cite journal |author1=Miller |journal=Chance Magazine 20 (2007), no. 1, 40–48 |pages=9698 |title=A Derivation of the Pythagorean Won-Loss Formula in Baseball |year=2005 |arxiv=math/0509698 |bibcode=2005math......9698M}}</ref> under some assumptions about baseball games: if runs for each team follow a [[Weibull distribution]] and the runs scored and allowed per game are statistically independent, then the formula gives the probability of winning.<ref>{{cite journal|first=Steven J|last=Miller|title=A Derivation of the Pythagorean Won-Loss Formula in Baseball|journal=Chance Magazine 20 (2007), no. 1, 40--48|pages=9698|year=2005|arxiv=math/0509698|bibcode=2005math......9698M }}</ref>
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| ==Use in basketball==
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| American sports executive [[Daryl Morey]] was the first to adapt James' Pythagorean expectation to professional basketball while a researcher at [[STATS, Inc.]]. He found that using 13.91 for the exponents provided an acceptable model for predicting won-lost percentages:
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| :<math>\mathrm{Win} = \frac{\text{points for}^{13.91}}{\text{points for}^{13.91} + \text{points against}^{13.91}}.</math>
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| Daryl's "Modified Pythagorean Theorem" was first published in ''[http://morey.org/pythbook.gif STATS Basketball Scoreboard, 1993-94]''.<ref>{{cite book |last= Dewan |first= John |coauthors= Don Zminda, STATS, Inc. Staff |title= [[STATS Basketball Scoreboard, 1993-94]] |publisher= [[STATS, Inc.]] |year= 1993 |month= October |isbn= 0-06-273035-5|page= 17}}</ref>
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| Noted basketball analyst [[Dean Oliver (statistician)|Dean Oliver]] also applied James' Pythagorean theory to professional basketball. The result was similar.
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| Another noted basketball statistician, [[John Hollinger]], uses a similar Pythagorean formula except with 16.5 as the exponent.
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| ==Use in pro football==
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| The formula has also been used in [[American football|pro football]] by football stat website and publisher [[Football Outsiders]], where it is known as '''Pythagorean projection'''. The 2011 edition of ''Football Outsiders Almanac''<ref>''Football Outsiders Almanac 2011'' (ISBN 1466246133), p.xviii</ref> states, "From 1988 through 2004, 11 of 16 [[Super Bowl]]s were won by the team that led the [[National Football League|NFL]] in Pythagorean wins, while only seven were won by the team with the most actual victories. Super Bowl champions that led the league in Pythagorean wins but not actual wins include the [[2004 New England Patriots season|2004 Patriots]], [[2000 Baltimore Ravens season|2000 Ravens]], [[1999 St. Louis Rams season|1999 Rams]] and [[1997 Denver Broncos season|1997 Broncos]]."
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| Although ''Football Outsiders Almanac'' acknowledges that the formula had been less-successful in picking Super Bowl participants from 2005–2008, it reasserted itself in 2009 and 2010. Furthermore, "[t]he Pythagorean projection is also still a valuable predictor of year-to-year improvement. Teams that win a minimum of one full game more than their Pythagorean projection tend to regress the following year; teams that win a minimum of one full game less than their Pythagoerean projection tend to improve the following year, particularly if they were at or above .500 despite their underachieving. For example, the [[2008 New Orleans Saints season|2008 New Orleans Saints]] went 8-8 despite 9.5 Pythagorean wins, hinting at the improvement that came with the [[2009 New Orleans Saints season|next year's]] [[2009 NFL season|championship season]]."
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| ==See also==
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| * [[Baseball statistics]]
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| * [[Sabermetrics]]
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| * [[Football Outsiders]]
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| ==Notes==
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| {{reflist|2}}
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| ==External links==
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| *{{cite journal|author1=Miller|journal=Chance Magazine 20 (2007), no. 1, 40--48|pages=9698|title=A Derivation of the Pythagorean Won-Loss Formula in Baseball|year=2005|arxiv=math.ST/0509698|bibcode=2005math......9698M}}
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| *[http://www.sportility.net/cgi-bin/pointspread.cgi?league=mlb Current Major League Baseball Pythagorean expectation.]
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| *[http://www.pro-football-reference.com/blog/?p=337 Adjusting football’s Pythagorean Theorem]
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| [[Category:Bill James]]
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| [[Category:Baseball terminology]]
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| [[Category:Baseball statistics]]
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| [[Category:Sports records and statistics]]
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