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Pythagorean Expectation
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. Based on a 162 game season, the Yankees should have won 101.07 games. The 2002 Yankees actually went 10358.[2] In efforts to fix this error, statisticians have performed numerous searches to find the ideal exponent. If using a single number exponent, 1.83 is the most accurate, and the one used by baseballreference.com , the premier website for baseball statistics across teams and time.[3] The updated formula therefore reads as follows: The most widely known is the Pythagenport formula[4] developed by Clay Davenport of Baseball Prospectus: 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 rootmeansquare error as opposed to a 4.126 rootmeansquare error for an exponent of 2.[5] Less well known but equally (if not more) effective is the Pythagenpat formula, developed by David Smyth.[6] 

"Secondorder" and "thirdorder" winsIn their Adjusted Standings Report,[8] 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.
Firstorder 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, sabermetricians can also calculate a team's expected runs scored and allowed via a runs createdtype 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. By plugging these expected runs scored and allowed into the pythagorean formula, one can generate secondorder 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. Thirdorder wins are secondorder wins that have been adjusted for strength of schedule (the quality of the opponent's pitching and hitting). Second and thirdorder winning percentage has been shown to predict future actual team winning percentage better than both actual winning percentage and firstorder winning percentage. 

Theoretical explanationInitially 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√&# 960;) where σ was the standard deviation of runs scored by all teams divided by the average number of runs scored.[9] In 2006, Professor Steven J. Miller provided a statistical derivation of the formula[10] 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.[11]


Use in basketballAmerican 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 wonlost percentages:
Daryl's "Modified Pythagorean Theorem" was first published in STATS Basketball Scoreboard, 199394.[12] Noted basketball analyst Dean Oliver also applied James' Pythagorean theory to professional basketball. The result was similar. Another noted basketball statistician, John Hollinger, uses a similar Pythagorean formula except with 16.5 as the exponent. 

Use in pro footballThe formula has also been used in pro football by football stat website and publisher Football Outsiders, where it is known as Pythagorean projection. The 2011 edition of Football Outsiders Almanac[13] states, "From 1988 through 2004, 11 of 16 Super Bowls were won by the team that led the 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 Patriots, 2000 Ravens, 1999 Rams and 1997 Broncos."
Although Football Outsiders Almanac acknowledges that the formula had been lesssuccessful 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 yeartoyear 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 went 88 despite 9.5 Pythagorean wins, hinting at the improvement that came with the next year's championship season." 

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As one of the darkest and most disturbing of the Greek dramatists, Euripides questions authority, and, in his plays, he reveals a fascination with the oppressed, including women, barbarians, and slaves. His complex representations of perverse, violent, and monstrous women demonstrate his interest in the role of women in society. He further questions hollow or hypocritical ideals. While Aeschylus depicts a vision of history and teleology and Sophocles portrays heroes, Euripides creates real men with alltoo human weaknesses. His is a voice of conscience, unafraid to reveal the world underneath Athens’ veneer of cultural and social advancement. The views expressed in Euripides’ tragedies seem almost prescient. After years of warfare (the Second Peloponnesian War began in 431 B.C.E.) and internal political strife, Athens fell to Sparta in 404 B.C.E., two years after the death of Euripides.

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Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law is a useful approximation are known as linearelastic or "Hookean" materials. Hookean materials is a necessarily broad term that may include the work of muscular layers of the heart. Hooke's law in simple terms says that stress is directly proportional to strain. Mathematically, Hooke's law states that
F=kx where x is the displacement of the spring's end from its equilibrium position (a distance, in SI units: metres); F is the restoring force exerted by the spring on that end (in SI units: N or kg·m/s2); and k is a constant called the rate or spring constant (in SI units: N/m or kg/s2). When this holds, the behavior is said to be linear. If shown on a graph, the line should show a direct variation. There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the displacement (for example, when a spring is stretched to the left, it pulls back to the right). 
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