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 runs they scored and allowed. The term is derived from the formula's resemblance to Pythagoras' formula to compute the length of the hypotenuse of a triangle from the lengths of its other two sides. This article or section does not cite its references or sources. ... A view of the playing field at Busch Stadium II St. ... Pythagoras of Samos (Greek: ; circa 582 BC – circa 507 BC) was an Ionian (Greek) mathematician, astronomer, scientist and philosopher,[1] founder of the mathematical, mystic, religious, and scientific society called Pythagoreans. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

The basic formula is:

where Win% is the winning percentage generated by the formula. You can then multiply this by the number of games played by a team (today, a season in the Major Leagues is 162 games) to compute how many wins one would expect them to win based on their runs scored and runs allowed. This article or section does not adequately cite its references or sources. ...

Empirical origin

Empirically, this formula correlates fairly well with how baseball teams actually perform, although an exponent of 1.81 is slightly more accurate. This correlation is one justification for using runs as a unit of measurement for player performance. Efforts have been made to find the ideal exponent for the formula, the most widely known being the Pythagenport formula[1] developed by Clay Davenport of Baseball Prospectus (1.5 log((r + ra)/g) + 0.45) and the less well known but equally effective Pythagenpat formula ((r + ra)/g)^0.287), developed by David Smyth.[2] Indeed, Davenport has endorsed the Smyth/Patriot or pythagenpat formula as "a better fit to the data. In that, X = ((rs + ra)/g)^0.285, 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."[3] In baseball, a run is scored when a player advances safely around all three bases and returns safely to home plate. ... Clay Davenport is a baseball sabermetrician and a writer for the Baseball Prospectus. ... Baseball Prospectus, sometimes abbreviated as BP, is a think-tank focusing on the statistical analysis of the sport of baseball, which is also known as sabermetrics. ... Pythagenpat is a formula created by David Smyth and Patriot which attempts to find the optimal exponent to use in the for the Pythagorean expectation formula. ...

However, these formulas are only useful when dealing with extreme situations in which the average amount of runs scored per game is either very high or very low. For most situations, simply squaring each variable yields accurate results.

It is important to keep in mind that Pythagorean estimation is an empirical observation that correlates with winning percentage. It was not theoretically derived; there was no known reason why a team's winning percentage correlates to this formula, except that it does. However, Steven J. Miller provided a statistical derivation of the formula under some assumptions about how runs are distributed.

That said, there are statistical deviations between actual winning percentage and expected winning percentage, which include a quality bullpen and luck. Teams that win a lot of games tend to be underrepresented by the formula (meaning they "should" have won less), and teams that lose a lot of games tend to be overrepresented (they "should" have won more).

Use in basketball

When noted basketball analyst Dean Oliver applied James' pythagorean theory to his own sport, the result was similar, except for the exponents: Dean Oliver is a prominent contributor to the statistical evaluation of basketball, or APBRmetrics. ...

Another noted basketball statistician, John Hollinger, uses a similar pythagorean formula except with 16.5 as the exponent. John Hollinger is an influential figure in the field of quantitative analysis in basketball. ...

See also

• Baseball statistics
• Sabermetrics

Statistics are very important to baseball, perhaps more than any other sport. ... Sabermetrics is the analysis of baseball through objective evidence, especially baseball statistics. ...

External links

• A Derivation of the Pythagorean Won-Loss Formula in Baseball, by Steven Miller PDF