From Wiki Gonzalez
In its simplest form, the Pythagorean formula states that if a team scores "RS" runs and allows "RA" runs, it has the following "Pythagorean" winning percentage:
PWP% = RS^2 / (RS^2 + RA^2)
Multiply this value by 162 (or whatever number of games have been played), and you have an "expected" win total that can be compared to a team's actual win total.
James's hypothesis was that team RS and RA were all that was needed to derive a team's winning percentage. He then tried to come up with a formula that came close to matching actual team performance. In other words, he developed an initial formula for PWP%, regressed his calculations of PWP% against multiple seasons worth of teams' actual WP% in order to assess how close his formula came to predicting actual win totals, and adjusted his PWP% formula to try again.
In James's empirical testing, described in one of the Baseball Abstracts, 2 was the exponent that seemed to work best, hence the "Pythagorean" tag in honor of the resemblance to the geometrical formula. In typical modern baseball, the formula works most accurately with an exponent of around 1.85, but 2 is still quite good.
A team that beats its Pythagorean winning percentage is most likely lucky (i.e., their actual team quality is not as impressive as it appears measured purely by winning percentage). Analogously, a team with unexpectedly low win totals may be considered unlucky by the Pythagorean formula (i.e., their Pythagorean winning percentage may be superior to their actual winning percentage).
Some things that can influence disparity between Pythagorean win totals and actual win totals are performance in one-run games, performance in blowouts, ballpark effects, etc. In most cases, teams' Pythagorean win totals closely match actual win totals; i.e., the factors mostly cancel each other out.
Another conclusion of James's research is that a team's win total in a following season is more accurately predicted by its Pythagorean win percentage than by its actual win percentage. Large disparities between Pythagorean wins and actual wins are generally not carried over from season to season.
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 Appearances on BBTF and Elsewhere
Rob Neyer used to put updated Pythagorean standings on his ESPN.com index page; i.e., standings based on team's Pythagorean records. He used the simple formula, the one with an exponent of 2.
Baseball Prospectus has developed a custom spin on the James formula called Pythagenport, where the exponent varies depending on the run environment (Runs Per Game). Pythagenport appears on the BP Adjusted Standings page.
As a final note, Primates generically refer to these calculations as "pythags" and find them useful tools to (a) forecast future team performance, and (b) have a little fun on the side by bashing "lucky" teams.