Pythagorean Expectation Calculator
Pythagorean Expectation Calculator
Expected wins from runs scored and runs allowed — at Bill James’s original exponent of 2, at the 1.83 that fits baseball better, and at the Pythagenpat exponent fitted to the run environment itself. The exponent is an empirical fit, not a law, and the three disagree.
Expected wins at three exponents
Runs in, wins out800 runs scored, 700 allowed, in a 162-game season
One formula, three published exponents, no derivation
Pythagenpat: x = ((RS + RA) ÷ games)^0.287
- x = 2, Bill James
- the original. Baseball-Reference’s Bullpen calls the squared version “The more commonly used, and simpler version”. The resemblance to Pythagoras is in the shape of the expression and nothing else
- x = 1.83, and a source disagreement
- Baseball-Reference’s own Bullpen says “More accurate versions of the formula use 1.81 or 1.83 as the exponent” and marks that sentence “[citation needed]”, without saying which exponent that site uses; Wikipedia says 1.83 “is the one used by baseball-reference.com”. Both are printed here rather than reconciled, because the stronger source is the one declining to make the claim
- Pythagenpat, credited to David Smyth
- makes the exponent a function of scoring: runs per game by both teams, to the power 0.287. On the default line that is 1.894. The attribution and the 0.287 come from Wikipedia, the weakest source here
- the exponent is FITTED, and other sports prove it
- the same expression is used elsewhere with wildly different exponents — roughly 13.9 to 16.5 for basketball, 2.37 for American football, slightly above 2 for ice hockey. A quantity varying by a factor of seven between sports is a curve-fitting parameter, not a property of competition
- accuracy, and the famous exceptions
- Baseball-Reference’s Bullpen puts the average gap between actual and expected records at “a bit more than 3 games at the end of a season”, with the 2005 and 2007 Arizona Diamondbacks beating theirs by 11
Worked example
800 runs scored, 700 allowed, in a 162-game season
At the 1.83 exponent: 800^1.83 ÷ (800^1.83 + 700^1.83) = .561, and .561 × 162 = 90.8 expected wins
At Bill James's exponent of 2: 640,000 ÷ (640,000 + 490,000) = .566, which is 91.8 wins — 0.90 wins more
Pythagenpat: (800 + 700) ÷ 162 = 9.26 runs a game, and 9.26^0.287 = 1.894, giving 91.2 wins
The direction matters. With more runs scored than allowed, a LARGER exponent gives MORE expected wins. Reverse the two figures — 700 scored, 800 allowed — and it inverts: the 1.83 exponent gives .439 and the exponent of 2 gives .434, so the larger exponent now gives FEWER wins
Set the runs equal, 750 each, and all three exponents give exactly .500. That is the only place they agree, and it is why testing this formula on an even run differential proves nothing at all
Double both run totals, to 1,600 and 1,400, and the two fixed exponents do not move: .561 and .566 again, because the expression is scale-free in the runs. Pythagenpat DOES move, to an exponent of 2.311 and a winning percentage of .577, because its exponent reads the run environment and you have just doubled it
The same season at three published exponents
| Exponent | Who fitted it | Expected win% | Expected wins in 162 |
|---|---|---|---|
| 1.83 | attributed to a better single-number fit; see the disagreement below | .561 | 90.8 |
| 1.894 | Pythagenpat, David Smyth — computed from this season’s own scoring | .563 | 91.2 |
| 2 | Bill James, the original | .566 | 91.8 |
The same idea in other sports, with its exponent refitted
| Sport | Exponent | Fitted by |
|---|---|---|
| Baseball | 2 | Bill James, the original |
| Baseball | 1.83 | a better single-number fit, attribution disputed |
| Basketball | 13.91 | Daryl Morey, at STATS, Inc. |
| Basketball | 16.5 | John Hollinger |
| American football | 2.37 | Football Outsiders |
| Ice hockey | slightly above 2 | Dayaratna and Miller, 2013 |
The exponent is fitted, and three published fits give three answers
Bill James noticed that a baseball team’s winning percentage tracks its runs scored and runs allowed closely enough to be worth a formula, and wrote one: runs scored squared, over runs scored squared plus runs allowed squared. The name is a joke about the shape of the expression — there is no triangle and no theorem.
The exponent of 2 is not derived from anything. It is a round number that happens to work, and when people looked for a better one they found it: a single exponent nearer 1.83 fits the historical record more closely, and the Pythagenpat variant does better still by making the exponent a function of the run environment. This page computes all three, because printing one would hide the fact that the choice is a choice.
There is a genuine source disagreement about the 1.83, printed here rather than tidied away. Baseball-Reference’s own Bullpen says “More accurate versions of the formula use 1.81 or 1.83 as the exponent” and marks that sentence “[citation needed]”; it never says which exponent Baseball-Reference itself uses. Wikipedia states that 1.83 “is the one used by baseball-reference.com”. Those cannot both be the last word, and the attribution of Pythagenpat to David Smyth and of the 0.287 power rests on Wikipedia alone. The arithmetic here is this page’s own and is exact; whose exponent is whose is not settled by anything read for it.
The strongest argument that the exponent is an empirical fit is not statistical — it is that the same expression is used in other sports with completely different exponents, around 13.9 to 16.5 for basketball, about 2.37 for American football and slightly above 2 for ice hockey. A quantity that varies by a factor of seven between sports, and by a fifth between two people fitting the same sport, is a curve-fitting parameter. Basketball’s version belongs on a basketball page and is not computed here.
The number of games in a season is a regulation and this page takes it as an input rather than assuming 162. Major League Baseball has played 162 games since 1961 in the American League and 1962 in the National League, both on expansion, having played 154 before that; it played 60 in 2020. The minor leagues, Nippon Professional Baseball, the KBO League, the NCAA and US high school baseball all schedule different numbers, and a calculator that bakes in 162 is wrong for every one of them. It matters twice here: once to turn a winning percentage into a win total, and again inside the Pythagenpat exponent, which divides total runs by games. That second dependence has a consequence worth stating — Pythagenpat is NOT scale-free in the runs. Double both run totals and the two fixed exponents give exactly the same winning percentages, while Pythagenpat’s exponent climbs from 1.894 to 2.311 and its answer moves from .563 to .577.
Nothing here is a price or a probability of anything you can stake on. This page does not convert odds, imply a market probability or evaluate a wager, and no page on this site does. An expected win total from runs scored and runs allowed is a description of a season that has already been played, not a forecast and not a quotation. And what the gap between a team’s actual and expected record is usually taken to mean — luck, a bullpen, a run distribution concentrated in blowouts — is not something this arithmetic can distinguish between. The football version of the question is the points per game and projected finish calculator, the rating-based way of turning team strength into a win probability is the Elo rating change calculator, and how far a lead actually is in the standings is the magic number and games behind calculator.
Frequently asked questions
What is the Pythagorean expectation formula in baseball?
Runs scored raised to an exponent, divided by runs scored plus runs allowed each raised to the same exponent, multiplied by the number of games. Bill James’s original used an exponent of 2; 800 runs scored and 700 allowed in 162 games gives .566 and 91.8 expected wins at that exponent.
Why do different sites give different Pythagorean win totals?
Because they use different exponents, and the exponent is a fitted parameter rather than a derived one. On 800 runs scored and 700 allowed the answers are 90.8 wins at 1.83, 91.2 at the Pythagenpat exponent of 1.894 and 91.8 at Bill James’s 2 — a spread of 0.90 wins on one season.
Which exponent is correct?
None of them is correct in the sense of being derived; they are fits. A single exponent near 1.83 fits baseball better than 2, and Pythagenpat fits better still by letting the exponent follow the run environment. Note that Baseball-Reference’s own Bullpen page marks the 1.83 claim “[citation needed]” and does not say which exponent that site uses, while Wikipedia says it uses 1.83 — this page prints both positions.
Does the same formula work in other sports?
The shape does; the exponent does not carry over. Basketball needs roughly 13.91 as fitted by Daryl Morey or 16.5 as used by John Hollinger, American football 2.37 at Football Outsiders, and ice hockey slightly above 2 per Dayaratna and Miller. That range is the clearest evidence there is that the exponent is an empirical fit.
What does it mean if a team beat its expected record?
Less than it is usually taken to mean. The average gap is about three games, and Baseball-Reference’s Bullpen notes the 2005 and 2007 Arizona Diamondbacks beating theirs by eleven. The arithmetic cannot separate a one-run-game bullpen from a run distribution concentrated in blowouts from chance, and it is not a forecast of anything.
Related calculators
References
- Baseball-Reference Bullpen. Pythagorean Theorem of Baseball — “is a creation of Bill James”; “The more commonly used, and simpler version uses an exponent of 2 in the formula”; “More accurate versions of the formula use 1.81 or 1.83 as the exponent”, which the page itself marks “[citation needed]”; “The average difference between the actual and the Expected W-L is a bit more than 3 games at the end of a season”, the 2005 and 2007 Arizona Diamondbacks beating theirs by 11. It does NOT say which exponent Baseball-Reference uses (read 12 Oct 2026).
- Wikipedia. Pythagorean expectation — cited for attributions nothing stronger supplied: 1.83 “is the one used by baseball-reference.com”; Pythagenpat, credited to David Smyth, exponent ((RS + RA) / games)^0.287; and the other sports’ fits — Morey 13.91 and Hollinger 16.5 for basketball, Football Outsiders 2.37 for the NFL, Dayaratna and Miller (2013) slightly above 2 for ice hockey. The weakest source cited here, and it CONTRADICTS Baseball-Reference’s Bullpen on whether that site uses 1.83 (read 12 Oct 2026).
- Wikipedia. 1961 Major League Baseball season — “Due to expansion of the American League, the AL schedule consisted of 162 games for all 10 teams”, the National League keeping 154 that year and adopting 162 in 1962 after its own expansion (read 12 Oct 2026).