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Kalkulator Peratusan Kemenangan Pythagoras

Modelkan jangkaan kemenangan pasukan daripada jurang jaringan mata dan kenal pasti faktor nasib dalam sukan.

Bill James Sabermetric Expectation Engine

Kalkulator Peratusan Kemenangan Pythagoras

Modelkan jangkaan kemenangan pasukan daripada jurang jaringan mata dan kenal pasti faktor nasib dalam sukan.

Pratetap:
PTS
OPP
Bill James traditional = 2.0 | Modern MLB = 1.83
Pythagorean Expectation +6.0 Luck Residual
.667

Expected Record: 108.0 W - 54.0 L (66.7%)

Expected Wins vs Losses +309 Run Differential
Pyth Wins: 108.0 Pyth Losses: 54.0
Points Ratio (PS/PA) 1.47 : 1
Luck Variance +6.0 Wins

In modern sports analytics and sabermetrics, a team’s win-loss record does not always tell the whole story. Bill James famously revolutionized baseball analysis by inventing Pythagorean Expectation—a formula that predicts a team’s true winning percentage based on the relationship between points or runs scored and points or runs allowed. Teams that significantly outscore their opponents but lose close games are diagnosed as 'unlucky,' while teams that overperform their point differential are primed for statistical regression.

Who Relies on Pythagorean Expectation?

  • Major League Baseball Front Offices: Evaluate whether midseason contenders are sustainable division winners or statistical flukes riding one-run game luck.
  • NBA Analytics Departments: Use Daryl Morey's modified Pythagorean exponent (k=13.91) to evaluate offensive and defensive net efficiency.
  • NFL Betting Syndicates: Identify overvalued and undervalued teams based on point differential versus actual record.
  • Soccer Analysts & Bettors: Model goal expectation curves in low-scoring leagues like the Premier League, La Liga, and Champions League.

Why Run Differential Predicts the Future Better Than Wins

  • Eliminates Random One-Score Noise: Performance in one-run or one-possession games behaves like a coin flip over small sample sizes.
  • True Talent Grounding: Outscoring opponents by 150 runs over a full season requires systemic offensive and pitching superiority.
  • Regression to the Mean: Teams with large Pythagorean luck residuals almost always regress toward their expected win percentage the following season.

The Pythagorean Expectation Formula & Exponent Variations

Bill James modeled team winning percentage as a power formula resembling the geometric Pythagorean theorem:

Bill James Pythagorean Formula
Pyth Win% = (Points Scored)^k ÷ [(Points Scored)^k + (Points Allowed)^k]

Where k is the sport-specific power exponent. Baseball traditionally used k = 2.0 (modern sabermetrics uses 1.83), while basketball uses 13.91.

Pythagorean Residual (Luck Factor)
Luck Residual = Actual Wins − Expected Pythagorean Wins

A positive residual indicates over-performance (good luck in close games); a negative residual identifies an under-performing club experiencing bad luck.

Sabermetric Expectation Curve

Interactive Pythagorean Scoring Ratio & Curve Inspector

Toggle sport exponents and adjust scoring dominance to inspect how non-linear mathematical powers convert point ratios into expected victory rates.

Scoring Ratio (Points Scored ÷ Points Allowed) 1.23 : 1
0.70 (Heavy Underdog) 1.00 (Even Parity) 1.25 (Contender) 1.50 (Historic Juggernaut)
Exponent Transformation
W% = Ratiok ÷ (Ratiok + 1)

Exponent k = 1.83 reflects baseball run-environment variance across 162 games.

Expected Win Rate
.602
97.5 Expected Wins (162 Games)
True Talent Diagnosis
Championship Contender

Teams outscoring opponents at a 1.23 ratio historically capture division titles and high postseason seeds.

Step-by-Step Calculation Breakdown & Historic Sports Examples

Follow these 4 mathematical steps to compute any team's Pythagorean expectation:

  1. Determine Exponent (k): Select 1.83 for MLB, 13.91 for NBA, 2.37 for NFL, or 1.25 for soccer.
  2. Raise Points to the Power k: Compute (Points Scored)^k and (Points Allowed)^k.
  3. Divide Scored Power by Total Power: Calculate PS^k ÷ (PS^k + PA^k) to find expected win percentage.
  4. Multiply by Total Contests: Multiply expected win percentage by total games to calculate expected wins.
Example 1: 1998 New York Yankees (Dominant MLB Record)
965 Runs Scored, 656 Runs Allowed (162 Games). Exponent k = 1.83.
965^1.83 ÷ (965^1.83 + 656^1.83) = .667 Expected Win%. Expected wins = 0.667 × 162 = 108.0 W.
Expected: 108-54 (.667). Actual: 114-48 (+6.0 Luck Residual)
Example 2: 2015-16 Golden State Warriors (73-Win NBA Record)
9,421 Points Scored, 8,539 Points Allowed (82 Games). Exponent k = 13.91.
9421^13.91 ÷ (9421^13.91 + 8539^13.91) = .818 Expected Win%. Expected wins = 0.818 × 82 = 67.1 W.
Expected: 67.1 Wins (.818). Actual: 73 Wins (+5.9 Luck Residual in clutch games)
Example 3: 2007 New England Patriots (16-0 NFL Regular Season)
589 Points For, 274 Points Against (16 Games). Exponent k = 2.37.
589^2.37 ÷ (589^2.37 + 274^2.37) = .888 Expected Win%. Expected wins = 0.888 × 16 = 14.2 W.
Expected: 14.2 Wins (.888). Actual: 16 Wins (+1.8 Luck Residual)
Example 4: Rebuilding Baseball Team (600 RS, 850 RA)
600 Runs Scored, 850 Runs Allowed (162 Games). Exponent k = 1.83.
600^1.83 ÷ (600^1.83 + 850^1.83) = .346 Expected Win%. Expected wins = 0.346 × 162 = 56.1 W.
Expected: 56.1 Wins (.346). Projected 106-loss season

Pythagorean Exponents Across Global Sports Leagues

Optimal empirical power exponents (k) derived by sports statisticians and sabermetricians:

Sport / LeagueStandard Exponent (k)Pioneering SabermetricianMathematical Characteristics
Baseball (MLB Modern) k = 1.83 Bill James & Pythagenpat Moderate scoring variance; 1 run ≈ 10 runs for 1 win
Baseball (Classic) k = 2.00 Bill James (Original 1980s) Traditional geometric formula; slightly over-estimates extremes
Basketball (NBA) k = 13.91 Daryl Morey (Houston Rockets) High-scoring environment creates steep logistic win probability
Football (NFL) k = 2.37 Aaron Schatz (Football Outsiders) Touchdown (6) and field goal (3) scoring increments
Ice Hockey (NHL) k = 2.15 Puck Prospectus / Sabermetrics Overtime points system compresses decimal variance
Association Football (Soccer) k = 1.25 Howard Hamilton / Opta Low goal volume and frequent 0-0/1-1 draws compress exponent

Kalkulator Peratusan Kemenangan Pythagoras FAQ

Knowledge Base

Search analytical answers or browse questions by topic.

What is the Pythagorean theorem in sports?

Created by baseball statistician Bill James, Pythagorean Expectation is a formula that calculates a sports team's expected winning percentage based on runs (or points) scored versus allowed: W% = Points_Scored^k ÷ (Points_Scored^k + Points_Allowed^k).

Why is it called 'Pythagorean' if it doesn't involve right triangles?

Bill James named the formula Pythagorean because his original formulation used an exponent of 2: RS² ÷ (RS² + RA²), bearing a visual resemblance to the classic Pythagorean theorem a² + b² = c².

What does it mean if a team has a positive luck residual?

A positive luck residual means the team won more games than their point differential suggests they should have. This typically happens when a team possesses an exceptional record in close, one-run, or one-possession games, which statistically tends to regress to .500 the following year.

Why is the basketball exponent (13.91) so much higher than baseball (1.83)?

Because basketball teams score between 80 and 130 points per game, a team outscoring opponents by just 5% plays dozens of possessions that compound their advantage. The high exponent reflects the non-linear relationship between point margin and win probability in high-scoring sports.

Can Pythagorean Expectation predict postseason success?

Yes. Numerous sabermetric studies have demonstrated that run differential and Pythagorean win percentage are better predictors of playoff success and next-season win totals than actual regular-season win-loss records.

How does the Pythagorean formula handle soccer draws?

Because association football has draws and low goal counts, statisticians use a lower exponent (around 1.25 to 1.30) or model Pythagorean points (3-1-0 points system) rather than binary wins.