Pythagorean Wins: Expected Standings from Run Differential

A team's win–loss record is the number everyone quotes, and it is one of the noisiest summaries in sports. Win a fistful of one-run games and your record races ahead of how you actually played; lose those coin-flips and it hides a good team. Pythagorean expectation cuts through the noise by asking a different question: given how many runs a team scored and allowed, how many games should it have won? Compute that for all 30 teams from the bundled 2023 standings, plot expected wins against real ones, and the luckiest and unluckiest teams in baseball fall straight out of the scatter.

This builds on Pandas for Sports Data: you should be comfortable loading a CSV and making a new column. The data is the same real sample_standings.csv (2023 MLB regular season, MLB Stats API, retrieved June 2026), so this runs offline.

1. Load the standings

   The bundled file already has everything we need: games played (G), wins (W), runs scored (RS), and runs allowed (RA).

   import pandas as pd
   
   df = pd.read_csv("sample_standings.csv")
   print(df[["Team", "G", "W", "RS", "RA"]].head().to_string())

   Run differential — runs scored minus runs allowed — is the raw material here. The whole idea of Pythagorean expectation is that the ratio of runs scored to runs allowed maps cleanly onto winning percentage.

2. The Pythagorean formula

   Bill James named it after the Pythagorean theorem because his first version squared each term. The expected winning percentage is:

   EXP = 1.83  # the refined exponent that best fits modern baseball
   
   df["ExpWinPct"] = df["RS"]**EXP / (df["RS"]**EXP + df["RA"]**EXP)

   The original exponent was a clean 2; analysts later found that 1.83 fits real baseball outcomes a little better, so that's what we use. The structure is intuitive: if a team scores exactly as many runs as it allows, the formula returns .500. The more RS outweighs RA, the closer the expected percentage climbs toward 1.000.

3. Turn it into expected wins, and measure luck

   Multiply the expected winning percentage by games played to get expected wins, then subtract from real wins. That gap — we'll call it Luck — is how many games a team won beyond what its run differential earned.

   df["ExpW"] = (df["ExpWinPct"] * df["G"]).round(1)
   df["Luck"] = (df["W"] - df["ExpW"]).round(1)   # +ve = won more than expected
   
   ranked = df.sort_values("Luck", ascending=False)
   print(ranked[["Team", "W", "ExpW", "Luck", "RS", "RA"]].head().to_string())

   Luckiest and unluckiest teams, 2023

   Luckiest teams in 2023 (most wins above Pythagorean expectation):
          Team    W  ExpW  Luck   RS   RA
   11  Marlins   84  74.9   9.1  666  723
   1   Orioles  101  93.8   7.2  807  678
   18   Tigers   78  72.7   5.3  661  740
   21  Pirates   76  71.2   4.8  692  790
   12  D-backs   84  79.5   4.5  746  761
   
   Unluckiest teams (most wins below expectation):
          Team   W  ExpW  Luck   RS   RA
   14   Padres  82  92.0 -10.0  752  648
   28   Royals  56  63.5  -7.5  676  859
   13     Cubs  83  90.2  -7.2  819  723
   10    Twins  87  93.2  -6.2  778  659
   7   Rangers  90  96.2  -6.2  881  716

   The 2023 Marlins are the poster child: they were outscored on the season (666 runs for, 723 against) yet went 84–78, winning about nine more games than their run differential expected. At the other extreme, the Padres scored 752 and allowed just 648 — the profile of a ~92-win team — but finished 82–80, ten wins short. A useful reminder that Pythagorean luck is about the regular season: the Rangers also underperformed their expectation here, then got hot in October and won the World Series.

4. Plot expected vs. actual wins

   A scatter with a 45-degree reference line makes luck visible at a glance: teams above the line won more than expected, teams below it less.

   import matplotlib.pyplot as plt
   
   fig, ax = plt.subplots(figsize=(8, 6))
   ax.scatter(df["ExpW"], df["W"], color="#B23A3A", s=45, zorder=3)
   lo, hi = 45, 110
   ax.plot([lo, hi], [lo, hi], color="#6C7079", ls="--", label="expected = actual")
   ax.set_aspect("equal")
   ax.set_xlabel("expected wins (from runs scored and allowed)")
   ax.set_ylabel("actual wins")
   fig.savefig("expected_vs_actual.png", dpi=144, bbox_inches="tight")

   

   Setting ax.set_aspect("equal") matters here: because both axes are in the same unit (wins), an equal aspect ratio keeps the 45-degree line at a true 45 degrees, so "distance from the line" reads honestly as games of luck. Most teams hug the line — over a 162-game season, luck mostly washes out — which is exactly why the few that stray are worth a closer look.

Troubleshooting