Standardize Stats with Z-Scores: Compare Columns on One Scale

How do you compare a team that scores a lot of runs with one that prevents a lot of runs? Runs scored and runs allowed sit on the same numeric scale here, but plenty of stats don't — on-base percentage lives near 0.3, strikeouts in the hundreds. The fix is the z-score: rescale any column to "how many standard deviations from the mean", and suddenly +1.0 means the same thing everywhere. We'll standardize three columns of the 2023 standings and read the result straight off a diverging bar chart.

This builds on Summary Statistics and Distributions, where mean() and std() first showed up. The data is the bundled sample_standings.csv (real 2023 MLB standings), so it runs offline.

1. The formula in one function

   A z-score is just (value − mean) ÷ standard deviation. Wrap it in a tiny function and it applies to any column. We standardize runs scored, runs allowed, and run differential in a single loop.

   import pandas as pd
   
   df = pd.read_csv("sample_standings.csv")
   
   def zscore(s):
       return (s - s.mean()) / s.std()   # pandas std() uses ddof=1 (sample std)
   
   for col in ["RS", "RA", "RunDiff"]:
       df[col + "_z"] = zscore(df[col])
   
   print(df.sort_values("RunDiff_z", ascending=False)
           [["Team", "RS", "RS_z", "RA", "RA_z", "RunDiff_z"]].round(2).head(8).to_string())

   Standardized leaders (and the tell-tale mean/std)

          Team   RS  RS_z   RA  RA_z  RunDiff_z
   0    Braves  947  2.40  716 -0.39       1.65
   2   Dodgers  906  1.91  699 -0.59       1.48
   3      Rays  860  1.35  665 -1.01       1.39
   7   Rangers  881  1.61  716 -0.39       1.18
   5    Astros  827  0.96  698 -0.61       0.92
   1   Orioles  807  0.72  678 -0.85       0.92
   10    Twins  778  0.37  659 -1.08       0.85
   14   Padres  752  0.05  648 -1.21       0.74
   
   mean RunDiff_z = 0.000  (≈0 by construction)
   std  RunDiff_z = 1.000  (=1 by construction)

   Read the Braves' row: RS_z of about +2.4 means their offense was roughly two and a half standard deviations above league average, while RA_z of −0.4 means a slightly better-than-average defense. The two columns are now directly comparable even though raw runs scored and runs allowed answer opposite questions.

2. Why the mean is 0 and the std is 1

   Standardizing always re-centers a column to mean 0 and spread 1 — that's the whole point, and it's a free sanity check. If your "z-score" column doesn't come out that way, you standardized the wrong thing.

   print(df["RunDiff_z"].mean())   # ≈ 0
   print(df["RunDiff_z"].std())    # = 1

   This is also why z-scores are the backbone of composite ratings: because every input is on the same 0-centered, unit-1 scale, you can average a hitter's standardized power, discipline, and speed into one number without one stat drowning out the others.

3. A diverging bar chart reads off zero

   Once a column is centered on zero, a diverging bar chart is the natural picture: bars to the right are above average, bars to the left below, and length is the magnitude in sigma. Color the two directions differently and the league splits in half at a glance.

   import matplotlib.pyplot as plt
   
   s = df.sort_values("RunDiff_z")
   colors = ["#2E7D4F" if v >= 0 else "#B23A3A" for v in s["RunDiff_z"]]
   
   fig, ax = plt.subplots(figsize=(8, 9))
   ax.barh(s["Abbr"], s["RunDiff_z"], color=colors)
   ax.axvline(0, color="#20242B", lw=1)
   ax.set_xlabel("run differential (z-score)")
   fig.savefig("zscore_bars.png", dpi=144, bbox_inches="tight")

   

   The shape is symmetric around zero because the average team, by definition, sits at the line. What the chart adds over raw run differential is interpretability: a bar at +2 isn't just "good", it's "top-of-the-distribution rare", and the same +2 would mean the same thing if the axis were on-base percentage or ERA.

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