Box Plots: Comparing Distributions Across Groups

An average is a single number standing in for a whole group, and it hides everything interesting: the spread, the shape, the outliers. When you want to compare groups — American League versus National, one division versus another, this era versus that one — a box plot shows all of that at once. Can the AL and NL really be told apart by run differential, or do two averages just look different by chance? Plot the run differential of both leagues side by side and the five numbers a box plot draws will answer that better than any mean ever could.

This builds on Summary Statistics and Distributions, where we met describe() and the histogram. The data is the bundled sample_standings.csv (real 2023 MLB standings), so it runs offline.

1. Start with the averages — then distrust them

   A quick groupby gives the mean of each league. It's a fine first look, and also a trap.

   import pandas as pd
   
   df = pd.read_csv("sample_standings.csv")
   print(df.groupby("League")[["W", "RS", "RA", "RunDiff"]].mean().round(1).to_string())

   League averages

   Mean by league:
              W     RS     RA  RunDiff
   League                             
   AL      79.9  737.7  740.7     -2.9
   NL      82.1  757.7  754.8      2.9

   The averages are nearly mirror images — the AL at about −3 run differential, the NL at about +3 — which makes sense, since one league's interleague runs are the other's to concede. But "the average AL team was roughly break-even" tells us nothing about whether the league was tightly bunched or split between juggernauts and cellar-dwellers. For that, we need the distribution.

2. What a box plot shows

   A box plot draws five things for each group: the median (the line in the box), the quartiles (the box spans the middle 50% of teams, from the 25th to the 75th percentile), the whiskers (the reach of the bulk of the data), and any outliers (points beyond the whiskers, drawn individually). One glance gives you center, spread, and extremes — the things an average leaves out.

   import matplotlib.pyplot as plt
   
   leagues = ["AL", "NL"]
   data = [df.loc[df["League"] == lg, "RunDiff"] for lg in leagues]

   The key move is building a list of arrays — one run-differential series per league. boxplot draws one box per item in that list.

3. Draw and read the box plot

   fig, ax = plt.subplots(figsize=(7, 5))
   ax.boxplot(data, tick_labels=leagues, patch_artist=True)
   ax.axhline(0, color="#6C7079", ls="--")   # break-even reference
   ax.set_ylabel("run differential")
   fig.savefig("league_boxplot.png", dpi=144, bbox_inches="tight")

   

   Now the comparison is honest. Both leagues center near the break-even line, with similar-sized boxes — the middle half of each league lived within a few dozen runs of even. The story the average buried is in the tails: the AL stretches much further downward, pulled by a team whose run differential sat far below everyone else's (the kind of outlier the box plot draws as its own point). Same rough average, very different bottoms.

Troubleshooting