The ECDF: Read Any Percentile Off a Cumulative Curve

A histogram bins your data and smooths away detail; the bin width is a choice that can hide or invent structure. The empirical cumulative distribution function — ECDF — makes no such choice. It plots every value against the fraction of the data at or below it, so you can read "what share of teams fall below this run differential?" straight off the curve. It's two lines of code and the most underrated distribution chart there is.

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

1. Two lines to build it

   Sort the values, then give each one its running share of the data: the smallest is at 1/n, the largest at 1.0. That pair of arrays is the ECDF.

   import numpy as np
   import pandas as pd
   
   df = pd.read_csv("sample_standings.csv")
   x = np.sort(df["RunDiff"].values)         # values, ascending
   y = np.arange(1, len(x) + 1) / len(x)     # cumulative share: 1/n ... 1.0

   No binning, no bandwidth, no tuning. Every data point is preserved exactly, which is why two analysts will always draw the identical ECDF from the same numbers — not true of histograms or density curves.

2. Read percentiles off the curve

   The y-axis is percentiles. Find 0.75 on the y-axis, slide right to the curve, drop down — that x is the 75th percentile. np.quantile gives the same numbers exactly.

   for q in (0.25, 0.50, 0.75, 0.90):
       print(int(q*100), "th pct:", round(np.quantile(df["RunDiff"], q), 1))
   
   print("share below zero:", round((df["RunDiff"] < 0).mean(), 3))

   Percentiles, and the share below zero

   25th percentile run differential:  -87.2
   50th percentile run differential:  -13.5
   75th percentile run differential:  102.8
   90th percentile run differential:  168.0
   
   Share of teams with a negative run differential: 0.567

   Here the median run differential is below zero even though differentials must sum to zero league-wide — a sign the distribution is right-skewed, with a few dominant teams piling up big positive numbers while most teams sit a little under water. A histogram hints at that; the ECDF lets you quantify it precisely.

3. Draw the staircase

   step() draws the ECDF as a staircase; where="post" holds each level until the next value. Guide lines from the median show how to read a percentile geometrically.

   import matplotlib.pyplot as plt
   
   fig, ax = plt.subplots(figsize=(8, 5.5))
   ax.step(x, y, where="post", lw=2)
   ax.scatter(x, y, s=18)                       # mark the actual data points
   median = np.quantile(df["RunDiff"], 0.5)
   ax.axhline(0.5, ls="--"); ax.axvline(median, ls="--")
   ax.set_xlabel("run differential"); ax.set_ylabel("cumulative share (≤ x)")
   fig.savefig("ecdf.png", dpi=144, bbox_inches="tight")

   

   Steep stretches of the staircase are where values cluster (many teams close together); long flat stretches are gaps in the data (the lonely outliers at the top). The curve's shape is the distribution's shape, read without ever choosing a bin.

Troubleshooting