Bootstrap distribution
Resampled correlation with percentile and BCa 95% confidence intervals.
The bootstrap re-estimates a statistic on thousands of samples drawn with replacement from the data; the spread of those estimates is its sampling uncertainty. The histogram is the bootstrap distribution of a correlation, the orange line the value observed in the full sample. Two 95% intervals are drawn above it: the simple percentile interval and the BCa (bias-corrected and accelerated) interval, which shifts to account for the bias and skew visible here — the distribution is far from symmetric.
Data are the 50 US states + DC from the statsmodels statecrime dataset:
the share of single-parent households vs the violent-crime rate.
scipy.stats.bootstrap (paired, 9,999 resamples) provides both intervals.
import matplotlib.pyplot as plt
import numpy as np
import statsmodels.api as sm
from scipy import stats
# Data: violent crime vs single-parent households, 51 states
df = sm.datasets.statecrime.load_pandas().data
x, y = df["single"].to_numpy(), df["violent"].to_numpy()
r_obs = stats.pearsonr(x, y)[0]
# Paired bootstrap of Pearson's r: percentile and BCa intervals from the same resamples
r = lambda a, b, axis=-1: stats.pearsonr(a, b, axis=axis)[0]
boot = {m: stats.bootstrap((x, y), r, paired=True, vectorized=True, n_resamples=9999,
method=m, random_state=np.random.default_rng(7))
for m in ["percentile", "BCa"]}
dist = boot["BCa"].bootstrap_distribution
fisher = np.tanh(np.arctanh(r_obs) + np.array([-1, 1]) * 1.96 / np.sqrt(len(x) - 3))
fig, ax = plt.subplots(figsize=(8, 5))
lo, hi = boot["BCa"].confidence_interval
counts, edges, bars = ax.hist(dist, bins=60, color="#C7C7CC", edgecolor="white", lw=0.4)
for b, left in zip(bars, edges[:-1]):
if lo <= left + (edges[1] - edges[0]) / 2 <= hi:
b.set_facecolor("#007AFF"); b.set_alpha(0.75)
# Intervals stacked above the histogram
top = counts.max()
for i, (m, c, name) in enumerate([("percentile", "#8E8E93", "Percentile"),
("BCa", "#007AFF", "BCa")]):
a, b = boot[m].confidence_interval
yy = top * (1.12 + 0.12 * i)
ax.plot([a, b], [yy, yy], color=c, lw=3, solid_capstyle="butt")
ax.vlines([a, b], yy - top * 0.03, yy + top * 0.03, color=c, lw=2)
ax.text(a - 0.01, yy, f"{name} 95% CI [{a:.2f}, {b:.2f}]", ha="right", va="center",
fontsize=9.5, color=c, fontweight="semibold")
ax.vlines(r_obs, 0, top * 1.34, color="#FF9500", lw=2, zorder=3)
ax.text(r_obs, top * 1.36, f"observed r = {r_obs:.2f}", ha="center", va="bottom",
fontsize=9.5, color="#FF9500", fontweight="semibold")
ax.set_xlim(0.15, 1)
ax.set_ylim(0, top * 1.46)
ax.set_xlabel("Bootstrap estimate of Pearson's r")
ax.set_ylabel("Resamples")
ax.set_title("Crime and single parenthood: a strong but skewed correlation", pad=26)
ax.text(0, 1.02, f"Paired bootstrap, B = 9,999 · n = {len(x)} states · bias = "
f"{dist.mean() - r_obs:+.3f} · Fisher-z 95% CI [{fisher[0]:.2f}, {fisher[1]:.2f}] for comparison",
transform=ax.transAxes, fontsize=9.5, color="#6E6E73")
plt.show()
Figures use the optional PyStatlas style: add plt.style.use("pystatlas.mplstyle") to match the look exactly.
Data source
- US state crime and social indicators, 2009. U.S. Census Bureau. Statistical Abstract of the United States, state tables for 2009 (crime rates from the FBI Uniform Crime Reporting program). https://www.statsmodels.org/stable/datasets/generated/statecrime.html
Compiled for statsmodels (sm.datasets.statecrime); public domain. See the statsmodels page for each variable's table and footnotes.
The examples illustrate statistical methods. Their titles describe what each figure shows in these particular data — often a small, dated or partial sample — not the author's position on the substantive question. For interpretation, context and data quality, rely on the original sources cited here.
How to cite
PyStatlas is created and maintained by Thierry Warin. If it helped your work, please cite it.
Warin, T. (2026). Bootstrap distribution. In PyStatlas: A Python Gallery of Statistical Graphics for the Social Sciences. https://warint.github.io/pystatlas/inference-uncertainty/bootstrap-distribution/
@misc{warin2026statlas,
author = {Warin, Thierry},
title = {PyStatlas: A Python Gallery of Statistical Graphics for the Social Sciences},
year = {2026},
howpublished = {\url{https://warint.github.io/pystatlas/}},
note = {Source code: https://github.com/warint/pystatlas}
}More in Inference & uncertainty
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