Bootstrapping
Use cases, how it works, and types of bootstrap confidence intervals
TRCP Education
March 06, 2026
In one sentence
Bootstrapping is a resampling method: we treat our sample as the population, draw many new samples with replacement from it, compute the statistic of interest in each new sample, and use that bootstrap distribution to estimate standard errors and confidence intervals—without assuming a particular distribution (e.g. normality).
When is bootstrapping useful?
Bootstrapping is especially helpful when:
| Situation | Why bootstrap helps |
|---|---|
| Non-normal data | You want a CI for the median, IQR, or another non-mean statistic where formulas are messy or assume normality. |
| Complex statistics | The statistic has no simple formula for its SE (e.g. ratio of medians, custom estimator). |
| Small samples | Large-sample (e.g. normal) approximations may be unreliable; the bootstrap uses the data’s shape. |
| Checking robustness | You can compare bootstrap CIs to model-based CIs to see if normality assumptions matter. |
You can use the bootstrap for the mean too—it will usually agree with the usual normal-based CI when the sample is reasonable. The real power is for anything else (median, trimmed mean, regression coefficients, etc.).
How it works (step by step)
- You have one sample of size \(n\) (e.g. \(n = 12\) measurements).
- Resample with replacement: Draw \(n\) observations from your sample, putting each back after drawing. So some original values appear 0 times, some once, some twice or more.
- Compute the statistic (e.g. mean, median) on this bootstrap sample. That gives one bootstrap replicate.
- Repeat steps 2–3 many times (e.g. \(B = 2000\) or \(B = 10000\)). You get \(B\) values of the statistic.
- The bootstrap distribution of the statistic is the
histogram (or density) of those \(B\)
values. From it you get:
- Standard error ≈ standard deviation of the bootstrap distribution.
- Confidence interval ≈ e.g. 2.5th and 97.5th percentiles (percentile method), or other methods below.
The idea: if we could repeat the study many times we’d get a distribution of the statistic; we can’t, so we simulate that by resampling from our one sample.
Visual: one sample and one bootstrap sample
Below we use a tiny sample of \(n = 12\) values so you can see what “resample with replacement” means. The top panel shows the original sample (each value once). The bottom panel shows one bootstrap sample: we drew 12 times with replacement, so some values appear more than once and some not at all. Each dot is one draw; repeated values are stacked vertically so you can see multiplicity.
Top: original sample. Bottom: one bootstrap sample—same n, drawn with replacement; some values repeat, some are omitted.
In the bootstrap sample, the mean (and other statistics) will be slightly different from the original. Repeating this thousands of times gives the bootstrap distribution of the mean.
The bootstrap distribution of the mean
We now take \(B = 5000\) bootstrap samples from the same data and compute the mean of each. The histogram below is the bootstrap distribution of the sample mean. The vertical dashed line is the mean of the original sample; the shaded region is the 95% percentile interval (2.5th to 97.5th percentiles of the bootstrap distribution).
Bootstrap distribution of the sample mean (B = 5000). Dashed line: original sample mean. Green vertical lines: 95% percentile CI (2.5th–97.5th).
So from one sample we get an entire distribution of possible values of the mean. The spread of that distribution is the bootstrap standard error; the 2.5th and 97.5th percentiles give a 95% confidence interval that does not assume normality.
Same idea for a non-normal statistic: the median
Formulas for the standard error of the median are awkward and often assume normality. The bootstrap treats the median like any other statistic: resample, compute the median, repeat. Below we show the bootstrap distribution of the median for the same 12 observations. Again, the dashed line is the original sample median; the shaded region is the 95% percentile interval.
Bootstrap distribution of the sample median. Green vertical lines: 95% percentile CI. No normality assumption.
The bootstrap distribution of the median is discrete (with only a few possible values when \(n\) is small); the percentile interval still gives a plausible range. For larger \(n\), the bootstrap distribution of the median becomes smoother.
Types of bootstrap confidence intervals
From the same bootstrap distribution you can build different 95% confidence intervals:
| Type | How it’s built | When it’s useful |
|---|---|---|
| Percentile | Take the 2.5th and 97.5th percentiles of the bootstrap distribution. | Default choice; easy to explain and often works well. |
| Normal approximation | Mean of bootstrap distribution ± 1.96 × (bootstrap SE). | When the bootstrap distribution is roughly symmetric; then similar to percentile. |
| BCa (bias-corrected and accelerated) | Adjusts the percentiles for bias (bootstrap distribution shifted from original) and skewness (acceleration). | When the statistic is biased or the bootstrap distribution is skewed; gives better coverage in theory. |
In R, boot (package boot)
can compute percentile, normal, and BCa intervals. For teaching, the
percentile method is the one we used in the figures
above.
Summary: one picture
Conceptually, bootstrapping does this:
- One sample → treat it as the only information about the population.
- Resample with replacement many times → each time you get a “new sample” of the same size.
- Compute the statistic each time → you get the bootstrap distribution.
- Use that distribution → SE = sd(bootstrap values); 95% CI = e.g. 2.5th and 97.5th percentiles (percentile method).
The next figure summarises the flow: from original sample to many bootstrap samples to one bootstrap distribution (here, of the mean), with the 95% percentile CI marked.
From one sample to bootstrap distribution and 95% CI. Green vertical lines: 2.5th and 97.5th percentiles (percentile interval).
Takeaway
- Bootstrap = resample from the sample with replacement, many times; compute the statistic each time; use that distribution for SE and CIs.
- Use it when you want CIs (or SEs) without assuming normality—e.g. for the median, ratios, or other non-standard statistics—or when the sample is small.
- Percentile method: 95% CI = 2.5th and 97.5th percentiles of the bootstrap distribution. Other options (normal approximation, BCa) adjust for bias or skew when needed.
- Figures in this lesson showed: (1) one sample vs one bootstrap sample, (2) bootstrap distribution of the mean with percentile CI, (3) bootstrap distribution of the median, (4) the same flow as a single summary plot.