Bootstrap Confidence Intervals

Distribution-free confidence intervals for any statistic — no normality assumption required.

Bootstrap confidence intervals (CIs) are distribution-free intervals that do not rely on the assumption that your data follow a normal distribution. They work by repeatedly resampling (with replacement) from your observed data — typically 2,000 times — computing the statistic of interest each time, and using the resulting distribution of estimates to construct the CI.

The Bias-Corrected and Accelerated (BCa) method is the gold standard for bootstrap CIs. It adjusts for both the bias of the bootstrap estimate relative to the observed statistic and for the acceleration (skewness of the sampling distribution). This makes BCa CIs more accurate than percentile bootstrap CIs, especially with small samples or skewed distributions.

StatRyx computes BCa bootstrap CIs for means, medians, standard deviations, and other common statistics, with optional grouped analysis (separate CIs per group). A histogram of the bootstrap distribution is included so you can see the sampling variability directly.

Bootstrapping estimates the uncertainty of a statistic by resampling your own data. It draws thousands of samples with replacement, computes the statistic in each, and uses the spread of those values as the sampling distribution. The percentile CI takes the middle 95% of the bootstrap values; the BCa interval also corrects for bias and skew and is the usual recommendation. Bootstrapping assumes your sample represents the population; it cannot fix a biased sample.

When to use the Bootstrap Confidence Intervals

Your data are skewed or clearly non-normal and you need a valid CI for the mean or median
You want a CI for the median, standard deviation, or any statistic that lacks a simple closed-form formula
Your sample size is small (n < 30) and parametric CI assumptions are questionable
You want to compare CI widths across groups without assuming equal variance
You need a confidence interval for an indirect effect, a difference in medians or a correlation in a small sample

When not to use it

Your data are approximately normal and n > 30 — use Standard t-interval (parametric CI) — simpler and equally valid instead
You need an exact test, not just a CI — use Permutation test or nonparametric test instead

Worked example

Variable	Statistic	Observed	95% CI Lower	95% CI Upper	Resamples
reaction_time	Median	412 ms	387 ms	441 ms	2000
score	Mean	74.3	71.1	77.6	2000
errors	SD	8.2	6.9	9.8	2000

How to report it (APA 7)

Bootstrap confidence intervals (BCa, 2,000 resamples) were computed for the median reaction time. The median was 412 ms (95% BCa CI [387, 441]). Parametric CIs were not used because the distribution was strongly right-skewed (skewness = 2.4).

Frequently asked questions

Is bootstrap CI free on StatRyx?

Yes. You can run it without an account (3 free analyses), and a free account adds 10 more — a one-time allowance, no payment needed. Every result includes the APA 7 write-up. Paid plans (Student from $9/month) add more analyses and Word, PDF and Excel export.

How many bootstrap samples does StatRyx use?

2,000 resamples by default, which is sufficient for 95% CIs. The number can be adjusted by specifying it in your request.

What is BCa bootstrap?

BCa (bias-corrected and accelerated) bootstrap is the most accurate bootstrap CI method. It adjusts for bias and skewness in the sampling distribution using a jackknife-based acceleration estimate.

How do I report a bootstrap confidence interval in APA 7?

Name the method and the number of resamples, then the estimate with its interval: "The median reaction time was 412 ms, 95% BCa CI [387, 441], based on 2,000 bootstrap resamples."

Is a bootstrap CI better than a normal CI?

Not always. For roughly normal data and moderate samples they agree closely, and the standard interval is simpler. The bootstrap earns its place when the data are skewed or the statistic (median, ratio, indirect effect) has no reliable formula for its standard error.

Run a Bootstrap Confidence Intervals on your data →