StatReason

Bootstrap Confidence Interval Calculator

Get a confidence interval for a statistic that has no simple textbook formula for its uncertainty, or when you would rather not rely on one, by resampling your own data directly.

What this answers

This calculator answers "what range plausibly contains the true value of this statistic?" for statistics like the median that have no simple closed-form confidence interval formula the way a mean does, or for any statistic when you would rather build the interval directly from your own data's resampling behavior.

How it is calculated

Your data is resampled with replacement 9,999 times by default, computing the chosen statistic on each resample. The confidence interval is read directly from the percentiles of that resampled distribution: for a 95% interval, the 2.5th and 97.5th percentiles of the resampled statistics. A seeded, fully reproducible pseudorandom generator is used throughout; the same seed always reproduces the exact same interval.

Worked example

For the values 1, 2, and 100, the observed median is 2, but with only 3 values the bootstrap resample distribution is genuinely wide, giving a confidence interval that reflects real uncertainty about the true median rather than a falsely precise number a small-sample formula might otherwise suggest.

Assumption audit

Calculated from your data: the observed statistic, its bootstrap standard error, and the resulting percentile confidence interval, alongside the exact seed and resample count used.
Evidence to review: your sample size; a bootstrap interval built from a very small sample reflects that smallness honestly (a wide interval), it does not manufacture precision a small sample does not actually have.
You must verify: that your original sample was collected in a way that represents the population you care about; resampling the same biased sample many times over does not fix a biased sampling method.

What this interval does not mean

A bootstrap confidence interval does not correct for a poorly collected original sample; it only describes the sampling variability present within the data you actually have, applied many times over through resampling.

Common mistakes

Treating a bootstrap interval as automatically superior to a formula-based interval is a common misconception; for well-behaved statistics like the mean with a reasonably large sample, the two typically agree closely, and the bootstrap earns its keep specifically when a closed-form formula is unavailable or untrustworthy. A second mistake is running the bootstrap on a sample too small to say anything useful; with only a handful of values, every resample is built from the same few points, and the resulting interval, however wide it looks, cannot manufacture information the original sample never contained.

Limitations

This calculator reports the percentile interval only; a bias-corrected and accelerated (BCa) interval, sometimes preferred for skewed statistics, is not implemented here, a documented simplification consistent with the Bootstrap Calculator this engine shares its resampling method with.