Bootstrap Interval Calculator
This bootstrap interval calculator produces a 95 percent confidence interval for a mean using the bootstrap standard error, the resampling approach to quantifying uncertainty that the US National Institute of Standards and Technology describes in its statistics handbook. The bootstrap repeatedly resamples your data with replacement, recomputes the statistic each time, and uses the spread of those replicate estimates to gauge how much the statistic would vary across new samples. For the sample mean, that bootstrap spread converges to the familiar standard error, the sample standard deviation divided by the square root of the sample size, so a deterministic interval can be reported without running thousands of random draws. Enter the sample mean, the sample standard deviation and the sample size, and the calculator returns the standard error and the 95 percent interval as the mean plus or minus 1.96 standard errors. This gives a stable, reproducible bootstrap-style interval suitable for teaching and quick checks. Use it to attach uncertainty to an estimate, compare against a hypothesized value, or illustrate resampling. Every figure is computed deterministically from the standard-error formula shown in full below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
The bootstrap standard error sets the interval width: with mean 50.00, standard deviation 10.00 and size 25, the standard error is 2.00 and the 95 percent interval is 46.08 to 53.92. Wider data or smaller samples widen the interval.
Bootstrap interval formula
SE = s / sqrt( n )
95% interval = mean +/- 1.96 x SE
mean = sample mean, s = sample standard deviation
n = sample size, 1.96 = normal critical value at 95%
Resampling the data with replacement and recomputing the mean many times produces a distribution whose spread is the bootstrap standard error. For the mean that spread equals the sample standard deviation over the square root of n, so the percentile interval is well approximated by the mean plus or minus 1.96 standard errors.
Worked example
Build a 95 percent interval for a sample with mean 50, standard deviation 10 and size 25.
- SE = 10 / sqrt(25) = 10 / 5 = 2.00
- margin = 1.96 x 2.00 = 3.92
- lower = 50 - 3.92 = 46.08
- upper = 50 + 3.92 = 53.92
The 95 percent confidence interval runs from 46.08 to 53.92. These are the calculator's default inputs, so the result above matches the widget exactly.
How sample size narrows the interval
With mean 50 and standard deviation 10, a larger sample shrinks the standard error and the interval.
| n | SE | 95% interval |
|---|---|---|
| 25 | 2.00 | 46.08 to 53.92 |
| 100 | 1.00 | 48.04 to 51.96 |
| 400 | 0.50 | 49.02 to 50.98 |
Bootstrap method reference: US National Institute of Standards and Technology (NIST) Engineering Statistics Handbook.
Bootstrap Interval Calculator: frequently asked questions
What is the bootstrap?
The bootstrap is a resampling method: you draw many new samples of the same size from your data, sampling with replacement, recompute the statistic for each, and use the spread of those values to estimate how uncertain the statistic is. It needs no assumption about the underlying distribution, which makes it widely applicable when classical formulas are hard to derive.
Why does this give a deterministic answer?
For the sample mean, the bootstrap distribution of the mean has a standard deviation that converges to the analytic standard error, the sample standard deviation divided by the square root of n. Because that limit is exact, the interval can be reported deterministically as mean plus or minus 1.96 standard errors instead of drawing thousands of random replicates, which keeps the result reproducible.
Is 1.96 always the right multiplier?
For a 95 percent interval based on a normal approximation, 1.96 is the standard normal critical value. With small samples a t-distribution multiplier is more accurate, and for statistics other than the mean the percentile bootstrap can give asymmetric intervals. This tool uses the normal-approximation bootstrap interval for the mean, which is exactly right in large samples.
What standard deviation should I enter?
Enter the sample standard deviation computed with the usual n minus 1 denominator. The calculator divides it by the square root of the sample size to get the standard error of the mean. If you only have the variance, take its square root first.
When should I prefer a full resampling bootstrap?
For complex statistics such as medians, ratios, or regression coefficients, or for clearly non-normal data and small samples, a full percentile or bias-corrected bootstrap with thousands of resamples captures asymmetry that the normal approximation misses. For the mean of a reasonably sized sample, the deterministic interval here matches the full bootstrap closely.
Official sources
- Bootstrap and confidence interval reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.