Analyze sample uncertainty with thousands of flexible resamples. Inspect interval methods, distributions, and summary diagnostics. Export results, validate assumptions, and explain findings with confidence.
| Observation | Value | Observation | Value |
|---|---|---|---|
| 1 | 12 | 7 | 25 |
| 2 | 15 | 8 | 27 |
| 3 | 18 | 9 | 29 |
| 4 | 21 | 10 | 31 |
| 5 | 22 | 11 | 34 |
| 6 | 23 | 12 | 36 |
This example uses twelve observations. Paste them into the data box to test the calculator quickly and inspect how interval estimates change across methods.
The calculator first computes the chosen sample statistic, written as θ. It then creates many resamples of size n with replacement from the original data. Each resample produces a bootstrap estimate, written as θ*.
Percentile interval: lower = percentile of θ* at α/2, upper = percentile of θ* at 1 − α/2.
Basic interval: lower = 2θ − percentile at 1 − α/2, upper = 2θ − percentile at α/2.
Normal interval: lower = θ − z × SE*, upper = θ + z × SE*. Here SE* is the standard deviation of the bootstrap estimates and z is the standard normal critical value.
Bias: mean of θ* − θ. Interval width: upper − lower.
Bootstrap confidence intervals estimate the uncertainty of a statistic without requiring a strict parametric distribution. They are especially useful for skewed samples, small studies, complex estimators, and statistics where analytic standard errors are difficult to derive.
The percentile method is direct and intuitive. The basic method reflects the interval around the original estimate. The normal method is quick when the bootstrap distribution is nearly symmetric. Comparing all three can reveal instability or asymmetry in the resampled estimates.
It estimates a confidence interval for a chosen sample statistic by repeatedly resampling the original data with replacement and recomputing that statistic.
Use them when formulas are unavailable, assumptions are uncertain, or your statistic has a complicated sampling distribution.
Many analysts start with 1,000 to 5,000 resamples. Higher counts usually stabilize the interval but take longer to compute.
The percentile method follows the empirical bootstrap distribution directly. The normal method assumes approximate symmetry around the estimate.
Bias is the difference between the average bootstrap estimate and the original sample estimate. Large bias may signal instability.
Yes. Enter binary data coded as 0 and 1, then choose Proportion. The estimate equals the sample mean.
A seed makes the resampling sequence reproducible. Reusing the same seed recreates the same bootstrap estimates.
Bootstrap methods use random resampling. Without a fixed seed, each run draws a different set of resamples.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.