# Bootstrap Mean Interval Calculator

> Resample your data thousands of times to get a percentile bootstrap confidence interval and standard error for the mean, with a seed for repeatable runs.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/bootstrap-mean-interval-calculator

The bootstrap estimates how much a sample mean would vary by resampling your own data with replacement. The middle 95% of the resampled means is a 95% percentile interval.

Also called: Bootstrap CI calculator, Resampling interval calculator.

## Inputs
- Data (default 12, 15, 9, 20, 17, 14, 11, 25, 16, 13, 18, 10)
- Confidence level (unit %; default 95)
- Bootstrap resamples (B) (default 2000)
- Random seed (default 12345)

## Results
- Lower bound
- Upper bound
- Sample mean
- Bootstrap standard error
- Bootstrap bias estimate
- Observations

## Formula

```
Repeat B times: draw n values with replacement and take their mean x̄*. Interval = [q(α/2), q(1 − α/2)] of the sorted x̄*;  SE = SD of the x̄*;  Bias = mean(x̄*) − x̄
```

Where:
- x̄*: the mean of one resample
- B: number of resamples
- α: 1 − confidence level
- q: sample quantile with linear interpolation

## Worked example
1. The sample mean is 180 ÷ 12 = 15.
2. Draw 12 values with replacement from the data and average them; repeat 2,000 times.
3. Sort the 2,000 means; the 2.5th and 97.5th percentiles are the interval bounds.

**Result:** For these 12 values the mean is 15 and the 95% bootstrap interval runs from 12.58 to 17.75 with seed 12345; another seed moves the bounds slightly.

## What the result means

The interval shows the range of means consistent with the variation in your sample. It makes no normality assumption, but it is only as representative as the sample itself. If two seeds give noticeably different bounds, increase the number of resamples.

## Assumptions and limits
- Observations are independent and identically distributed.
- The sample is representative of the population.
- With small samples (under about 10–15 values) percentile intervals are too narrow; a t-interval is usually better.
- This is the plain percentile method, not the bias-corrected (BCa) interval.
- Not suitable for time series or clustered data without a block or cluster bootstrap.

## Questions

### How many resamples should I use?

About 1,000 is the usual minimum for a 95% interval and 2,000–10,000 gives stable bounds. The standard error settles with fewer, around 200.

### How does it compare with a t-interval?

For roughly symmetric data the two are close. The bootstrap interval can be asymmetric, which helps with skewed data, but it is less reliable than the t-interval when n is small.

## Related calculators
- [Confidence Interval Calculator](https://calculators.geekmidst.com/calculators/confidence-interval-calculator)
- [Standard Deviation Calculator](https://calculators.geekmidst.com/calculators/standard-deviation-calculator)
- [Statistics Calculator](https://calculators.geekmidst.com/calculators/statistics-calculator)
- [Percentile Rank Calculator](https://calculators.geekmidst.com/calculators/percentile-rank-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/bootstrap-mean-interval-calculator