How to use this calculator
- Enter or choose data.
- Enter or choose confidence level.
- Enter or choose bootstrap resamples (B).
- Enter or choose random seed.
- The lower bound updates as you type. Use Copy result or Share to keep it.
Method
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
- x̄*
- the mean of one resample
- B
- number of resamples
- α
- 1 − confidence level
- q
- sample quantile with linear interpolation
Worked example
- The sample mean is 180 ÷ 12 = 15.
- Draw 12 values with replacement from the data and average them; repeat 2,000 times.
- Sort the 2,000 means; the 2.5th and 97.5th percentiles are the interval bounds.
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.