Confidence Interval Simulator

Draw many samples from a population with a known true mean and watch how often the resulting confidence intervals actually capture it.

Set up the simulation

Samples are drawn from a population with a known true mean of 100 and standard deviation 15. Each sample builds a confidence interval for the mean.

Confidence level

Sample size (n)

Predict the coverage, score points

Before drawing, guess what % of your intervals will actually contain the true mean.

Each line is one sample's 95% confidence interval

Press a Draw button to build some intervals

How to use this tool

Choose a confidence level (90%, 95%, or 99%) and a sample size.
Press a Draw button to take that many samples, each producing its own confidence interval for the mean.
Green lines are intervals that contain the true mean; red lines are the ones that missed it.
Watch the coverage percentage settle near the confidence level as more intervals are drawn.

Formula Reference

Confidence interval

xbar +/- t* x (s / sqrt(n))

Long-run interpretation

over many samples, about (confidence level)% of intervals contain the true mean

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Frequently Asked Questions

What does a 95% confidence interval actually mean?+

It means that if you repeated the sampling process many times and built an interval the same way each time, about 95% of those intervals would contain the true population mean. It does not mean there is a 95% chance the true mean is inside any one particular interval you have already calculated.

Why do some intervals miss the true mean even at 95% confidence?+

Sampling variation means some samples, just by chance, produce an estimate far enough from the truth that even a correctly built interval misses it. At 95% confidence, missing is expected to happen about 5% of the time in the long run.

What happens to the intervals as sample size increases?+

They get narrower, because the standard error shrinks as n grows. Coverage should stay close to the chosen confidence level regardless of n; only the width changes.

Why does the observed coverage bounce around instead of landing exactly on 95%?+

With a small number of simulated intervals, the observed proportion is itself a random quantity. Draw more intervals and watch the coverage percentage settle closer to the true 95%, exactly as the law of large numbers predicts.