Confidence Interval Coverage
Draw many samples from a population whose true mean you can see, build a confidence interval from each sample, and count how often the interval captures that true mean. That long-run capture rate is what "95% confidence" means.
Predict first
Before any intervals appear, a prompt asks how many of the next batch you think will miss the true mean. Enter a guess and choose Lock & draw to commit and compare against what actually happens, or Skip . Predicting first makes the result stick.
Set up the population
Population — the shape sampled from: Normal, Right-skewed, or Uniform. Its true mean is drawn on the chart as the target line.Sample size (n) — how many observations go into each sample (and each interval).Confidence level — 90%, 95%, or 99%. Higher confidence makes wider intervals that capture the mean more often.CI method — how each interval is built: t-interval (the formula-based interval) or Percentile bootstrap (resampled from the sample). Compare how their coverage behaves, especially for skewed populations or small n.
Draw samples
Use +1 / +10 / +25 / +100 to draw that many samples; each adds an interval to the chart. Intervals that miss the true mean are marked. The running coverage rate below settles toward the confidence level as you draw more. Reset clears them.
Compare the four interval methods
The CI method selector builds every interval a different way from the same samples, so you can measure — not assert — which one actually delivers its advertised 95%:
t-interval — x̄ ± t*·s /√n . The traditional interval.Bootstrap percentile — the middle 95% of the bootstrap distribution.Bootstrap ±z ·SE — the bootstrap distribution's centre, stepped out z of its own standard errors. Always symmetric.Bootstrap BCa — percentile with the cut-points shifted to correct for bias and skew.
Add ?method=t, ?method=bootstrap, ?method=se, or ?method=bca to link straight to one.
What you will find, on a right-skewed population at nominal 95% (measured here, 1200 intervals per cell):
n t percentile ±2·SE BCa
8 88.4% 83.7% 85.3% 85.8% 15 91.9% 88.9% 90.3% 90.1% 25 92.4% 90.3% 92.0% 92.3% 50 92.9% 92.5% 93.9% 93.8%
Three things worth saying out loud. Every method under-covers here — the gap to 95% is larger than any gap between methods. BCa does help , but by a point or two, not by closing the gap; it estimates its correction from the same short sample. And the t-interval holds up best at very small n , which is a genuine result worth showing students rather than hiding. On a Normal population, or once n is large, the four converge.
Because bootstrap bounds come from the replicate distribution rather than a stored standard error, changing the confidence level on a bootstrap method redraws the run instead of re-widening the existing intervals.
Plausibility shape (cat's-eye)
Tick Show plausibility shape (cat's-eye) , then hover an interval. The bulge shows that values near the sample mean are more plausible than values near the interval's ends — the interval is not a band of equally likely values.
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1 +1 sample
2 +10 samples
3 +25 samples
4 +100 samples
0 Reset
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