Confidence Interval Calculator
A confidence interval turns one sample into an honest range for the truth: instead of a single guess, you get the full band of values your data actually support. Pick what you are estimating, enter your numbers, and read the bounds with the working shown. Everything runs in your browser.
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How this is computed
✓ No data leaves your browser
✓ Matches R: t.test(), prop.test(), binom.test(), poisson.test(), cor.test(), confint()
✓ Free, no sign-up
Which interval do I need?
Match your question to the interval type.
| Your question | Interval type | Built on |
|---|---|---|
| What is the average, with margin? | Mean | t distribution |
| How different are two group averages? | Two means | Welch t |
| How sure am I of a percentage? | Proportion | Wilson score / exact |
| How different are two rates? | Two proportions | Wald (unpooled) |
| Events per unit of time or space? | Poisson rate | Exact (chi-square) |
| How variable is the process? | Variance / SD | Chi-square |
| How strong is the association, really? | Correlation | Fisher z |
| How precise is my model's slope? | Regression β | t distribution |
Which proportion interval should you trust?
| Method | How it behaves | Use it when |
|---|---|---|
| Wilson score | Accurate even for small n and proportions near 0 or 1 | Default choice, always safe |
| Exact (Clopper-Pearson) | Guaranteed coverage, slightly wide | Regulated or clinical settings |
| Wald (p ± z·SE) | Breaks down for small n or extreme p; can leave [0, 1] | Only as a classroom stepping stone |
This calculator offers Wilson and exact for a single proportion. The Wald interval is deliberately absent there: it is the one most likely to mislead, exactly when the stakes are high. (For a difference of two proportions the unpooled Wald form is standard and matches R's prop.test.)
Frequently asked questions
What does a 95% confidence interval actually mean?
Run the same study 100 times, building an interval each time, and about 95 of those intervals will contain the true value. The 95% belongs to the method, not to any single interval. That distinction trips up nearly everyone at first, and matters in exams.
Zero events in my sample. Is the risk zero?
No. Zero observed events puts an upper bound on the rate, not a zero. With 0 events in 15 trials the exact 95% interval runs from 0 to about 22%; with 0 events over 10 hours a Poisson rate can still be as high as 0.37 per hour. Try the "Zero events" scenario above.
How do I make my interval narrower?
Three levers: collect more data (width shrinks with the square root of n, so 4x the sample halves the width), accept less confidence (90% is narrower than 99%), or reduce measurement noise.
Why does the calculator use the t distribution for means?
Because the standard deviation is estimated from the same sample as the mean, which adds uncertainty. The t distribution accounts for that extra uncertainty; with large samples it becomes indistinguishable from the normal.
Why is the correlation interval asymmetric?
Correlations are trapped between -1 and 1, so their uncertainty cannot be symmetric near the edges. The Fisher z transform maps r to an unbounded scale, builds a symmetric interval there, and maps back, producing the correct lopsided interval.
Does this match R?
Yes: every interval type is verified against the corresponding R function to at least six decimal places, including boundary cases like x = 0, x = n and zero Poisson events.