This confidence interval calculator computes intervals for a population mean (using z or t, as your data allows) and for a proportion — with the margin of error broken out, at any confidence level.
What a Confidence Interval Actually Promises
The procedure, repeated over many samples, captures the true value in the stated share of runs — 95% of 95% intervals contain the parameter. The popular misreading (“95% probability the mean is in this interval”) treats the fixed parameter as random; frequentist machinery puts the randomness in the sampling instead. Practical reading: the interval’s width is your precision, and its location is your best map of where the truth plausibly sits. Shrink the width with larger n (planning: our Sample Size Calculator and Margin of Error Calculator). Part of our Statistics Calculators collection.
Worked Examples (Verified)
- Mean, σ unknown: x̄ = 100, s = 15, n = 36, 95% → t(35) = 2.030, margin = 2.030 × 15/6 = 5.08 → (94.92, 105.08). The z-version (crit 1.960) gives a slightly tighter (95.10, 104.90) — t’s extra width is the honest price of estimating σ from the same data
- Proportion: 54 of 100 → 54% ± 9.8% → (44.2%, 63.8%) — the width that makes poll headlines humbler than they read
- Level trade-off: the same mean data at 99% (t = 2.724) widens to ±6.81: more confidence costs more width — there is no free certainty
z or t? The One Decision That Matters
- σ genuinely known (rare — calibrated instruments, huge historical data): z.
- σ estimated by s (almost always): t, with n−1 degrees of freedom; by n ≈ 100+ the difference is cosmetic.
- Non-normal populations with small n: neither rescues you — transform, bootstrap, or gather more data. The t-interval’s robustness grows with n via the central limit theorem.
Related Tools
- Margin of Error Calculator — the ± on its own
- P-Value Calculator — the hypothesis-testing mirror (a 95% CI excluding the null value ⟺ p < 0.05)
- Statistics Calculators — the full collection
Frequently Asked Questions
How do you calculate a confidence interval?
Estimate ± (critical value × standard error). For a mean with s = 15, n = 36: SE = 15/√36 = 2.5; at 95% the t-critical is 2.030, so the interval is x̄ ± 5.08.
What does 95% confidence really mean?
If the same sampling-and-interval procedure ran forever, 95% of the intervals would contain the true parameter. It is a property of the method, calibrated over repetitions — the standard interpretation working scientists use when reading any single interval.
Why is my interval so wide?
Three levers: variability (s), sample size (√n in the denominator — quadrupling halves width), and confidence level (99% costs ~35% more width than 95% here). Of these, only n is usually yours to choose — in advance.
When should I use t instead of z?
Whenever σ is estimated from the sample — which is nearly always. t widens intervals appropriately for small n and converges to z as n grows. Using z with a small sample understates your uncertainty.
Can a confidence interval be one-sided?
Yes — “at least X” bounds use a one-sided critical value (1.645 for 95% one-sided vs 1.960 two-sided). This calculator builds the standard two-sided interval; halve the stated two-tail α logic by choosing the 90% level to mimic a 95% one-sided bound.
Is this confidence interval calculator free?
Yes — free, instant, no sign-up.