Summary: This confidence interval calculator computes mean and proportion intervals with live steps, formulas and a chart. It accepts labeled numeric inputs, works offline through file:// and includes source-backed explanations for students, analysts and researchers.
Confidence interval calculator for means and proportions using z or t critical values, margin of error, formula and steps. The calculator works offline, updates instantly and includes a worked example, plain-text formula, MathML, references and structured data.
Default values are loaded. Click any field and edit it; results and chart update automatically.
| Input or setting | Result or interpretation | Use this when |
|---|---|---|
| 90% confidence | z* = 1.645 | interval estimates |
| 95% confidence | z* = 1.960 | interval estimates |
| 99% confidence | z* = 2.576 | interval estimates |
mean: estimate +/- critical value * standard error; proportion: phat +/- z*sqrt(phat(1 - phat)/n)A confidence interval is an estimate plus and minus a critical value times standard error.
mean=50, s=10, n=25, 95%
45.87 to 54.13 using t* about 2.064
This page uses the shared statistics core for distribution functions, quantiles and exact integer counting where needed. The formula is shown in plain text so screen readers and search engines can parse it reliably.
Use Load example in the calculator to reproduce this reference result.
{
"tool": "Confidence Interval Calculator",
"input": "mean=50, s=10, n=25, 95%",
"output": "45.87 to 54.13 using t* about 2.064",
"formula": "mean: estimate +/- critical value * standard error; proportion: phat +/- z*sqrt(phat(1 - phat)/n)"
}| Calculator | Example input | Expected output |
|---|---|---|
| Confidence Interval Calculator | mean=50, s=10, n=25, 95% | 45.87 to 54.13 using t* about 2.064 |
What does p = 0.03 mean? If the null hypothesis and model assumptions were true, a result at least this extreme would occur about 3% of the time. The American Statistical Association cautions that a p-value alone does not measure effect size, practical importance or the probability that Hâ‚€ is true.3
For most classroom and professional reports, pair the calculator result with the question you are answering. A mean or median summarizes location, but spread explains consistency. A confidence interval estimates plausible values, while a hypothesis test evaluates compatibility with a null model. Regression and correlation describe association, so they should be reported with a chart and residual or outlier review. When a result is statistically significant, still ask whether the effect is large enough to matter in the real setting.
| Statistic | Small | Medium | Large | Use |
|---|---|---|---|---|
| Cohen's d | 0.2 | 0.5 | 0.8 | t-test effect size |
| Cramér's V | 0.1 | 0.3 | 0.5 | chi-square association |
| |r| | 0.10 | 0.30 | 0.50 | correlation strength |
| R² | 0.01 | 0.09 | 0.25 | variance explained |
The standard formula is: sample mean ± (critical value × standard error). For a 95% CI on a mean using a known population SD, the critical value is z = 1.96, and the standard error is σ/√n. So if your sample mean is 50, σ = 10, and n = 100, the SE is 1, giving 50 ± 1.96 × 1 = (48.04, 51.96). When the population SD isn't known, swap z for the t-critical value at df = n − 1 and use the sample SD.
Use a t interval whenever you don't know the population standard deviation and you're estimating it from your sample — which honestly is most of the time in practice. The t distribution has heavier tails than the standard normal, accounting for the extra uncertainty introduced by estimating SD. This matters especially when n is small (say under 30). With very large samples, t and z give nearly identical results, but the t interval is the technically correct choice whenever σ is unknown, regardless of sample size.
Higher confidence demands a bigger safety net. To be 99% confident the interval captures the true parameter, you need to extend further from your sample mean than at 95%. The critical value jumps from z = 1.96 (for 95%) to z ≈ 2.576 (for 99%), which directly inflates the margin of error. So precision drops as confidence rises — you can't have both at the same sample size. The only way to keep a tight interval at 99% confidence is to collect more data and shrink the standard error.
Easy method: take half the width of the interval. If your CI is (62, 78), the margin of error is (78 − 62)/2 = 8. The interval is centred on the sample estimate, so you can also write it as estimate ± MOE — meaning estimate = (62 + 78)/2 = 70, with MOE = 8. This works for any symmetric interval, whether built from z or t. If the CI isn't symmetric (some bootstrap intervals aren't), this shortcut doesn't apply and you'd need the original calculation.
The required n depends on three things: your desired confidence level, the margin of error you can tolerate, and an estimate of variability. For a mean, n ≈ (z × σ / E)², where E is your target MOE. For a proportion, n ≈ z²p(1−p) / E². At 95% confidence with a 5% MOE and worst-case p = 0.5, you'd need about 385. Tighter MOE or higher confidence both push n up — and quickly. Halving the MOE roughly quadruples the required sample.
The formula is p̂ ± z√[p̂(1 − p̂)/n], where p̂ is your sample proportion. Suppose 60 of 200 surveyed people support a policy: p̂ = 0.30, n = 200. At 95% confidence (z = 1.96), the SE is √[0.3 × 0.7 / 200] ≈ 0.0324, so the CI is 0.30 ± 0.0635, giving roughly (0.237, 0.363). One condition: np̂ and n(1 − p̂) should both be at least about 10 for the normal approximation to hold. For tiny samples, the Wilson or exact (Clopper–Pearson) interval is safer.
The strict interpretation is about the procedure, not a single interval. If you repeated the sampling many times and built a 95% CI each time, about 95% of those intervals would contain the true parameter. Once you've calculated one specific interval, it either contains the parameter or it doesn't — there's no probability attached to that fixed interval. Common pitfall: students often say "there's a 95% chance the true mean is in this interval", which is technically wrong. The 95% refers to long-run reliability of the method.
You need four inputs: sample mean (x̄), sample SD (s), sample size (n), and confidence level. Step one: compute the standard error, SE = s/√n. Step two: pick the critical value — z for known population SD, t with df = n − 1 otherwise. Step three: build the interval as x̄ ± (critical × SE). For example, x̄ = 75, s = 12, n = 25, 95% CI: t0.025,24 ≈ 2.064, SE = 12/5 = 2.4, so 75 ± 4.95 → (70.05, 79.95).