How do I calculate a confidence interval for a proportion?

For large samples, use the Wald formula: p +/- z * sqrt(p(1-p)/n). For small samples or extreme proportions, use Wilson or Clopper-Pearson, they give better coverage. The calculator runs all four (Wald, Wilson, Agresti-Coull, exact) so you can compare; Wilson is the recommended default in R's prop.test.

What does a 95% confidence interval actually mean?

A 95% CI means that if you repeated the same study many times, 95% of the resulting intervals would contain the true parameter. It does NOT mean there is a 95% probability the parameter lies in this specific interval, that is a Bayesian credible interval. Use the Bayes Theorem or Bootstrap CI tools when you need probability statements.

When should I use a t-interval versus a z-interval?

Use the t-interval whenever the population standard deviation is unknown, which is almost always in practice. The z-interval applies only when sigma is known. The two converge as n grows; the difference matters most when n is below 30. R's t.test is the practical default.

Confidence Interval Calculator

A confidence interval gives a range of plausible values for what is really true in the population, not just a single point estimate. Pick a flavor (mean, proportion, difference, correlation, regression coefficient, Poisson rate, variance), drop in your stats, and get the interval, half-width, and reproducible R for every method.

8 flavors · one tool · Wilson · Clopper-Pearson · Newcombe · Welch · exact Poisson · Runs in your browser

Try a real-world example to load.

📐 Mean test score

A class of 30 students took a test; mean = 72, SD = 11. We want a 95% CI for the true class-level mean.

Output

Pick a flavor and enter inputs.

Z-scores and tail areas → Translate to effect size → How big a sample do I need? →

R code RUNNABLE

R Reproduce in R

Confidence interval INTERACTIVE

Inference

Read more Anatomy of a confidence interval

Live recap - your inputs plugged in

Enter inputs above to see the derivation chain.

CI = estimate ± critical · SE

The general form. Most CIs decompose into a point estimate, a critical value from a sampling distribution (z, t, χ², F), and a standard error. The critical value depends on confidence level and (for some flavors) df; the SE depends on the data.

INα from your Confidence level (α = 1 − conf), critical from the appropriate distribution, SE from the formula below.OUTLower/upper bounds shown in the big CI display; half-width shown as the "± margin of error" in the result panel.

SE_mean = s / √n df = n − 1 critical = t_{1−α/2, df} CI = x̄ ± critical · SE

Mean (one sample). If you don't know σ - and you almost never do - use the t-distribution. For n > 30 the t and z critical values are essentially identical.

INx̄ = your Sample mean x̄, s = your Sample SD s, n = your Sample size n.OUTSE and t-critical printed under "point = … · half-width = …"; CI bounds in the headline.

Welch: SE = √(s₁²/n₁ + s₂²/n₂) Pooled: sp² = ((n₁−1)s₁² + (n₂−1)s₂²)/(n₁+n₂−2) SE = sp · √(1/n₁ + 1/n₂) df_Welch ≈ Satterthwaite CI = (x̄₁ − x̄₂) ± t_{1−α/2, df} · SE Paired: d̄ ± t · sd/√n, df = n−1

Difference of means. Welch (default) doesn't assume equal variances and adjusts df. For paired data (same subject measured twice), it reduces to a one-sample t-CI on the within-pair differences.

INIndependent: Group 1 + Group 2 with the Use Welch toggle. Paired: Number of pairs n, Mean difference, SD of difference.OUTMean diff is the point shown; SE and df in the aux line below the bounds.

p̂ = x / n Wilson score CI: center = (p̂ + z²/(2n)) / (1 + z²/n) half = z · √(p̂(1−p̂)/n + z²/(4n²)) / (1 + z²/n) CI = center ± half

Wilson is the right default for proportions. Stays inside [0, 1], performs well when p̂ is near 0 or 1, and matches what prop.test(..., correct = FALSE) reports. Wald is the textbook formula and is misleading at extremes - we still show it, but boxed for comparison.

INx = Successes x, n = Trials n, z from your Confidence level.OUTWilson row in the "All methods compared" table is the highlighted recommended one.

Clopper-Pearson exact: lower = qbeta(α/2, x, n−x+1) upper = qbeta(1−α/2, x+1, n−x)

Clopper-Pearson is the exact CI for a proportion via the binomial distribution. Always covers at least the nominal level (sometimes more) - conservative but never anti-conservative. Use when guaranteed coverage matters (regulatory, medical).

INSame Successes x and Trials n; α = 1 − conf level.OUTSecond row of the methods table.

Newcombe (independent): w₁ = Wilson(x₁, n₁); w₂ = Wilson(x₂, n₂) lo = (p̂₁ − p̂₂) − √((p̂₁−w₁_lo)² + (w₂_hi−p̂₂)²) hi = (p̂₁ − p̂₂) + √((w₁_hi−p̂₁)² + (p̂₂−w₂_lo)²) McNemar (paired): p̂_diff = (b − c) / n SE = √((b+c) − (b−c)²/n) / n CI = p̂_diff ± z · SE

Difference of proportions. For independent groups, Newcombe's score CI stays in [-1, 1]. For paired (same subjects under two conditions), use the McNemar-style discordant-pair formula.

INIndependent: Group 1 x₁, n₁ + Group 2 x₂, n₂. Paired: b (yes-no) + c (no-yes).OUTRecommended row of the methods table; for paired, also the McNemar χ² and p-value in the summary.

CI = β̂ ± t_{1−α/2, df} · SE(β̂)

Regression coefficient. Same skeleton, but the SE comes from the model's variance-covariance matrix and df is the residual df. confint() in R does exactly this for an lm fit.

INβ̂ = Estimate β̂, SE = Standard error SE(β̂), df = Residual df.OUTIf 0 lies outside [lo, hi], the coefficient is "significant" at α; the summary states this in plain English.

Exact (Garwood / χ² inversion): λ̂ = x / T lower = qchisq(α/2, 2x) / (2T) upper = qchisq(1−α/2, 2(x+1)) / (2T)

Poisson rate. The exact CI inverts the χ² distribution and works at any count, including x = 0. Wald (= λ̂ ± z·√x/T) is fine only for large x.

INx = Event count x, T = Exposure (person-time) T.OUTExact row (highlighted) in the methods table; λ̂ is the point estimate displayed.

CI for σ²: [(n−1)s² / χ²_{1−α/2, n−1}, (n−1)s² / χ²_{α/2, n−1}] CI for σ = √CI for σ²

Variance / SD. Built on the χ² distribution; assumes underlying normality. Highly sensitive to that assumption - a single outlier can dominate s². Don't trust this CI if a Q-Q plot looks bad.

INn = Sample size n, s = Sample SD s.OUTTwo rows: σ² and σ in the methods table; the SD row is highlighted as the more interpretable one.

Fisher z-transform: z_r = atanh(r), SE_z = 1/√(n−3) CI_z = z_r ± z_{1−α/2} · SE_z CI_ρ = tanh(CI_z)

Correlation. r's sampling distribution is skewed near ±1, so we transform to Fisher's z (where it's roughly normal), build the CI there, then back-transform.

INr = Sample correlation r, n = Sample size n.OUTBounds (asymmetric on the r-scale) in the headline; Fisher z and SE_z appear in the aux line.

Caveats When this is the wrong tool

If you have…: Use instead
Bootstrap CIs from raw data: R: boot::boot.ci() with type = "bca" - coming as a separate tool in Batch 8.
Bayesian credible interval: Different paradigm; we don't include here. Try a brms tutorial.
CI for ratio of means / variances: Fieller's theorem (means) or F-based (variances) - coming in Batch 6.
Survival times - Kaplan-Meier confidence band: Survival analysis tool - out of scope here.
Median CI: Order-statistic-based; needs raw data. Add as a v2 paste-mode option.

Confidence Interval Calculator

How we got there

Anatomy of a confidence interval