When should I use a t interval vs a z interval for the mean?

When the population standard deviation is unknown and you use the sample SD s, the standard exact small-sample approach is the t interval: x̄ ± t_{α/2, n−1} · s/√n, assuming approximate normality. The z multiplier is often used as a large-sample normal approximation (sometimes taught as n ≥ 30 as a rough rule).

What formula does the proportion Wald interval use?

It uses p̂ = x/n and SE = √(p̂(1−p̂)/n), then p̂ ± z_{α/2} · SE, clipped to [0,1]. When p̂ is 0 or 1 the SE is 0, so the tool directs you to Wilson instead.

What is the Wilson score interval?

It inverts a score test on p and typically performs better than Wald for small n or p near 0 or 1. The calculator uses the standard closed form centered on an adjusted proportion with a widened margin that keeps endpoints in [0,1].

How do I enter the confidence level?

Use a decimal like 0.95 for 95% or type 95 and the tool treats it as 95%. The level must be strictly between 0 and 100%.

Does the calculator assume a simple random sample?

Yes—interpretations assume independent observations and a design where these textbook formulas apply. Cluster sampling, stratification, or finite-population corrections may need different SE formulas.

Are these intervals the same as credible intervals?

No. These are frequentist confidence intervals: over repeated sampling, a chosen fraction of such intervals would contain the parameter under the model. Bayesian credible intervals use posterior probabilities and can differ numerically.

What is the margin of error?

For symmetric intervals around x̄ or p̂, the margin is half the interval width: critical value × SE (with Wilson, the half-width is not always exactly “z × SE” of p̂ because the center is adjusted).

Can I use this for difference of two means?

This page handles one mean or one proportion at a time. For μ₁−μ₂ you need the appropriate SE for the difference (pooled or unpooled) and often a t reference—use dedicated two-sample tools or software.

How accurate is the t critical value?

t quantiles are found by inverting the Student t CDF computed via the regularized incomplete beta function (standard relationship to the beta distribution). This matches common tables and software to high practical accuracy.

Where are related statistics calculators on thecalcs?

See statistical power, effect size, t-test, z-score percentile, standard error, and inverse normal calculators in Math & Science—linked below.

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Statistics

Confidence Interval Calculator - Mean & Proportion CIs

Free confidence interval calculator on our Math & Science hub: t and z style intervals for a mean, plus Wald and Wilson intervals for a binomial proportion—with SEs, critical values, and df where relevant.

Last updated: April 13, 2026

Mean: t (unknown σ) or z multiplier

Proportion: Wald or Wilson score

Flexible confidence: 0.95 or 95

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Confidence Interval Calculator

Mean intervals (t or z) and proportion intervals (Wald or Wilson). Enter confidence as 0.95 or 95.

Interval type

Sample mean (x̄)

Sample SD (s)

Sample size (n)

Confidence level

Uses t_{df=n−1} with x̄ ± t·s/√n. Assumes approximate normality or moderate n for the t procedure.

Results

95% confidence interval

[47.0128, 52.9872]

Point estimate: 50

1.4606

Critical

t_{0.9750, 29} ≈ 2.0452

df = 29 (t distribution)

Assumptions

Random sampling, correct model (e.g. Binomial for proportions), and for means roughly normal data or enough n for the CLT when using z-style intervals.

Confidence interval modes

Mean — t

x̄ ± t·s/√n

Uses df = n − 1. Default choice when σ is unknown and the population is not strongly skewed (or n is moderate/large).

Mean — z

Same SE, z multiplier

Applies the normal critical value with s/√n. Useful for large-sample teaching parallels; prefer t for small n when σ is unknown.

Proportion — Wald

Classic p̂ ± z·SE

Simple and familiar; avoid when p̂ is 0 or 1, or when n is small—Wilson is preferred there.

Proportion — Wilson

Score-type interval

Better coverage behavior near 0 and 1; endpoints are automatically constrained to [0,1].

What you see

Interval, SE, critical value

Every mode reports the point estimate, standard error (mean or p̂ Wald SE), and the z or t multiplier used.

Not for every design

Scope limits

Regression coefficients, differences of means with complex designs, and exact Clopper–Pearson limits need specialized tools.

Quick example

Mean t-interval: x̄ = 50, s = 8, n = 30, 95% confidence

[47.0128, 52.9872]

How this confidence interval calculator works

Mean intervals use the standard error s/√n. The t implementation inverts the Student t cumulative distribution using the beta function relationship. Proportion Wilson uses the usual closed form with the standard normal quantile for tail area α/2. For more statistics tools, browse our Math & Science calculators.

Core formulas

Mean t: x̄ ± t_{1-α/2, n-1} · s/√n
Mean z: x̄ ± z_{1-α/2} · s/√n
Wald: p̂ ± z_{1-α/2} √(p̂(1-p̂)/n)
Wilson: closed-form score interval (reported as [L, U])

References

Agresti & Coull — Approximate is better than “exact” for interval estimationWilson and related binomial intervals
Confidence intervalDefinitions and frequentist interpretation
Moore, McCabe & Craig — Introduction to the Practice of Statisticst intervals and one-proportion intervals

Next: t-test calculator and standard error.

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Worked example

Wilson 95% for a survey item

x = 45, n = 100

p̂ = 0.45; Wald would use SE ≈ √(0.45×0.55/100).
Wilson adjusts the center and width for better small-to-moderate n behavior.
Select “Proportion — Wilson” in the form and enter 45, 100, 0.95 to see [L, U].

Frequently Asked Questions

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Statistics

Confidence Interval Calculator - Mean & Proportion CIs

Last updated: April 13, 2026

Mean: t (unknown σ) or z multiplier

Proportion: Wald or Wilson score

Flexible confidence: 0.95 or 95

Need survey or analytics calculators for your product? Get a Quote

Confidence Interval Calculator

Mean intervals (t or z) and proportion intervals (Wald or Wilson). Enter confidence as 0.95 or 95.

Interval type

Sample mean (x̄)

Sample SD (s)

Sample size (n)

Confidence level

Uses t_{df=n−1} with x̄ ± t·s/√n. Assumes approximate normality or moderate n for the t procedure.

Results

95% confidence interval

[47.0128, 52.9872]

Point estimate: 50

1.4606

Critical

t_{0.9750, 29} ≈ 2.0452

df = 29 (t distribution)

Assumptions

Random sampling, correct model (e.g. Binomial for proportions), and for means roughly normal data or enough n for the CLT when using z-style intervals.

Confidence interval modes

Mean — t

x̄ ± t·s/√n

Uses df = n − 1. Default choice when σ is unknown and the population is not strongly skewed (or n is moderate/large).

Mean — z

Same SE, z multiplier

Applies the normal critical value with s/√n. Useful for large-sample teaching parallels; prefer t for small n when σ is unknown.

Proportion — Wald

Classic p̂ ± z·SE

Simple and familiar; avoid when p̂ is 0 or 1, or when n is small—Wilson is preferred there.

Proportion — Wilson

Score-type interval

Better coverage behavior near 0 and 1; endpoints are automatically constrained to [0,1].

What you see

Interval, SE, critical value

Every mode reports the point estimate, standard error (mean or p̂ Wald SE), and the z or t multiplier used.

Not for every design

Scope limits

Regression coefficients, differences of means with complex designs, and exact Clopper–Pearson limits need specialized tools.

Quick example

Mean t-interval: x̄ = 50, s = 8, n = 30, 95% confidence

[47.0128, 52.9872]

How this confidence interval calculator works

Core formulas

Mean t: x̄ ± t_{1-α/2, n-1} · s/√n
Mean z: x̄ ± z_{1-α/2} · s/√n
Wald: p̂ ± z_{1-α/2} √(p̂(1-p̂)/n)
Wilson: closed-form score interval (reported as [L, U])

References

Agresti & Coull — Approximate is better than “exact” for interval estimationWilson and related binomial intervals
Confidence intervalDefinitions and frequentist interpretation
Moore, McCabe & Craig — Introduction to the Practice of Statisticst intervals and one-proportion intervals

Next: t-test calculator and standard error.

Get Custom Calculator for Your Platform