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Confidence interval calculator (mean or proportion)

Get a CI straight from summary numbers — no raw data needed. Means use the t distribution; proportions use the Wilson score interval, which behaves well even for small samples and extreme percentages.

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When to use this

Use this whenever you have summary statistics but not raw data: reporting a CI next to a mean or percentage in your own results (APA 7 and most journals now expect intervals, not just point estimates), reconstructing a CI from a published paper that only reported M, SD and n, or checking a claim in a manuscript you are reviewing. Confidence intervals communicate precision — the same 62% approval means something very different from n = 100 than from n = 10,000.

Key assumptions

  • Mean CI: observations are independent and roughly normal (or n is large enough for the CLT, usually n ≥ 30).
  • Proportion CI: independent binary outcomes; the Wilson interval is accurate even for small n and p near 0 or 1.
  • The stated coverage is about the procedure, not any single interval.

Common mistakes

  • "There is a 95% probability the true mean is in this interval" — the true mean is fixed; the interval is what varies.
  • Using the Wald (normal approximation) interval for proportions with small n — it can extend past 0% or 100% and undercovers badly.
  • Judging a group difference by whether two CIs overlap — overlapping CIs can still be significantly different.

Frequently asked questions

Why does this use the Wilson interval for proportions?
The textbook Wald interval p ± 1.96√(p(1−p)/n) fails exactly where CIs matter most: small samples and extreme proportions (with 0 events it collapses to a width of zero). The Wilson score interval maintains close-to-nominal coverage across the board and is what statisticians actually recommend (Agresti & Coull, 1998).
Can I get a CI for the difference between two groups here?
Not directly — a difference needs both groups’ summary statistics at once. Run the independent samples t-test calculator (it reports the CI for the mean difference) or the effect size calculator (it reports the CI for Cohen’s d).
What if my data are skewed?
For clearly skewed outcomes (reaction times, income, counts) the mean CI from summary statistics can be misleading at small n. Options: report the median with a bootstrap CI from raw data, or analyze on a log scale and back-transform.

Related free tools

Sample size calculatorSample size for two groupsEffect size (Cohen's d)Descriptive statisticsOne-sample t-testIndependent t-testPaired t-testz-table (standard normal)t-table (critical values)Chi-square table

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