Free calculator
Confidence interval calculator
Build a two-sided confidence interval for a population mean (population σ known → z; σ unknown → t), a binomial proportion (Wald), or paste raw data and we compute x, s, and n then a t-interval. Pick a confidence level, read margin of error, standard error, and bounds—plus Google Sheets and Excel patterns for CONFIDENCE.NORM vs CONFIDENCE.T.
When to use this calculator
Quick interval estimates that mirror what you would set up in a spreadsheet—without turning this page into a full statistics platform.
- Homework or review: z vs t mean intervals and Wald proportion intervals at common confidence levels.
- Check CONFIDENCE.NORM vs CONFIDENCE.T in Sheets or Excel against the same α, σ or s, and n you type here.
- Paste a short lab or ops list and read x, s, n, and a t interval in one pass—then compare with
STDEV.S/AVERAGEcells. - Need variance or SD only? Open the standard deviation or variance calculator from the tools directory—same paste rules as the raw-data field here.
A confidence interval is a range of plausible values for a population parameter based on sample information, at a chosen confidence level (how often the method would cover the true parameter across repeated samples—not a probability that this interval “contains” the truth).
Mean — σ known (z)
SE = σ / √n; margin of error = z\* × SE where z\* is the two-sided normal critical value; bounds x ± MOE.
Mean — σ unknown (t)
SE = s / √n with sample s; MOE = t\* × SE where t\* uses df = n − 1; bounds x ± MOE.
Proportion (Wald)
p̂ from data; SE = √(p̂(1 − p̂)/n); MOE = z\* × SE; bounds clamped to [0, 1]. This is the common textbook interval—not a substitute for Wilson or exact binomial intervals when n is small or p̂ is near 0 or 1.
Sheets / Excel alignment
Map σ known to CONFIDENCE.NORM and σ unknown to CONFIDENCE.T, each with α = 1 − confidence and your σ or s and n. Your workbook’s language pack may show localized names—use Insert function to match.
We do not compute two-sample differences, paired tests, prediction intervals, Wilson/Agresti–Coull/Clopper–Pearson intervals, finite-population corrections, or bootstrap CIs in v1.
For spread only on the same list shape (no interval), open the standard deviation calculator.
For variance-first summaries, open the variance calculator.
For z-scores and normal tail areas from μ and σ, open the z-score calculator.
For median, mode, and range on the same pasted list (without interval math on this page), open the mean median mode range calculator.
To plan a minimum n and margin of error for one proportion before you collect data, open the sample size calculator (this page stays on intervals from data you already have).
FAQs cover frequentist interpretation, z vs t, Wald limits, and CONFIDENCE.NORM vs CONFIDENCE.T.
Google Sheets & Excel
CONFIDENCE.NORM (or legacy CONFIDENCE) matches σ known setups: it returns the half-width of a two-sided interval at significance α = 1 − confidence when you supply σ and n. CONFIDENCE.T matches σ unknown with sample s: same half-width role with t on n − 1 degrees of freedom. Replace A1:A99 with your data range where examples use a block reference.
=CONFIDENCE.NORM(0.05,2.7,100)First argument is α for a two-sided interval (0.05 → 95% confidence). Second is σ; third is n. Multiply by 2 if you mistakenly need a naive total width check—usually you add/subtract this value from x.
=CONFIDENCE.T(0.05,15,25)α is 1 − confidence; second argument is s; third is n. Pairs with T.INV.2T thinking on df = n − 1.
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Frequently asked questions
What is a confidence interval?
A confidence interval is an estimate of a plausible range for a population parameter (often a mean or proportion) using sample data. The confidence level describes how the method behaves across repeated samples, not the odds that any one interval is “right” on its own.
Does 95% confidence mean there is a 95% probability the parameter is inside this interval?
Not in the textbook frequentist sense. 95% means that if you could repeat the same sampling process many times and build a 95% interval each time, about 95% of those intervals would capture the true parameter. After you see the data, the parameter is fixed—the interval either covers it or it does not.
When should I use z vs t for a mean?
Use z when the population σ is known (rare in practice) or when a course explicitly gives σ. Use t with sample s and df = n − 1 when σ is unknown—including small samples. This page always uses t for the σ unknown and raw data paths (no “switch to z after 30” shortcut).
What makes a confidence interval wider or narrower?
Wider intervals come from higher confidence (larger critical values), a smaller sample size n, or more variability (σ or s in the mean case; p̂ near 0.5 in the Wald proportion case). A larger n and lower variability tighten the interval, all else equal.
Why might Wald proportion intervals look odd for a small n or p̂ near 0 or 1?
Wald uses a normal approximation to the sampling distribution of p̂. With a small n or extreme p̂, better tools use Wilson, Agresti–Coull, or exact (Clopper–Pearson) intervals. We do not ship those in v1—this page is transparent about the Wald choice.
How is standard error related to the confidence interval?
The margin of error is critical value × standard error. For a mean, SE is σ/√n (z path) or s/√n (t path). For a Wald proportion, SE = √(p̂(1−p̂)/n).
What is the difference between CONFIDENCE.NORM and CONFIDENCE.T?
Both return the half-width for a two-sided interval at significance α = 1 − confidence with arguments (α, stdev, n). CONFIDENCE.NORM treats the third inputs as population σ and uses a normal critical value. CONFIDENCE.T uses the sample standard deviation s and a t distribution on n − 1 degrees of freedom.
Can I use this for two-sample or paired differences?
No. v1 is for one-sample mean or proportion setups (or raw data for one sample). Two-sample, paired, regression, or bootstrap intervals belong in dedicated workflows or software.
How is this different from your standard deviation calculator?
The standard deviation page summarizes spread (s or σ conventions) for a pasted list. This page targets interval estimation for means and proportions and shows critical values, SE, and MOE.
Which Google Sheets or English Excel functions line up with this page?
CONFIDENCE.NORM and CONFIDENCE.T return the half-width at α = 1 − confidence. Pair them with AVERAGE / STDEV.S on the same range when you are rebuilding a t interval from raw data.
What are the German Excel names for these functions?
Common Excel (Deutsch) names include KONFIDENZ.NORM and KONFIDENZ.T—confirm in Formulas → Insert function on your build.
What are the French Excel names for these functions?
Common Excel (français) names include INTERVALLE.CONFIANCE.NORM and INTERVALLE.CONFIANCE.T—verify on your install.
Is this professional statistical advice?
No. It is a free educational calculator. For regulated work, research protocols, or product decisions, follow qualified experts and your organization’s standards.