Confidence Interval Calculator — Mean, Proportion & Difference
Sample summary
Confidence interval result
Your interval will appear here.
Choose a parameter, enter sample summaries, and calculate.
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Confidence interval formulas and methods
| Parameter | Method | Two-sided interval |
|---|---|---|
| Mean, unknown σ | One-sample Student t | x̄ ± t1−α/2,n−1 × s/√n |
| Mean, known σ | Normal z | x̄ ± z1−α/2 × σ/√n |
| One proportion | Wilson score | Center and half-width are adjusted by 1 + z²/n, keeping limits in [0, 1]. |
| Difference in means | Welch t | (x̄₁ − x̄₂) ± t1−α/2,ν × √(s₁²/n₁ + s₂²/n₂) |
| Difference in proportions | Newcombe method 10 | Combines the two uncorrected Wilson score intervals by the square-and-add method. |
α is 1 minus the confidence level. Welch degrees of freedom use the Welch–Satterthwaite approximation. Wilson and Newcombe limits can be asymmetric, so the calculator reports separate distances from the estimate to each limit.
How to use the calculator
- Choose Mean, Proportion, or Difference.
- Enter summary statistics from one sample or two independent groups.
- Enter the confidence percentage. For a mean, identify whether the SD is a sample estimate or known population σ.
- Calculate and review the limits, method, standard error, critical value, assumptions, and graphic.
- Use Copy or Download CSV to save the result without uploading the input values.
How to interpret the interval
A 95% confidence procedure produces intervals that contain the true parameter in about 95% of repeated samples when its assumptions hold. After one interval has been calculated, the parameter is fixed; avoid saying there is a 95% probability that it lies inside this particular frequentist interval.
Width reflects sampling uncertainty, not every source of error. Larger samples usually narrow intervals, while greater variability and higher confidence widen them. Bias, nonresponse, confounding, dependence, measurement error, and poor sampling are not repaired by a narrow interval.
Assumptions and limits
- Observations and the two groups, where applicable, are independent. Clustered, repeated, matched, or paired data require methods that model that dependence.
- Mean intervals assume a representative sample. Exact small-sample t coverage additionally relies on an approximately normal population; larger samples are often more robust, but severe skew and outliers still matter.
- Proportion intervals assume a binomial count with a fixed number of trials and a common success probability. Wilson and Newcombe intervals are approximate, not exact.
- The Wald proportion option is included for comparison and can have poor coverage or exceed 0%–100%; its displayed limits are not silently clipped.
- For paired differences, calculate one difference per pair, then use the Mean tab with the mean and sample SD of those differences.
Statistical limit: This calculator summarizes sampling uncertainty under the selected model. It does not establish causality, eliminate bias, assess study quality, or determine whether an effect is practically important.
Frequently asked questions
What does a 95% confidence interval mean?
Under repeated sampling, about 95% of intervals produced by the same procedure would contain the true parameter when the assumptions hold. It is not a 95% posterior probability for the fixed parameter.
Should I use z or t for a mean?
Use Student t when the population standard deviation is unknown and you enter the sample SD. Use z only when population σ is genuinely known or established independently.
Why is Wilson the default for a proportion?
The Wilson score interval usually has better coverage than the simple Wald interval, especially for smaller samples and estimates near 0 or 1, and its limits remain between 0 and 1.
What method is used for two independent means?
The Welch t interval. It does not assume equal variances and uses Welch–Satterthwaite degrees of freedom.
Can I use Difference for paired observations?
Not directly. Calculate the difference within every pair, then treat that set of differences as one sample in the Mean tab.
Why can a Wald proportion interval exceed 0% or 100%?
The Wald method is a symmetric normal approximation around the observed proportion and does not respect the parameter boundaries. Wilson is the recommended default.
Does a confidence interval prove practical importance?
No. Compare the interval with a defensible subject-matter threshold and consider design quality, measurement, bias, costs, and consequences.
Are my inputs tracked?
No. Calculations run locally in your browser, and entered values are not sent, stored, or attached to analytics events.
Methodology and references
Last reviewed: August 4, 2026. The mean formulas and t critical-value definition were checked against NIST. Proportion method selection and difference-of-proportions construction were checked against Newcombe's comparative studies.