Confidence Interval Tool
Calculate confidence intervals for population means, paired differences, proportions, differences between groups, and variance. Choose confidence levels, statistical methods, and interval direction, then examine numerical bounds, sampling distribution visuals, interpretation, and exportable results in your browser.
Table of Contents
Highlights
- Six interval types covering means, proportions and variance
- Multiple methods including Welch, Wilson and exact binomial
- Preset and custom confidence levels with one-sided options
- Interactive confidence-region chart and detailed statistics table
- Export CSV, JSON, SVG or PNG, or copy a summary
How Confidence Interval Tool Works?
A confidence interval uses a sample estimate and its sampling uncertainty to describe plausible values of a population parameter. Select an interval type, a confidence level (80%, 90%, 95%, 98%, 99%, 99.9%, or a custom percentage), a supported method, and either a two-sided interval or a one-sided lower or upper bound.
For a sample mean, enter the sample mean, standard deviation and sample size. With a known population standard deviation, the normal z method is available; with an estimated standard deviation, Student’s t uses n − 1 degrees of freedom. The general two-sided construction is:
Here c is the chosen normal or t critical value, s is the entered standard deviation, n is the sample size, and α = 1 − confidence level. For paired observations, enter the mean and standard deviation of within-pair differences and the number of pairs.
For two independent means, the point estimate is the first group mean minus the second. Welch’s t accommodates unequal variances, pooled t assumes equal population variances, and z treats the entered group standard deviations as known population values. For a single proportion, enter successes and trials; for a difference in proportions, enter both groups’ successes and trial counts.
| Quantity | Available method(s) | Important input(s) |
|---|---|---|
| One mean | Student’s t; normal z | Mean, standard deviation, sample size |
| One proportion | Wilson, Wald, Agresti–Coull, Clopper–Pearson exact | Successes, trials |
| Two means | Welch t, pooled t, normal z | Both means, standard deviations, sample sizes |
| Paired mean difference | Paired t | Mean difference, difference SD, number of pairs |
| Two proportions | Newcombe–Wilson, Wald | Successes and trials in both groups |
| Population variance and SD | Chi-square | Sample SD, sample size |
For normal-population variance, chi-square quantiles define the variance limits, and square roots of these limits give standard deviation limits:
Results show the interval bounds, estimate, standard error, critical value, applicable degrees of freedom, an interactive sampling-distribution chart, and explanatory notes. You can export result data as CSV or JSON, save the chart as SVG or PNG, or copy a text summary.
Practical Use Cases
Medical researchers: Estimate a mean treatment response, a paired before-and-after change, or the difference between two independent treatment groups.
Marketing analysts: Use successful conversions and total visitors to bound a campaign conversion proportion, or compare the conversion proportions of two campaigns.
Quality engineers: Quantify uncertainty around the variability of a normally distributed manufacturing process using a chi-square interval for variance or standard deviation.
Teachers and education researchers: Examine uncertainty around mean test scores or compare average scores between two independent classes.
Pollsters and survey researchers: Build intervals for the proportion of respondents selecting an answer, particularly using Wilson or Clopper–Pearson when counts are small.
Product teams: Compare the proportions of users who complete a task in two independent user groups using a Newcombe–Wilson interval.
Accuracy and Limitations
Normal z and t intervals rely on representative independent samples and appropriate distributional conditions; pooled t also assumes equal population variances. Chi-square variance intervals require normally distributed observations. Wilson and Newcombe–Wilson generally handle small or extreme proportions better than Wald, while Clopper–Pearson can be conservative. The tool accepts summary statistics rather than raw observations, so it cannot diagnose skewness, dependence, outliers or sampling bias. The displayed sampling curve is an illustrative normal-shaped visual, not an exact t, beta or chi-square density; its shading should not be read as a numerical coverage calculation. Numerical quantiles are approximated in JavaScript, and displayed results are rounded.
