Confidence Interval Calculator

Mastering Confidence Intervals: Complete Guide

A confidence interval (CI) gives an estimated range of values which is likely to include an unknown population parameter. The width of the interval reflects the precision of the estimate—narrower intervals indicate more precise estimates. This Professional Confidence Interval Calculator computes CIs for population means (using z or t critical values) and for population proportions, with a clear visual display and step‑by‑step derivation.

Confidence Interval for a Mean

When the population standard deviation σ is known, or the sample size is large (n ≥ 30), the CI is x̄ ± z* · (σ/√n). When σ is unknown and estimated by the sample standard deviation s, and the sample size is small, the t‑distribution is used: x̄ ± t* · (s/√n). The calculator automatically selects the appropriate distribution, or you can override it.

Confidence Interval for a Proportion

For a population proportion, the Wald interval is p̂ ± z* · √[p̂(1−p̂)/n], where p̂ = x/n is the sample proportion. This method assumes the sample is large enough for the normal approximation (np̂ ≥ 10 and n(1−p̂) ≥ 10). The calculator displays the interval in both decimal and percentage forms.

Interpreting the Confidence Level

The confidence level (e.g., 95%) indicates the long‑run success rate of the method. If you were to draw 100 different samples and compute a 95% CI from each, about 95 of those intervals would capture the true parameter. It does not mean there is a 95% chance the true value lies within a single computed interval.

Factors Affecting Interval Width

• Sample size (n): Larger n → smaller standard error → narrower interval.
• Confidence level: Higher confidence → larger critical value → wider interval.
• Variability (σ or s): Greater variability → larger standard error → wider interval.