Confidence Intervals Explained

The correct interpretation

A 95% confidence interval means: if you repeated the sampling procedure many times and built an interval each time, about 95% of those intervals would contain the true population parameter. It does NOT mean there is a 95% probability that the true value lies inside this particular interval — the true value is fixed, and the interval either contains it or it does not.

This subtle distinction matters because the confidence is in the procedure, not in any single interval.

Interval for a mean

For a mean with known population standard deviation, the interval is x̄ ± z·σ/√n. When σ is unknown and estimated from the sample, replace z with the t critical value and σ with the sample standard deviation s: x̄ ± t·s/√n, with n−1 degrees of freedom.

The PanelRoster confidence interval calculator applies the normal approximation for proportions; the t-based mean interval is the standard textbook construction.

Interval for a proportion

For a proportion, the Wald interval is p̂ ± z·√(p̂(1−p̂)/n), where p̂ is the sample proportion and z is the confidence-level z-score (1.96 for 95%). The interval is most reliable when both p̂n and n(1−p̂) are reasonably large (commonly at least 5–10); for small samples, exact or score methods are preferable.

What changes the width

Three things widen or narrow an interval: the confidence level (higher confidence → wider), the sample size (larger n → narrower, in proportion to 1/√n), and the variability of the data (more spread → wider). You cannot fix precision by raising confidence; you fix it by collecting more data or reducing measurement error.