Confidence Interval Calculator

estimate ± critical value × standard error.

Estimate — your sample mean or proportion.

Standard error — the SD of the estimate. For a sample mean, SE = s/√n; for a proportion, SE = √(p̂(1−p̂)/n).

Critical value — how many SEs out you need to reach to capture the target confidence. Comes from the t (small n, unknown σ) or normal (large n or known σ) tables. For 95% it's about 1.96 for z, slightly larger for t.

If you know σ exactly, use z. This almost never happens in practice.

If you're estimating σ from the sample (using s), use t with df = n − 1. The t critical values are bigger than z critical values, which correctly widens the interval to account for uncertainty in s.

As n grows, t converges to z. By n = 60 the two are essentially identical; the choice mostly matters for n < 30.

Survey of 30 commuters: mean = 32 minutes, s = 8 minutes. 95% CI for the population mean?

SE = 8 / √30 ≈ 1.461.

t(0.975, df = 29) ≈ 2.045. Margin of error = 2.045 · 1.461 ≈ 2.99 minutes.

95% CI: 32 ± 2.99 → (29.0, 35.0).

Interpretation: if we repeated this survey many times, about 95% of such intervals would cover the true average commute time. We do NOT say there is a 95% probability the true mean lies in this specific interval (that's a Bayesian credible interval).

Wald formula: p̂ ± z · √(p̂(1−p̂)/n). Works well when np̂ ≥ 10 and n(1−p̂) ≥ 10.

Edge cases (p̂ near 0 or 1, small n) are better handled by Wilson or Agresti-Coull intervals. The Wald interval can dip below 0 or above 1, which is impossible.

Example: 240 of 400 customers say yes (p̂ = 0.60). 95% Wald CI = 0.60 ± 1.96 · √(0.60·0.40/400) = 0.60 ± 0.048 = (0.552, 0.648).

Saying 'there's a 95% chance the parameter is in this interval'. The parameter is fixed; the interval is random. The 95% refers to the long-run capture rate.

Using a small-sample interval when the data is heavily skewed. CIs assume the sampling distribution of the mean is roughly normal — true for any shape when n is large, but failing for small skewed samples.

Confusing CI width with effect size. Wider intervals just mean more uncertainty; they don't change the point estimate.

Treating overlapping intervals as 'no difference'. Two CIs can overlap and the difference still be statistically significant — use a CI on the difference instead.