Confidence Interval Calculators

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A confidence interval (CI) is a range of values calculated from sample data that, with a specified probability (the confidence level, typically 95%), contains the true population parameter. A 95% CI means: if we repeated the study many times and calculated a CI each time, 95% of those intervals would contain the true parameter. The CI is calculated as: estimate ± (critical value × standard error). For a sample mean with large n: 95% CI = x̄ ± 1.96 × SE. Confidence intervals are more informative than p-values alone because they show both statistical significance and the precision of the estimate.

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CI Formulas

For a mean (large n): 95% CI = x̄ ± 1.96 × SE = x̄ ± 1.96 × (s/√n)

For small n: use t* (from t-distribution with df = n−1) instead of 1.96: 95% CI = x̄ ± t* × SE.

For a proportion: 95% CI = p̂ ± 1.96 × √(p̂(1−p̂)/n).

Example: sample mean = 42.5, SD = 8.2, n = 36: SE = 8.2/√36 = 1.367. 95% CI = 42.5 ± 1.96 × 1.367 = 42.5 ± 2.68 = (39.8, 45.2).

Interpreting Confidence Intervals

Correct interpretation: 'We are 95% confident that the true population mean lies between 39.8 and 45.2.' NOT: 'There is a 95% probability that the true mean is in this interval' (the true mean is fixed; only the interval varies). CI and hypothesis testing: if the 95% CI for a mean difference does not include 0 → statistically significant at α = 0.05. If the 95% CI for a ratio (OR, RR) does not include 1 → statistically significant.

Width of Confidence Interval

Narrow CI: precise estimate (large n or small variability). Wide CI: imprecise estimate (small n or large variability). CI width = 2 × 1.96 × SE. Reduce CI width: increase n (most effective); reduce measurement variability; use more efficient study design.

CI vs. P-value

CI and p-value are complementary: CI shows effect size and precision; p-value shows whether the effect is 'statistically significant.' A CI that includes the null value corresponds to p > α.

Glossary

Confidence Interval (CI)
A range of values calculated from sample data likely to contain the true population parameter; 95% CI = estimate ± 1.96 × SE (large n); excludes null value → significant at α = 0.05.
Standard Error (SE)
SE = s/√n; the variability of the sample mean as an estimator of the population mean; determines CI width; CI width = 2 × z* × SE; halving SE requires 4× larger n.
Critical Value (z* or t*)
The multiplier for SE in CI calculation; z* = 1.96 for 95% CI (large n); z* = 2.576 for 99% CI; t* from t-distribution for small n (df = n−1); higher confidence level → larger critical value → wider CI.

Frequently Asked Questions

A confidence interval (CI) is a range of values calculated from sample data that is likely (at the confidence level, e.g., 95%) to contain the true population parameter. Formula for a mean: 95% CI = x̄ ± z* × (s/√n), where z* = 1.96 for 95% CI (large n); t* from t-distribution for small n. Example: 50 patients, mean blood pressure = 135 mmHg, SD = 20 mmHg. SE = 20/√50 = 2.83. 95% CI = 135 ± 1.96 × 2.83 = 135 ± 5.5 = (129.5, 140.5 mmHg). This interval is likely to contain the true mean blood pressure of the population from which these patients were sampled.

Correct interpretation: 'We are 95% confident that the true population parameter lies within this interval.' In frequentist statistics: the parameter is fixed (not random); the interval is random (changes from sample to sample). If we took 100 random samples and calculated a 95% CI from each, approximately 95 of those intervals would contain the true parameter. Common misinterpretation to avoid: 'There is a 95% probability that the true value is in this interval' — the true value is fixed (either in the interval or not); the 95% refers to the procedure, not the specific interval. From a Bayesian perspective, the analogous 'credible interval' can be interpreted probabilistically for the parameter.

CI width = 2 × z* × SE = 2 × z* × σ/√n. Since SE ∝ 1/√n: doubling n reduces the CI width by a factor of 1/√2 ≈ 0.71 (a 29% narrower CI). To halve the CI width: n must be quadrupled. Examples: n = 25: SE = 20/5 = 4.0; 95% CI width = 2 × 1.96 × 4.0 = 15.7. n = 100: SE = 20/10 = 2.0; width = 2 × 1.96 × 2.0 = 7.8. n = 400: SE = 1.0; width = 3.9. This quadratic relationship is why: very small samples give very wide, uninformative CIs; increasing n from 100 to 400 is needed to halve the CI width; precise clinical research requires large samples even when effects are large.

CI and p-value are mathematically related: A 95% CI that does NOT include the null value (0 for differences, 1 for ratios) → result is statistically significant at α = 0.05. A 95% CI that DOES include the null value → p > 0.05. Equivalence: p = 0.05 corresponds exactly to the 95% CI just touching the null value. p < 0.05 → 95% CI excludes null; p > 0.05 → 95% CI includes null. Advantages of reporting CI over p-value alone: CI shows effect size and direction (not just significance). CI shows precision of estimate. CI allows clinical interpretation — a statistically significant result with a very small CI may still be too small to matter clinically. Best practice: always report both the CI and the p-value, along with the effect size.