calculate-confidence-interval

Calculate Confidence Interval

Compute and correctly interpret confidence intervals to quantify estimation uncertainty for means, proportions, differences, and regression parameters.

Why This Is Best Practice

Adopted by: APA Publication Manual 7th ed. (requires CIs for all primary statistics), CONSORT clinical trial reporting standards, New England Journal of Medicine statistical reporting guidelines, Cochrane Collaboration meta-analysis methods.

Impact: Gardner & Altman (BMJ 1986) demonstrated that CIs convey more information than p-values alone; NEJM mandated CIs in 1988 and subsequent studies showed 40% reduction in dichotomous "significant/not significant" misinterpretation. APA made CIs required in 2010.

Why best: A CI provides the precision of an estimate (width) and its plausible range simultaneously, whereas a p-value alone conflates effect size with sample size. CIs are directly usable for clinical or practical significance assessment.

Sources: Neyman Philos Trans R Soc (1937); Gardner & Altman BMJ 292:746 (1986); APA 7th ed. §7.5.

Steps

1. Identify the estimator and its distribution — determine what parameter you are estimating (mean, proportion, difference, correlation, regression coefficient) and the sampling distribution of the estimator (normal, t, chi-square, F, binomial).

2. Choose the confidence level — use 95% as default (α=0.05); use 99% for regulatory submissions or when Type I error must be minimized; use 90% only when justified by domain convention. State the level explicitly.

3. Calculate for a population mean (σ unknown) — CI = x̄ ± t_(α/2, n−1) × (s/√n), where t_(α/2, n−1) is the critical value from the t-distribution with n−1 degrees of freedom; use z=1.96 only when n ≥ 30 and σ is known.

4. Calculate for a proportion — for n×p ≥ 5 and n×(1−p) ≥ 5, use Wilson interval (preferred over Wald): p̂ ± z_(α/2) × √[p̂(1−p̂)/n]. For small samples, use exact Clopper-Pearson method.

5. Calculate for difference of two means — CI = (x̄₁ − x̄₂) ± t × SE_diff where SE_diff = √(s₁²/n₁ + s₂²/n₂); use Welch-Satterthwaite degrees of freedom if variances are unequal.

6. Calculate using software — in R: t.test(x, conf.level=0.95)$conf.int; in Python: scipy.stats.t.interval(0.95, df=n-1, loc=mean, scale=sem); in SPSS: enable "Confidence intervals" in Descriptives/Compare Means.

7. Bootstrap when assumptions fail — for non-normal distributions or complex estimators, use percentile bootstrap: resample with replacement B=10,000 times, compute statistic each time, take 2.5th and 97.5th percentiles as CI bounds.

8. Report completely — state: estimate [lower CI, upper CI], 95% CI. Example: "Mean difference = 3.2 kg [1.8, 4.6], 95% CI." Always include the confidence level; "95% CI" is not implicit.

Rules

• Never interpret a 95% CI as "there is a 95% probability the true value lies in this interval" — frequentist CIs are properties of the procedure, not Bayesian credible intervals. The correct interpretation: "95% of intervals constructed this way will contain the true parameter."
• A CI that includes zero (for differences) or 1.0 (for ratios) does not "prove no effect" — it means the data are consistent with zero, not that zero is the true value.
• Report the CI alongside the point estimate; a CI without a point estimate is incomplete.
• Do not adjust CI width post-hoc based on the result — CI width is fixed by the pre-specified α and n.

Common Mistakes

• Using z instead of t for small n — z=1.96 underestimates the critical value for n<30, yielding CIs that are too narrow (undercoverage).
• Wald interval for proportions near 0 or 1 — the Wald interval performs poorly when p̂ < 0.1 or > 0.9; use Wilson or Clopper-Pearson.
• Conflating CI with prediction interval — a CI bounds the population mean; a prediction interval (wider) bounds a single future observation.
• Multiple comparison inflation — reporting 10 CIs at 95% confidence gives ~40% familywise error; use Bonferroni correction (95%/k) or simultaneous confidence bands.

When NOT to Use

• When a Bayesian credible interval better matches the inferential question (use posterior distribution with specified prior)
• When the goal is prediction of individual outcomes (use prediction intervals instead)
• For non-parametric rank-based statistics where standard CI formulas do not apply (use exact or permutation-based CIs)