A confidence interval gives a range of plausible values for a parameter, not just a single point estimate. A 95% CI for the mean is constructed so that, across many hypothetical repetitions of the experiment, 95% of such intervals would contain the true mean.
Construction (mean of normal data)
For a sample mean with sample standard deviation and size , the 95% CI is approximately:
(1.96 comes from the standard normal; use the t-distribution's for small samples.)
Interpretation
A specific 95% CI either contains the true parameter or it doesn't — there's no probability statement about the specific interval. The 95% refers to the long-run frequency: if you repeated the experiment many times, 95% of constructed intervals would cover the true parameter.
Width is information
A narrow CI says "we know this precisely"; a wide CI says "we don't." Width scales like — to halve the CI width, quadruple the sample size.
CI vs p-value
A 95% CI excludes some value iff the two-sided test against rejects at . They're two views of the same information. CIs are usually more useful — they convey effect SIZE, not just "significant or not."
Bayesian credible intervals
The Bayesian analog. A 95% credible interval is a range where the posterior probability is 0.95. The interpretation is what most people mistakenly attribute to frequentist CIs: "there's a 95% probability the parameter is in this range." Different framework, different math, similar-looking numbers for non-pathological cases.
Bootstrap CIs
For statistics without a clean analytical CI formula (median, ratios, model accuracy), bootstrap. Resample your data times with replacement, recompute the statistic on each resample, take the 2.5th and 97.5th percentiles of the resampled statistics. Distribution-free, computationally cheap, and increasingly the default.