A point estimate gives one number, but repeated random samples would produce different estimates. A confidence interval combines an estimate with its sampling uncertainty to give a range of plausible parameter values.

Many introductory intervals have the form

$$\text{estimate}\pm\text{critical value}\times\text{standard error}.$$

Repeated-sampling interpretation

A $95%$ confidence procedure is designed so that, under its assumptions, about $95%$ of the intervals produced across repeated samples contain the true parameter.

The parameter itself is treated as fixed. Once one particular interval has been calculated, the frequentist statement is not that there is a $95%$ probability the fixed parameter lies inside it; the $95%$ describes the long-run performance of the procedure.

Example structure

If an estimator is approximately normally distributed with standard error $SE$, an approximate $95%$ interval often uses

$$\hat\theta\pm1.96,SE.$$

The number $1.96$ is a critical value from the standard normal distribution.

What controls interval width

Larger standard errors produce wider intervals. Increasing sample size usually reduces the standard error and therefore narrows the interval.

A higher confidence level requires a larger critical value, which widens the interval. Greater confidence and greater precision therefore trade off when the data are fixed.

Assumptions matter

The advertised coverage applies only when the sampling model and approximation used to construct the interval are appropriate. A narrow interval from biased data can be precisely wrong.

A confidence interval quantifies sampling uncertainty; it does not correct poor measurement, selection bias or a misspecified model.