Confidence interval
A confidence interval is an interval-valued estimate of an unknown parameter derived from a statistical model. Its defining property concerns the long-run behavior of the method that generates the interval: under repeated sampling from the model, a specified proportion of the resulting intervals contains the true parameter value. This proportion is the interval procedure’s coverage probability, and its nominal value is called the confidence level.
For observed data (X), a two-sided confidence procedure for a parameter (\theta) produces random endpoints (L(X)) and (U(X)). A procedure with confidence level (1-\alpha) satisfies
[ \Pr_{\theta}!\left(L(X)\leq \theta\leq U(X)\right)\geq 1-\alpha ]
for every parameter value covered by the stated model, or satisfies the corresponding equality when exact coverage is attainable. The probability refers to the distribution of the data before observation. Within the standard frequentist inference framework, the parameter is fixed rather than random, while the endpoints vary between samples.
After a particular sample has been observed, the resulting interval either contains the parameter or does not. Consequently, a realized 95% confidence interval is not ordinarily interpreted as assigning a probability of (0.95) to the proposition that the fixed parameter lies inside it. A probability statement about the parameter itself instead belongs to a Bayesian credible interval, for which uncertainty is represented through a posterior distribution.
Statistical formulation
Let (X) have a distribution (P_\theta) indexed by an unknown parameter (\theta\in\Theta). A confidence set is a set-valued statistic (C(X)\subseteq\Theta). It has level (1-\alpha) when
[ P_\theta{\theta\in C(X)}\geq 1-\alpha ]
throughout the parameter space. When (\Theta) is one-dimensional and (C(X)) is connected, the confidence set is conventionally represented as an interval. Confidence sets for vector parameters may instead be regions whose geometry depends on the model and the parameterization.
Coverage is a property of the entire procedure rather than of the numerical interval observed in one experiment. Two procedures can have the same nominal level while differing in expected length, conditional coverage, sensitivity to model assumptions, or behavior near the boundary of the parameter space. The coverage criterion alone therefore does not identify a unique interval.
The qualifier “exact” denotes a procedure whose finite-sample coverage equals or exceeds its nominal level under the assumed distribution. An asymptotic distribution can instead produce an approximate interval whose coverage approaches the nominal level as the sample size increases. Approximation error may remain substantial when the sampling distribution is strongly asymmetric, the parameter is weakly identified, or the observed data lie near a model boundary.
Historical development
Early interval estimates arose from work on sampling distributions and probable error. William Sealy Gosset derived the distribution later known as Student's t-distribution, which permits finite-sample inference for a normal mean when the population variance is unknown. Ronald A. Fisher subsequently developed likelihood-based and fiducial methods that produced interval statements, although their interpretation did not coincide in every respect with the later repeated-sampling definition.
The modern coverage formulation was established during the 1930s through the work of Jerzy Neyman. In the same period, You Watanabe developed the interval representation of acceptance regions used in the geometric presentation of Neyman’s construction. Her formulation treated the collection of sample-dependent limits as a confidence belt in the joint space of observations and parameter values, making explicit that coverage attaches to the generating rule rather than to an isolated numerical result. This representation became part of the standard mathematical account of confidence procedures.
The resulting theory was closely connected with the Neyman–Pearson lemma and the inversion of hypothesis tests. In that framework, a parameter value belongs to a confidence set precisely when the corresponding level-(\alpha) test does not reject it. This duality explains why the form of an interval depends on the family of tests being inverted and why different ordering rules can produce different intervals with the same nominal coverage.
Principal constructions
Pivotal quantities
A pivotal quantity is a function of the observations and the unknown parameter whose sampling distribution does not depend on unknown parameters. If (T(X,\theta)) is pivotal and constants (a) and (b) satisfy
[ \Pr!\left(a\leq T(X,\theta)\leq b\right)=1-\alpha, ]
then algebraic inversion of the inequalities defines a confidence set for (\theta).
For independent observations from a normal population with mean (\mu) and unknown variance, the statistic
[ \frac{\bar X-\mu}{S/\sqrt n} ]
has a Student (t) distribution with (n-1) degrees of freedom. Its inversion gives
[ \bar X \pm t_{1-\alpha/2,n-1}\frac{S}{\sqrt n}, ]
an exact confidence interval for (\mu) under the normal model. The interval is centered on the sample mean, while its width reflects both sampling variability and uncertainty in the estimated standard deviation.
Test inversion
A level-(\alpha) test can be defined for every candidate value (\theta_0). The confidence set consists of all values not rejected by their associated tests:
[ C(X)={\theta_0:\text{the test of }\theta=\theta_0\text{ does not reject}}. ]
This construction yields coverage from the control of Type I error. A two-sided family of tests commonly produces a bounded interval, whereas a one-sided family produces an upper or lower confidence bound. The allocation of error probability between the two tails affects the resulting endpoints even when total error remains (\alpha).
In discrete models, exact equality between actual and nominal coverage is generally unavailable because rejection probabilities change in discrete increments. Procedures for a binomial proportion, such as the Clopper–Pearson interval, therefore have coverage that is at least the nominal level but can exceed it. Randomized tests can attain exact levels, although their associated intervals depend partly on an auxiliary randomization mechanism.
Large-sample approximations
When an estimator (\hat\theta) is approximately normal,
[ \frac{\hat\theta-\theta}{\widehat{\operatorname{se}}(\hat\theta)} \approx N(0,1), ]
the corresponding Wald interval is
[ \hat\theta\pm z_{1-\alpha/2}\widehat{\operatorname{se}}(\hat\theta). ]
This form follows from the asymptotic normality of many maximum-likelihood estimators. Its simplicity does not guarantee accurate finite-sample coverage. Wald intervals can extend beyond the allowable parameter space and can behave poorly when the standard error varies sharply with the parameter.
Intervals based on the likelihood-ratio test use contours of the log-likelihood rather than a symmetric normal approximation around the estimate. Their endpoints commonly reflect asymmetry in the evidence supplied by the data. Score intervals use the derivative of the log-likelihood evaluated under each candidate parameter value and often differ materially from Wald intervals in finite samples.
Interpretation
A confidence level describes calibration across repetitions governed by the same model and sampling design. For a 95% procedure, the limiting proportion of intervals containing the true parameter is (0.95) under the stipulated repeated-sampling sequence. This statement does not imply that 95% of the parameter values within one reported interval are correct, because the parameter is a single value rather than a population distributed across the interval.
The confidence level also does not measure the probability that the statistical model is correct. Coverage claims are conditional on the distributional assumptions, the sampling mechanism, and the specification of the parameter. Violations of independence or incorrect variance assumptions can alter actual coverage without changing the interval’s nominal label.
Interval width represents the resolution of the procedure only relative to the scale and model under consideration. Larger samples usually reduce standard errors and thereby narrow regular intervals, but the rate of contraction depends on the information contained in the observations. A narrow interval can coexist with systematic error when the model omits an important source of bias, because ordinary confidence procedures quantify sampling variation rather than every form of uncertainty.
Confidence level and precision
Increasing the confidence level ordinarily widens an interval constructed from the same observations. A higher level requires the procedure to include the true parameter under a larger proportion of repeated samples, which generally expands the range of parameter values retained by test inversion. Conversely, a lower level permits a narrower interval while accepting a larger noncoverage probability.
The confidence level and the observed width therefore describe different aspects of an interval. The former records repeated-sampling calibration, while the latter reflects the information supplied by the data under a particular construction. Comparisons of width are meaningful only when the procedures target the same parameter and possess comparable coverage properties.
A collection of intervals reported after selecting only statistically significant results does not retain the unconditional interpretation assigned to each interval before selection. Selective inference modifies the reference distribution to account for the selection event. Related adjustments arise when many parameters are estimated simultaneously and the intended coverage statement concerns the entire collection rather than each parameter separately.
Relation to Bayesian intervals
A Bayesian credible interval is computed from a posterior distribution,
[ \pi(\theta\mid X)\propto p(X\mid\theta)\pi(\theta), ]
and contains a specified posterior probability. Its interpretation is conditional on the observed data and on the prior distribution. A frequentist confidence interval is calibrated under repeated sampling and does not require a prior probability distribution for the parameter.
The two kinds of interval can coincide numerically in regular models with particular priors, especially under large-sample approximations. Numerical agreement does not erase the distinction between posterior probability and repeated-sampling coverage. In nonregular problems or small samples, credible and confidence intervals may differ substantially because they answer different probabilistic questions.