A confidence interval is an interval estimate of an unknown population parameter in statistics. Calculated from sample data, it expresses uncertainty about quantities such as a population mean or proportion. Its confidence level, commonly 95%, describes the procedure’s performance under repeated sampling: when its assumptions hold, the stated proportion of intervals constructed by that procedure contains the true parameter. The endpoints are called confidence limits or bounds. (itl.nist.gov)
Definition and interpretation
Let (X) represent the sample data and (\theta) an unknown parameter. An interval procedure assigns lower and upper bounds (L(X)) and (U(X)). Its coverage probability is
[ P_\theta!\left(L(X)\leq\theta\leq U(X)\right), ]
where the probability is calculated under the sampling model with parameter (\theta). An exact (100(1-\alpha)%) procedure has coverage (1-\alpha); a conservative procedure has coverage at least that level. Approximate procedures may achieve their nominal coverage only under suitable conditions or as sample size increases. (nvlpubs.nist.gov)
In the frequentist interpretation, the parameter is fixed while the interval’s endpoints vary between samples. After an interval has been calculated, it either contains the parameter or does not. Thus, a realized 95% interval does not, by itself, assign a 95% probability to the parameter being inside it. Nor does it contain 95% of individual observations: uncertainty about a population mean is different from variability within the population. (itl.nist.gov)
Intervals may be two-sided, with both lower and upper limits, or one-sided, providing only a lower or upper bound. The choice changes how the noncoverage probability (\alpha) is allocated. Equal-tailed two-sided intervals allocate (\alpha/2) to each tail, whereas one-sided bounds allocate it to one tail. (itl.nist.gov)
Construction for a population mean
A familiar construction assumes observations are independent and drawn from a normal distribution. If the population standard deviation (\sigma) is known, a confidence interval for the population mean (\mu) is
[ \bar{x}\pm z_{1-\alpha/2}\frac{\sigma}{\sqrt n}, ]
where (\bar{x}) is the sample mean, (n) the sample size, and (z_{1-\alpha/2}) the indicated standard-normal quantile. The expression following the (\pm) sign is the margin of error. (itl.nist.gov)
When (\sigma) is unknown, the corresponding interval uses the sample standard deviation (s):
[ \bar{x}\pm t_{1-\alpha/2,n-1}\frac{s}{\sqrt n}. ]
Here (t_{1-\alpha/2,n-1}) is a quantile of Student’s t-distribution with (n-1) degrees of freedom, and (s/\sqrt n) is the estimated standard error of the mean. For example, hypothetical data with (n=25), (\bar{x}=10), and (s=2) give a 95% interval of approximately (10\pm2.064(0.4)), or ([9.17,10.83]). (itl.nist.gov)
This t-based construction is exact under the independent normal-sampling model. Outside that model, it can provide an approximation, but small samples and substantial skewness can impair its performance. The central limit theorem explains why approximately normal sampling distributions arise in many sufficiently large-sample settings; it does not guarantee accuracy for every dataset. (itl.nist.gov)
Other interval methods
For a proportion modeled by a binomial distribution, methods include the Wald, Wilson score, and Clopper–Pearson intervals. The elementary Wald approximation can have poor coverage, particularly near proportions zero or one. Wilson intervals arise by inverting a score test. Clopper–Pearson intervals use binomial tail probabilities and are generally conservative because the distribution is discrete; “exact” does not mean coverage equals the nominal level for every parameter value. (itl.nist.gov)
Bootstrap sampling provides another route when an estimator’s sampling distribution is difficult to derive. Repeated resampling approximates that distribution, allowing percentile, basic, or studentized interval constructions. Bootstrap validity still depends on the resampling scheme and statistical setting. For dependent time-series observations, ordinary independent resampling can be inappropriate; block resampling preserves aspects of dependence. (stat.cmu.edu)
Precision and hypothesis testing
For the standard mean intervals, greater confidence produces a wider interval for the same data. Increasing sample size generally narrows it, while greater variability widens it. When the critical value and variability are approximately unchanged, width scales as (1/\sqrt n): roughly quadrupling sample size halves the margin of error. This relationship concerns sampling precision rather than eliminating systematic error. (itl.nist.gov)
Confidence intervals are closely connected to statistical hypothesis testing. An interval obtained by inverting tests contains the parameter values that those tests would not reject. Consequently, a matched two-sided 95% interval excludes a null value precisely when the corresponding test rejects it at the 5% significance level. This equivalence requires compatible methods and assumptions, not merely matching numerical labels. (itl.nist.gov)
Related intervals and simultaneous coverage
A credible interval in Bayesian inference contains a specified probability under a parameter’s posterior distribution, conditional on the observations, model, and prior distribution. Its interpretation therefore differs from repeated-sampling confidence coverage, even when the numerical endpoints coincide. (nvlpubs.nist.gov)
A prediction interval concerns a future random observation, rather than a fixed parameter. In linear regression, it incorporates both uncertainty in the fitted mean and variability of the new observation, making it wider than the corresponding confidence interval for the mean response. A tolerance interval instead covers a specified proportion of the population with a stated confidence level. (itl.nist.gov)
Several individually valid 95% intervals do not generally provide 95% probability of covering all their parameters simultaneously. Simultaneous confidence intervals address this joint requirement. For (m) intervals, the Bonferroni correction constructs each at confidence level (1-\alpha/m), giving joint coverage of at least (1-\alpha), provided the individual coverage guarantees hold. This guarantee does not require independence among the intervals. (itl.nist.gov)