About

A confidence interval is an interval that covers an unknown parameter of interest (e.g., the mean) with $(1-\alpha)\%$ confidence. Confidence intervals are a frequentist concept. A credible interval is a related concept in Bayesian statistics. People often incorrectly ascribe the meaning of credible intervals to confidence intervals.

In frequentist statistics, a confidence interval for a parameter, $\theta$, is an interval computed from a set of data whose distribution depends on that parameter in some way. The interval is computed such that, if the process of drawing a sample and computing the interval were repeated identically ad infinitum, the proportion of the intervals that included the true value of the parameter would converge to $(1-\alpha)\%$. This does not mean that the probability of a given interval including the true value of the parameter is $(1-\alpha)\%$. Each interval either does include the true value or it does not include the true value. The 'confidence' is a property of the procedure used to compute the interval and pertains to the theoretical infinite set of such intervals.

Some additional notes:

1. The confidence interval is a function of the data, $X$. Since the data are conceptualized as a random sample from a population, confidence intervals are random variables (although the confidence interval computed on a particular set of data is a realization).
2. Often one can only compute approximate confidence intervals, which may have the nominal coverage asymptotically.
3. It may not be possible to compute any exact confidence interval that might otherwise be preferred if the data or the parameter can only take discrete or otherwise limited values.
4. The same ideas can be applied to a set of parameters, e.g., $\vec{\theta} = [\mu\ \ \sigma^2]^T$. In that case, it is more correct to refer to the confidence region.
5. In a regression context, the set of confidence intervals for all possible conditional means ($\mu_Y|X=x$) is called a confidence band.