One application where confidence bands are used is in regression analysis, in the case of a simple regression involving a single dependent variable. Results can be presented in the form of a plot showing the estimated regression line. A first extension of this is to include lines showing point-wise confidence limits: the lines are drawn through the upper and lower limits obtained treating each possible value of the dependent variable separately. Two cases are commonly dealt with:
A second extension of the plot is to include confidence bands: these are lines derived from the observed dataset which define a region such that, across notional repetitions of the experiment, regions devised in the same way have a given probability that the true regression line lies entirely within such a region.
A confidence band typically has a similar shape to the region defined by the limits of the point-wise confidence intervals, but is rather wider.
A second type of application where confidence bands are available are those deriving from the empirical distribution function. Again, simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a confidence band for the distribution function as a whole. See Setting confidence limits for a distribution function for example, or Owen (1995).
Confidence bands have also been devised for applications to spectral density estimation.