Mean confidence intervals and prediction intervals

Write the target in the heading

In the independent normal model, a mean confidence interval targets the population mean. A prediction interval targets a single future independent observation from that population. Giving both the generic label “range” conceals which question has been answered.

Add the two sources of variation

With known sigma, a new observation has variance sigma². The sample mean has variance sigma²/n. If the future observation is independent of the estimation sample, their difference has variance sigma²(1 + 1/n). This produces a prediction scale sigma√(1 + 1/n), rather than sigma/√n.

Compare fictional half-widths

Read the conditions of the prediction

The comparison assumes the same stable distribution and independence. It is not a prediction interval for every future market return or a simultaneous guarantee for a sequence of observations. Unknown sigma and other models require their corresponding procedure.