The Art of Statistics: Learning from Data

by David Spiegelhalter

The procedure for deriving an uncertainty interval around our estimate, or equivalently a margin of error, is based on this fundamental idea. There are three stages: We use probability theory to tell us, for any particular population parameter, an interval in which we expect the observed statistic to lie with 95% probability. These are 95% prediction intervals, such as those displayed in the inner funnel in Figure 9.2. Then we observe a particular statistic. Finally (and this is the difficult bit) we work out the range of possible population parameters for which our statistic lies in their 95% prediction intervals. This we call a ‘95% confidence interval’. This resulting confidence interval is given the label ‘95%’ since, with repeated application, 95% of such intervals should contain the true value.fn5 All clear? If it isn’t, then please be reassured that you have joined generations of baffled students. Specific formulae are provided in the Glossary, but the details are less important than the fundamental principle: a confidence interval is the range of population parameters for which our observed statistic is a plausible consequence.

A confidence interval