Illustrating the Law of Large Numbers (and Confidence Intervals)

Instructors of introductory statistics classes tend to separate out abstract probability concepts because of the level of mathematics required to present them. The Law of Large Numbers, which explains why making repeated observations on a population is informative, seems especially vulnerable to this tendency. This is unfortunate because the Law validates students' intuitive understanding about why statistics works and reinforces the notion that abstract probability concepts have important, real-life consequences. Instead of presenting the Law in a mathematical fashion, we propose a graphical presentation designed to communicate and emphasize the Law's importance. Specifically, we illustrate the effect that the Law of Large Numbers has on a sequence of sample means by plotting a randomly generated sequence of sample means versus their sample size. Students can then observe (1) how the sample mean approaches the population mean as the sample size grows large, (2) the variability associated with that process, and (3) the implied mathematical result. The corresponding sequence of probability intervals for the sample mean can then be superimposed on the plot to show how the sample mean's variability is reduced as the sample size grows. Furthermore, the sequence of confidence intervals can also be superimposed on the plot to demonstrate that the concept of coverage probability applies only to the procedure for setting confidence intervals (which we refer to as a random probability interval) and not to the observed confidence interval itself.