Simulation of the Sampling Distribution of the Mean Can Mislead

Abstract

Although the use of simulation to teach the sampling distribution of the mean is meant to provide students with sound conceptual understanding, it may lead them astray. We discuss a misunderstanding that can be introduced or reinforced when students who intuitively understand that “bigger samples are better” conduct a simulation to explore the effect of sample size on the properties of the sampling distribution of the mean. From observing the patterns in a typical series of simulated sampling distributions constructed with increasing sample sizes, students reasonably—but incorrectly—conclude that, as the sample size, n, increases, the mean of the (exact) sampling distribution tends to get closer to the population mean and its variance tends to get closer to ś²/n, where ś² is the population variance. We show that the patterns students observe are a consequence of the fact that both the variability in the mean and the variability in the variance of simulated sampling distributions constructed from the means of N random samples are inversely related, not only to N, but also to the size of each sample, n. Further, asking students to increase the number of repetitions, N, in the simulation does not change the patterns.

• Simulated sampling distribution
• Sampling variability
• Variance of means
• Variance of variances
• Central Limit Theorem