A Simulation-based Approach to Understanding the Sampling Distribution of the Mean
Khairul Islam *
Department of Mathematics and Statistics, Eastern Michigan University, Ypsilanti, MI 48197, USA.
*Author to whom correspondence should be addressed.
Abstract
A longstanding debate surrounds the use of simulations of the sampling distribution of the mean (SSDM), with some researchers cautioning that such simulations may be misleading, while others argue they can enhance statistical learning. To address these conflicting perspectives, this study highlights the importance of a strategic approach that clearly distinguishes between the exact sampling distribution of the mean (SDM) and its simulated counterpart (SSDM), thereby clarifying the scope and limitations of simulation. Using the statistical software R, an SSDM was developed with targeted learning objectives and evaluated across three applied examples. In all cases, the SSDM closely approximated the mean and standard error of the corresponding SDM—for instance, 130.8 and 10.64 in Example 1; 14.74 and 0.44 in Example 2; and 1.5 and 2/√n across varying sample sizes n in Example 3—with T- and Z-test p-values indicating no significant difference at the 5% level. Additionally, both SSDM and SDM standard errors decreased with increasing sample size, aligning with theoretical expectations. These results suggest that, when thoughtfully designed, SSDMs can effectively support conceptual understanding while minimizing the risk of misconceptions. Although simulations are inherently approximate, they can provide meaningful insights into otherwise abstract statistical concepts, and instructors are encouraged to anticipate and address potential misunderstandings in their use.
Keywords: Simulation, sampling distribution of mean, simulated sampling distribution of mean, test of hypothesis