Students have difficulty consistently applying the Empirical Law of Large Numbers, with conflicting evidence regarding why these difficulties arise. Recent conceptual change research indicates that supporting students' reasoning about underlying mechanisms may improve their understanding. We developed a series of simulation tasks to support students’ visualization of two mechanisms related to increasing sample size: swamping, the decreasing influence of extreme values on the mean; and heaping, the increasing concentration of possible sample means. The mechanistic reasoning of five students was explored over six hours of clinical interviews using a framework from philosophy of science to examine their identification of mechanism components, reasoning strategies, and analogies. All five students displayed strong understanding of swamping; students' understanding of heaping was more fragile and appeared to depend on the extent to which they focused on the most extreme possible observations. Students generated powerful and pedagogically promising analogies for the mechanisms of the Empirical Law of Large Numbers, but also sometimes struggled to coordinate concepts at multiple levels.