You mean the Law of Large Numbers (LLN), not the Law of Averages, right? Both the Weak LLN and the Strong LLN presume all samples are independent and identically distributed. If we make a hierarchical model on the data of each paper, we can bind all the data into a single distribution, but assuming that each of these studies is independent is a _long_ shot. WLLN and SLLN _only_ apply to, roughly, sampling from the same process. Its scope is more applicable to things like sensor readings.
The Law of Large Numbers is an actual math theorem. The Law of Averages is a non-technical name for various informal reasoning strategies, some fallacious (like the gamblers fallacy), but mostly just types of estimation that are justified by more formal probability theory.
