At its most basic, empirical legal research is legal research that uses statistical techniques and analysis.
John Baldwin and Gwynn Davis in Chapter 39 of the Oxford Handbook of Legal Studies state, "...empirical research in law involves the study, through direct methods rather than secondary sources, of the institutions, rules, procedures, and personnel of the law, with a view to understanding how they operate and what effects they have."
Shari Seidman Diamond in her article Empirical Marine Life in Legal Waters: Clams, Dolphins, and Plankton, 2002 U. Ill. L. Rev. 803, 805 (2002) further states, "What makes research empirical is that it is based on observations of the world. These facts may be historical or contemporary, based on legislation or case law, the results of interviews or surveys, or the outcomes of secondary archival research or primary data collections."
Places to Start
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Try a keyword search for your desired topic along with the term "statistics" (e.g., housing statistics or criminal statistics) or try a subject search.