This study addresses the question, "What should graduate students know about research and statistics after completing an initial course?" Individuals who teach such courses at various Carnegie classifications of institutions were surveyed about the specific characteristics of an introductory graduate research course at their own institutions to see if a core of topics could be identified that was common to these courses. Responses were received from 80 institutions. Seventeen topics were identified that appear to comprise a knowledge base for this course across institutions. These topics may be described as those that deal with one's ability to understand and utilize research results reported in the literature. This core included topics such as the formulations of viable research problems and testable hypotheses, the scientific method, types of variables, types of research, null and alternative hypotheses, measures of center and dispersion, correlation coefficients, statistical significance, reliability, and validity, to name a few. Institutions differ, although not by much, on topics that do not comprise the common core. Mean ratings of importance for items not in the common core, most of which deal with technical aspects like inferential statistical procedures, were uniformly lower for masters' institutions, and somewhat lower for research institutions, than for doctoral institutions. Where institutions seem to differ most is in the degree to which topics of a more technical nature are addressed within the course and the extent to which such topics may be left for inclusion in followup courses. (Contains one figure, six tables and eight references.) (Author/SLD)