Understanding how teaching and research contribute to the overall efficiency of university operations is of great importance for universities to improve their performance. This paper adopts a holistic approach to evaluate the university efficiency from three perspectives including (a) the overall university operations efficiency, (b) the university teaching efficiency, and (c) the university research efficiency. It applies the technique of data envelopment analysis to thirty-six Australian universities in the period of 2011 to 2015 for evaluating their relative efficiency respectively from these perspectives. A strategic group analysis is further conducted for exploring the source of inefficiency of an individual university in its respective strategic group. Such an analysis provides individual universities with valuable information on how they can make full use of their resources for improving their efficiency in an increasingly competitive environment.