Bradley (1977) has shown that the mixed normal family of distributions is an important population model in many behavioural science research contexts. Researchers engaged in studies of the type discussed by Bradley (1977) might well wonder about the relative appropriateness of various statistical techniques that might be employed in data analyses. For example, in the case of a two‐sample test for shift, should one use the parametricttest or some non‐parametric counterpart such as Wilcoxon's rank‐sum test?The primary purpose of this paper is to examine some of the characteristics of the asymptotic relative efficiency (ARE) of the Wilcoxon rank‐sum test relative to the independent meansttest under various mixtures of two normal distributions. Pursuant to this goal, the equation for finding the ARE of the two tests under various mixtures of two distributions is developed, the equation is applied to various example situations, and certain limiting values of the equation are noted. As a result, it is concluded that the Wilcoxon statistic tends to have large ARE advantages over thettest in research contexts similar to those considered by Bradley (1977). It is concluded, therefore, that the Wilcoxon test is the more appropriate statistic for the research situations con