The power of the procedure of W. Stout to detect deviations from essential unidimensionality in two-dimensional data was investigated for minor, moderate, and large deviations from unidimensionality using criteria for deviations from unidimensionality based on prior research. Test lengths of 20 and 40 items and sample sizes of 700 and 1,500 were studied. The power of Stout's procedure was directly related to the deviation from unidimensionality based on deviation areas. Deviation areas were inversely related to the correlation between the dominant ability and the reference composite. When the sample size or test length increased, the power of Stout's procedure also increased. In general, Stout's procedure had sufficient power to reject the null hypothesis of essential unidimensionality if 10 to 20 percent of the items were dimensionally distinct from the rest of the items. Results indicate that for minor deviation from unidimensionality, the rejection rates of Stout's procedure were not near the nominal level of 5 percent. For moderate and large deviations from unidimensionality, Stout's procedure had power to reject the null hypothesis of essential unidimensionality, especially if the sample size was 1,500 and the test length was 40. Twelve figures and 10 tables illustrate the discussion. (SLD)