The analysis of variance is the most powerful technique to testing hypothesis about means when the assumptions are met. Failure of any assumption would impair the utility of the test which bring to the wrong and invalid conclusion. This research was proposed to investigate the robustness and compare the power of the ANOVA F-test and the Kruskal-Wallis test under violating the assumptions of normality and homogeneity of variance in terms of the probability of Type I error and the power when the distributions were normal and exponential. There were four groups. Total sample sizes were 20,40,80 and 120 which were defined in equal and unequal sample sizes in each group conditions. The variances for equal variances case were equal to unity and for unequal variances case were equal to .4,.8,1.2 and 1.6. The data were simulated by using Monte Carlo technique which was performed on the microcomputer IBM compatible 80486-DX repeated 1,000 times for each situation. The results in this study found that the ANOVA F-test could control Type I error for all sample sizes and provided less power than the Kruskal-Wallis test for violating the assumption of normality (exponential distribution). Under the assumption of homogeneity of variance was not satisfied, the ANOVA F-test could control Type I error in only the sample sizes of 80,120 and has more power than the Kruskal-Wallis test. When violating the assumptions of normality and homogeneity of variance, Type I error and power of the ANOVA F-test and the Kruskal-Wallis test were in contrast such that ANOVA F-test provided small Type I error but having low power while the Kruskal-Wallis test provided great Type I error but having high power. Thus which statistics would be used depend on the researcher focus on either Type I error or power of the test. sample sizes of 80,120 and has more power than the Kruskal-Wallis test. When violating the assumptions of normality and homogeneity of variance, Type I error and power of the ANOVA F-test and the Kruskal-Wallis test were in contrast such that ANOVA F-test provided small Type I error but having low power while the Kruskal-Wallis test provided great Type I error but having high power. Thus which statistics would be used depend on the researcher focus on either Type I error or power of the test.