I have a sample size that is more than 30 but one of my continuous variables is not normally distributed. The Shapiro-Willk test does say that my data is not normally distributed. I am going to compare the results of the subjects among a categorical variable with more than two levels and want to see if there are significant differences among the categorical levels. From my understanding the central limit theorem states that in even though it is not normally distributed, at a large enough sample size it would be normally distributed. The subjects only have one score (i.e. no pre and post score or test, just one score). Would greatly appreciate help in clarifying this. Thanks.
Yes I am aware that the Kruskal-Wallis test is for medians and not means as in ANOVA.I am also assuming that the sample is randomly chosen.