Hi, is it possible to have not normal data but there is no violation of homogeneity? And if it is possible, what test should I run for this data? Please help. Thanks!
If you mean "homogeneity of variance" or "equal variance" when you say "homogeneity", yes, it is possible to have samples with equal variance and not be normally distributed. However, technically, the terms "variance" and "standard deviation" have no meaning when the data are not normally distributed. If the assumptions of normality and/or equal variance are violated, you can always run nonparametric statistical tests. For example, the Mann-Whitney U test and Kruskal-Wallis test are the nonparametric equivalents of the independent samples t-test and one-way ANOVA, respectively. Alternatively, you could attempt to transform the data to make it normal. A log transformation is often attempted. You could then run the appropriate parametric test. The advantage of attempting a transformation followed by a parametric test is that parametric tests are typically more powerful statistically than are nonparametric tests.
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