# Statistical tests, when data are not normally distributed & variance is not equal

I have a data with two variables - one variable is quantitative, and one is nominal (4 categories). I ran the normality tests and it turned out that this quantitative variable is not normally distributed. When I checked Levene's test it turned out that the variance is not equal either. Therefore instead of Anova I thought about Kruskal-Wallis test, but still the test assumption is that the variance between the groups is equal. Can I simply run Anova with Welch statistics additionaly and that would be fine? (in SPSS).

What more: I have near 3000 observation, so on charts I see that the data are normally distrubuted but are right-skewed and kurtosis is bigger (that's probably why tests showed not-normal distribution).

Thanks for any suggestion! Mary

• Note that it suffices if the residuals are normally distributed. Furthermore, with that many observations, the tests have extreme power and can detect already minor deviations from normality that may not necessarily be a problem for the ANOVA. Commented May 11, 2018 at 13:18
• @hplieninger Hi! Thanks for your reply. That's true, I also read that it's possible to run Anova even if the quantitative variable is not normally distributed. But what about the equality of variance? If the Levene's test shows p < 0.001, how we can perform Anova (Analysis of variance) or the non-parametric test like Kruskal-Wallis? Is it OK then to run Anova but with this additional Welch option?
– Mary
Commented May 11, 2018 at 13:30
• You will find a lot of information about that on Cross Validated, you may transform your DV or you may try robust methods or you may ignore it if the differences in variances are small despite significant. Commented May 11, 2018 at 14:00
• Have a look at this question wrt the heterogeneity of variance problem.
– Joel
Commented May 11, 2018 at 15:15
• Just to emphasise what @hplieninger has already stated... your measured quantitative data need not be normal... the residuals of your model need to be normal. In fact, most data won't be normal if your conditions actually have some effect. For example, height is a very normally distributed variable, but if measured across a whole population might seem bimodal (two peaks) because men and women differ. After accounting for gender, the residuals will be closer to normal. Commented Jun 7, 2018 at 15:20