# R: Shapiro-Wilk test yields different results in descriptives and in One-Way ANOVA

I got a dataset of 60 people from a between-subject approach that I was trying to check for normality. (Since I need information about normality for ANOVA post-hoc tests.) They were split in three different groups. So, let's say I take the variable "visualDistance" (just as an example, since it really matters there), the descriptives of that variable say the following:

You can see, that for the group 2 and 3 the distribution seems to be clearly non-normal. However, when I do an ANOVA, it returns the following value:

And a One-Way ANOVA (which theoretically should be the same) yields:

I'm not completely solid with ANOVAs, but I thought the normality assumption has to apply to all test groups.

• Why is it then that the assumption checks in R for ANOVAs result in a confirmed assumption for normality, when clearly two out of three groups don't fit the assumption?
• Which results should I listen to? Can I savely use post-hoc tests based on normality, with such results?
• And how come that the results slightly differ between ANOVA and One-Way ANOVA? (in this case it seems minor, but in another case one results in a p-value of 0.472, while the other results in a p-value of 0.466)
• Typically normality is tested using the residuals when you test it within a formula like ANOVA or regression. Check out Kim (2013) doi.org/10.5395/rde.2013.38.1.52. The article discusses the alternative thresholds you can use for ANOVA with skew and kurtosis (minding whether you have kurt or excess kurt). It's definitely a good idea to reassess this. Either way, ANOVA is really pretty robust against violations of normality (for which it is equivalently weak against variance violations).
– Kat
Commented May 9, 2022 at 14:28
• Why do you say that groups 2 and 3 are clearly non-normal?
– Dave
Commented May 9, 2022 at 14:28