Have you graphed your data? If you can show us side-by-side box plots of the 5 groups of data, folks on this list will be able to make more useful comments.

Population data (which is what the assumption is about) are unlikely to be exactly normally distributed so the question is degree of ab-normality. The qq plots per groups should be useful to identify these.

I want to emphasize Daniel Wright's comment about assumptions applying to the populations from which you sampled. Textbooks often present the F-test for one-way ANOVA as if it is an exact test. But in order for it to be an exact test, you would need to have random samples from k populations that are perfectly normally distributed with exactly equal variances. (In addition to that, each observation would have to be perfectly independent of all other observations.) Even if it was possible to meet those conditions (which it is not if you are working with real data), the samples would not be perfectly normal, and would not have exactly equal sample variances.

Because it is not possible to meet the conditions described above (at least when you are working with real data, not simulated data), the F-test for ANOVA is really an approximate test. And when you are using an approximate test, the real question is whether the approximation is good enough to be useful.* That's how I see it. YMMV. ;-)

* Yes, I am borrowing "useful" from George Box's famous statement(s) about all models being wrong, but some being "useful". Several variations on that statement can be found here:

https://en.wikiquote.org/wiki/George_E._P._Box

Also borrowing from Box, a good quote for Bruce Weaver 's point is from Tukey's Sunset Salvo: we should ``move from [the idea that] all assumptions are right towards all assumptions are wrong'' (Tukey, 1986, p. 72). And then ask can you still get useful information using that procedure to help in your decision making.

Use quantile regression

I would just like to emphasize that since ANOVA is only cosmeticly different from OLS Regression it's not robust to departures from normality. David Booth

Normality of what, David?

David Eugene Booth's post reminds me of a great title by this Rand Wilcox paper.

Article ANOVA: A Paradigm for Low Power and Misleading Measures of E...

https://journals.sagepub.com/doi/10.3102/00346543065001051

Well Bruce we are talking about OLS Regression I assume we can all look that up if there are questions. David PS here's a hint outliers are found in residuals.

David, I was merely suggesting that you should be explicit, because as you know, many users of OLS models (and readers of these discussions) believe that the dependent variable must be normally distributed.

We have gone over this ground several times in other threads, and based on those discussions, I suspect you will disagree with some of what I say next, but here is how I see the normality requirement for OLS models.

• The F-test for the model and the t-tests for the coefficients would be exact tests if the errors were a random sample from a perfectly normal distribution--and if the errors were also "independently and identically distributed", of course.
• If George Box was correct in what he said about normal distributions in nature (see attached image), the error distribution is never perfectly normal--at least if one is using real world data (as opposed to simulated data).
• In other words, perfect normality of the errors is a sufficient, but unrealistic and unachievable condition.
• Therefore, the F- and t-tests are typically approximate tests. And in that case, we must think about the conditions that make the approximation good enough to be "useful" (another word George Box used).
• The necessary normality condition (for useful approximate tests) is that the sampling distributions of the parameter estimates (i.e., the coefficients) are approximately normal.
• The shape of those sampling distributions depends on both the shape of the error distribution and the sample size. As n increases, the sampling distributions of the parameter estimates approach the normal distribution, even if the errors are not normally distributed. See slides 9 and 10 in the attached PDF, for example.

I think there is another issue that is often not made explicit: I.e., users of OLS models will have different thresholds for judging when the approximations described above are good enough to be useful. I'll go out on a limb and guess that Rand Wilcox has a pretty high threshold for finding OLS models useful. And David, judging from things you have posted here and in other discussions, I suspect your threshold is higher than mine.

That's how I see it. YMMV. ;-)

Well thanks Bruce I am indeed indebted to you. I was fortunate enough to study with people who understood these fine points. Just a couple of minor points Rand Wilcox and several people in the robustne ss group have made this all clear since

Since at least 1980. Best wishes to you David Booth

I would add to Paul F Velleman 's very important suggestion – that you look at the data yourself by graphing it. While boxplots are useful in displaying summary statistics, adding dots to display the actual data is also very important. I've attached a plot from a study comparing normally-delivered babies with babies delivered by emergency section. You can see that the data and the density curves are telling a story about the latter group that you don't see from the boxplots alone. It looks like some babies get stuck because they are big, but there is also a group of small babies who get stuck – what's the story with them?

Sorry about the look of the plot. It's not the sort of thing I usually do!

It depends on the statistical test you are using. Some tests assume normality of the data while others do not. If you are using a test that assumes normality, then non-normality can be a problem. However, if you are using a test that does not assume normality, then non-normality is not necessarily a problem.