# What is not normal distribution?

I am running analyses for my dissertation and just got myself in a muddle. I am running descriptive stats to determine whether to use paired t-tests vs Wilcoxon signed rank tests. I've read that it all depends on the normality.

Looking at my histograms the data is not normally distributed, but when looking at skew and kurtosis I am undecided. My stats book says that if these values are above or below 0 then it is not normal. but I was just wondering if anything other than 0 is not normal or if there is a cut-off? As you can see below, some of these points are only slightly away from 0, so would I class this as normal or not normal?

        SKEW      KURTOSIS
V1     -.078       -1.460
V2      .267       -2.202
V3     -.735       -1.455
V4      .845        2.978
V5     1.082        1.041
V6     2.099        4.684

• – Tim Mar 28 '15 at 21:50
• You can try using both t-tests and Wilcoxon tests. If the conclusions are the same, then it probably doesn't matter much which one you use. One advantage of t-tests is that you can get confidence intervals easily. – mark999 Mar 28 '15 at 21:56
• Sample size is also very important. What is the size of data you are working with? – rnso Mar 29 '15 at 1:53
• "I've read that it all depends on the normality." -- can you say what you read, exactly, and where? – Glen_b -Reinstate Monica Mar 29 '15 at 3:24
• Sorry, but I can't help reading the title: stats.stackexchange.com/a/14356/35989 But seriously: you should change the title because it is misleading - the answer to question in the title is: "it is any distribution other then Normal". – Tim Mar 29 '15 at 9:40

A distribution can be non-normal and yet have identical skewness and kurtosis to a normal distribution

The fourth image in this answer is one example - a symmetric "double gamma" distribution with shape parameter around 2.303. Even symmetry is not guaranteed with skewness 0.

Even when population skewness and kurtosis are quite different from those for the normal, at small sample sizes it can be very hard to judge whether that's the case. In larger samples, you might be able to distinguish those cases ... but they may not matter so much then.

A few more specific points

1: skewness and kurtosis alone are not necessarily a good way to judge whether the sample is consistent with normality

2: using formal tests to assess normality for the purpose of choosing which test to apply answers the wrong question. Your data aren't going to be from a normal distribution. The relevant question might be something like "how non-normal might they be and how much impact will that have on significance level and power?"

In large samples, you'll reject normality when it doesn't matter and in small samples you'll fail to reject when it may matter. The test rarely tells you anything useful.

3: several papers advise against trying to choose which test to do based on goodness of fit tests, because it affects the behavior of the subsequently chosen test. If you don't have some reason to think the normal-theory procedures would be a reasonable choice, simply don't assume normality. There are more choices than t-tests and Wilcoxon-Mann-Whitney.

• Are you saying that there is no clear framework (workflow) for assessing analytically how close a distribution is to a normal one? If the opposite is true, it would be great, if you could specify the workflow steps or point to corresponding references. – Aleksandr Blekh Mar 29 '15 at 3:47
• @Aleksandr "How close" depends on the purposes of the analyst, the sample size, the test statistic, the loss function for changes in significance level and power, and so on. If you want a suggested sequence of steps, ask it in a question (if you can make something not overly broad, keeping in mind the many variables involved). As I read it, I don't think it's an answer to the question here -- and I think overly formulaic approaches are often less useful than they seem. But I do suggest a "workflow" of sorts in my answer already. – Glen_b -Reinstate Monica Mar 29 '15 at 5:05
• Good points, thank you for clarification. Regarding asking a separate question: I will think about it, since it is somewhat challenging to make such question narrow enough. – Aleksandr Blekh Mar 29 '15 at 5:18