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I have read multiple places that Shapiro-Wilk test should always be added with a QQ-plot, but no one has given a reason, and I do not see the intuition behind this. Can anyone explain why one need to confirm a Shapiro-Wilk test with QQ-plot?

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I don't think this Q is necessarily a duplicate, but you can get the idea from here: Is normality testing 'essentially useless'? – gung May 21 '14 at 22:28
I'd think that's actually not very good advice at all. Can you tell us some of those places you have seen it? I'd like to know who says such a thing and why they think it must always be done (my own advice would be to avoid doing it without a good reason to do it ... i.e. if you actually need a formal hypothesis test, which is rarely the case). It's most often done when assessing the suitability of the normality assumption for some other procedure, and frankly that's when I'd advise most strongly against using a goodness of fit test. It simply answers the wrong question. – Glen_b May 21 '14 at 23:12
@Glen_b which is it that you are saying should be avoided without a good reason? S-W test or QQ plot? If the former, I agree. – Peter Flom May 22 '14 at 10:03
@Peter Formal hypothesis testing of normality to assess whether to assume normality in some other procedure (not just Shapiro-Wilk, per se). On the other hand, I do use Q-Q plots, which indicate something of the extent of non-normality (i.e. something closer to "effect size") and give a more useful indication (i.e. is somewhat more related to the question of interest in that circumstance ... which is how much effect the non-normality will have). – Glen_b May 22 '14 at 10:08

At least two reasons:

1) A Shapiro Wilk test, at least if you base a decision on a p-value, is sample size dependent. With a small sample, you'll almost always conclude "normal" and with a large enough sample, even a tiny deviation from normal will be significant

2) A QQ plot tells you a lot about how the distribution is non-normal and may point to solutions.

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Citations would be helpful, but at face value, the claim is false. One of our favorite questions here (one of mine, anyway) is, "Is normality testing 'essentially useless'?" Answers to this question generally argue that Q–Q plots are more valuable than the Shapiro–Wilk test. I.e., if one of these is to be excluded, let it be the Shapiro–Wilk test, not the Q–Q plot.

Many analyses involve normality assumptions regarding distributions of interest, but these analyses vary in their sensitivity to violations of this assumption. As a significance test, the Shapiro–Wilk test does not indicate the degree of deviation from normality directly; it produces a significance estimate, which involves more than this effect size component. Another component involved somewhat infamously is sample size, which as @PeterFlom points out in his answer here, is potentially misleading. As a somewhat comical adaptation, throws an error when a user attempts to perform a shapiro.test on a sample larger than 5000 observations.

Furthermore, the Shapiro–Wilk test does not disambiguate skewness and kurtosis as different forms of deviation from the normal distribution. Some analyses may be more sensitive to skew than to kurtosis, or vice versa. Hence a given Shapiro–Wilk test statistic may not even reflect equivalently useful information about the invalidity of a normality assumption for two different analyses of the same sample. Conversely, as a data visualization technique (rather than a hypothesis test), a Q–Q plot may reveal much more to a trained eye about the specific nature of problems with a normality assumption, be it skew, kurtosis, a few particularly nasty outliers, etc.

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