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?
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.
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, r 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.