QQ Plot and Shapiro-Wilk Test Disagree My QQ Plot shows that the data is not normally distributed
qqplot(residual_values, fit = True, line = '45')
pylab.show()


It has a skewness of 0.54
residual_values.skew()  # 0.5469389365591185

But the p_value of Shapiro-Wilk test is greater than 0.05, telling me that it is normally distributed
shapiro(residual_values) # (0.9569438099861145, 0.2261517345905304)

What is the correct inference from this? Is it normally distributed or not?
 A: The q-q is consistent with (not "proving") approximate normality, more or less.
The Shapiro-Wilk is a formal test of normality and as such, it cannot confirm the null hypothesis of normality. The data may be reasonably consistent with normality yet still be from a different nonnormal underlying distribution. Frequentist hypothesis tests, as a general rule, cannot prove a hypothesis, and failure to reject (p>alpha) does not support the null hypothesis. 
@The Laconic gave some decent advice to interpret the q-q plot. However, large p-values do not lead you to accept the null hypothesis (therefore, you don't conclude normality based on this test; the best you can do is say insufficient evidence of nonnormality at the a priori chosen alpha level). 
A: The QQ plot is an informal test of normality that can give you some insight into the nature of deviations from normality; for example, whether the distribution has some skew, or fat tails, or there are specific observations that deviate from what you would expect from a normal distribution (outliers).  The QQ plot can often convince you that the distribution is definitely not normal, but this isn't such a case.  Here, the points fall more or less along the line, which is broadly consistent with normality--intuitively, the sort of variation you would expect to see in a small sample.  
The Shapiro-Wilk test is a formal test of normality.  I'm not familiar with the shapiro function's output, so I'm not sure which number, if either, is supposed to be the p-value, but if you say it's largish, then we are led to accept the null hypothesis of normality.  And this is consistent with what we see qualitatively in the QQ plot.
A: My understanding is that, given power issues with normality tests, they are not highly recommended. As a result I don't use them any more, preferring QQ plots (which are recommended in the literature I have seen).
A: The Shapiro-Wilk p-value being >0.05 indicates lack of evidence against normality. That is consistent with the QQ plot you showed, which is not too far off the line.  I don't see what the inconsistency is here. Also, you should give a CI for the skewness coefficient.
