On normality tests

I recently came across a very interesting problem. I wanted to verify if some of my data could be assumed to be normally distributed since I wanted to use statistical methods that assumed the same. One would assume that you would just run a statistical test for normality such as the Shapiro-Wilk or Anderson-Darling tests on the data and if the null hypothesis is not rejected, you would be okay to go ahead with the normality assumption.

I stumbled upon this blog post. What the author says is that these normality tests are

1. Dependent on sample size
2. Are null hypothesis tests against the assumption of normality As a result, for small sample sizes even a big departure from normality isn’t detected and for large sample sizes even the smallest departure is!

Conclusion: A Q-Q plot is required in addition to these tests for verification.

Keep in mind though that

1. You can never, no matter how much data you collect, conclusively determine that it was generated from an precisely normal distribution.
2. Your data is not generated from a precisely normal distribution (no real data is).

- Ian Fellows