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The following provides what seems to be the best non-statistician summary of normality tests. While the full paper is behind a paywall, the abstract is extensive and has references, for those who want to go further. It might enhance the posting, and where applicable in the website, if such were included..

Asghar Ghasemi and Saleh Zahediasl. Int J Endocrinol Metab. 2012 Spring; 10(2): 486–489. Normality Tests for Statistical Analysis: A Guide for Non-Statisticians