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When should we apply normality tests? For which types of the variables should we apply the normality test? For example dependent variables, independent variables, or control variables, etc?

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In many contexts (e.g., multiple regression) it is the distribution of the residuals that is the relevant assumption (see here for a discussion).

Your question also raises the general issue of the relevance of normality testing. See this previous question: Is normality testing 'essentially useless'?

However, more broadly, doing exploratory data analysis (e.g., plots, summary statistics, etc.) to understand the distribution of all your variables is important.

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  • $\begingroup$ I agree with Jeromy. There is no sense to categorize application of normality testing in terms of types of variables. What matters is when normality becomes a modeling issue. There are many pros and cons associated with normality testing that are discussed in other posts on this site including the one Jeromy mentioned. $\endgroup$ – Michael R. Chernick Aug 23 '12 at 10:57
  • $\begingroup$ +1 for "However, more broadly, doing exploratory data analysis (e.g., plots, summary statistics, etc.) to understand the distribution of all your variables is important". $\endgroup$ – gung - Reinstate Monica Mar 14 '13 at 4:53

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