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From my understanding if I'm looking at the relationship between two variables that are normal, I'd use Pearson's correlation coefficient, but if they are not normal I'd use Spearman's rank correlation coefficient. What would I do if one variable is normal and the other is not? I still expect for there to be some correlation between the two variables.

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One option is Kendall's tau. It's a nonparametric correlation coefficient and works well even when either or both of your variables are non-normal. As a sidenote, this answer gives a convincing argument for preferring Kendall's tau over Spearman's rho in cases where both variables are non-normal.

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