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How can I find the correlation between a categorical (dependent) variable and a continuous-scale (independent) variable? Is a Kruskal Wallis test appropriate? I'm a little confused as my independent variable is not nominal but scale!

Ach
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Correlation notions do not care about the direction of causality (if there is any). So it shouldn't matter which one is dependent and which independent. If assumptions of normality are not a problem, then one way ANOVA should be ok. If you want it to be non-parametric, then Kruska-Wallis should be appropriate.

highBandWidth
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