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The wikipedia link you give are using imprecise/confounded language, so you should maybe find some better tutorial! It also confuses assumptions and mathematical conclusions from those assumptions (it isn't an assumption that $S^2$ follows a $\chi^2$-distribution with $n − 1$ degrees of freedom, that can be concluded from the normal assumption and ...

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What you apparently want to do is to start by evaluating the relationship between the 3 classes and each of your features individually. For each continuous feature you are proposing a one-way ANOVA of that feature against the known classification (3 classes). ANOVA is not appropriate for a similar evaluation of the binary features, as interpretation of ANOVA ...

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In the words of Jacob Cohen the question of whether, when and how to correct for multiple corrections is one where "intelligent people disagree". One problem with the corrections is that, by lowering type 1 error they increase type 2 error. Another is how much correction to do. What is the number of comparisons? Everything in one paper? One table? One ...

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