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How can you do ANOVA between more than two groups? What would the results of that even mean?

As far as I understand it, ANOVA between A and B can tell you if the difference between A and B is statistically significant. What would a three-way comparison between A, B, and C tell you? Does it have multiple outputs?

Wouldn't it be more useful to do ANOVA between A and B, B and C, and C and A, to find out how different each one is from any other?

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