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Let's say that I am interested in how exam test score varies with a person's height. To do this, I take a sample from a school, and then I split this sample into two categories, (1) shorter than 165cm and (2) taller than or equal to 165cm. I then perform a one-way ANOVA test to compare the mean exam test score in the two subgroups.

With this setup, are the observations in the two groups independent?

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    $\begingroup$ You should not binarize the continuous heightvariable. Keep it as is, and use a regression model. $\endgroup$ – kjetil b halvorsen Dec 6 '19 at 4:08
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Yes, the groups are independent, but, as @kjetil pointed out, this is not the way to answer the question you have. Dichotomizing a continuous IV is nearly always a bad idea.

I wrote a blog post that shows, graphically, what happens when you categorize an IV. Also see Frank Harrell's book Regression Modeling Strategies which gives a list of the many things that go wrong when you do this.

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