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Say I have a continuous variable that for some reason needs to be binned into a dichotomous variable. In such a case, I pick some cutoff point and encode everything above that cutoff as 1 and everything below as 0.

Now I run a Pearson's point biserial correlation with both versions of that variable. Generally I'll expect a fairly high correlation, but still below 1.

Why is this and how should I interpret it?

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In effect you are asking why the chart below involves a correlation of about 0.83 rather than 1

The answer is that the points do not lie on a straight line, though high values of $x$ are still associated with high values of $y$

enter image description here

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  • $\begingroup$ Would then it be such that the lower the standard deviation within each bin (and thus the more centralized the data), the more you could draw a straight line, and thus the higher the expected correlation? $\endgroup$
    – Josh
    May 6, 2021 at 12:42
  • $\begingroup$ @Josh - if the standard deviations of the original data were much smaller within each group were much smaller than overall then the correlation would be higher: this would involve clustering near the top and bottom of the original data as opposed to being spread out $\endgroup$
    – Henry
    May 6, 2021 at 14:40

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