# Is there any logic to adding a threshold to see if two variables are related?

I have two variables $X$ and $Y$ given as tuples of $(x, y)$, and I want to see if there is a relationship between the two variables. I can do so by finding the correlation coefficient.

However, I found that by selecting an arbitrary subset of the data (e.g. $(x, y) | x > k$ ), I can get a higher correlation coefficient and a stronger result. Is doing so mathematically sound? I have no a priori reason to believe that certain data points are "more important" than others, to put it simply.

Try this thought experiment. Suppose you generate random points (no actual relationship). Then you compute the correlation coefficient of all the points; the correlation coefficient of all points where $x>k$; and the correlation coefficient of all points where $x<k$. Realistically, one of the two latter values will be larger than the correlation coefficient of all the points. So you'll always be in a position where if you pick a threshold you can increase the correlation coefficient. This is true even if the points are randomly generated.