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. $(x, y) | x &gt; k$ ), 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.
I think you may have a sample size effect on your correlation coefficient. You may find the following references useful:


At what sample size do correlations stabilize?
The effects of sample size and variability on the correlation coefficient.
And also this post in stats.stackexchange demonstrating the effect of subsampling on correlation coefficient.