How to find out if two variables are independent

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I have a dataset like this:scatterplot and I want to find out if there is a relationship between the two variables (x-axis and y-axis).

The r-value from applying linear regression is -0.221618778230968. After removing outliers, I've gotten it up to a point of -.23 but it doesn't seem like enough evidence to prove a relationship.

I want to either prove that the two variables are not related or are somewhat related. Even a solid conclusion that "I cannot prove anything" is good enough. I don't think I've tried everything I can because I don't know what there is out there that I can try.

What I've tried:

  1. I've read about how I can use p-values to reject/fail to reject a null hypothesis. I got stuck after getting the z-scores of each variable value- I'm not sure what to do from there.
  2. I've also tried applying the ACE algorithm (Alternating Conditional Expectation Algorithm) to the points to find some kind of transform. It didn't work- my dataset is too sparse for this.
  3. I've tried randomly square(x) and log(x) the values to find a correlation. I'm not sure if this is what I should be doing.

Are there any suggestions as to where I can go from here?