# How to find out if two variables are independent

I have a dataset like this: 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?
• "two variables are not related or are somewhat related" It should be pretty easy to prove either $A$ or $\not A$. :) Welcome to CV, unixcorn. Dec 26, 2021 at 1:52
• Thanks for the welcome :D Could you provide some keywords I could look into to prove 𝐴̸ ? Also, is it okay to assume A is true if I can prove 𝐴̸ ? Dec 26, 2021 at 2:44
• I was unclear in my attempt at humor: it should be pretty easy to prove the conjunction either $\boldsymbol{A}$ or $\boldsymbol{\not A}$: taken together they exhaust the sample space. :) Dec 26, 2021 at 2:52
• As to the question in your comment, I think one risks committing confirmation bias if one takes "I did not find evidence of $A$" as evidence for $\not A$, and vice versa. (I also combine inference in tests for difference and equivalence.) If you want to find evidence of no association (or no "effect of $x$ on $y$"), you might try a test for equivalence, like $\text{H}_{0}\text{: }|\beta| \ge \Delta$, with $\text{H}_{0}\text{: }|\beta| < \Delta$ where $\Delta$ is the smallest absolute value of slope size demarcating "a large enough slope to care about" and "a slope equivalent to zero". Dec 26, 2021 at 2:58
• I think this has been already answered a few times. Check this: stats.stackexchange.com/questions/73646/… Dec 26, 2021 at 3:58