# Proving non-correlation with very disperse distributions

I'm fairly new to statistics and came up with a problem.

I have a sample with a variation coefficient CV = 0.517 for variable x, and I want to prove this variable is not correlated with a second variable y.

In my college textbook it is said that with CV within 0 and 0.1 the mean of the sample is representative, so that would make the mean of my x variable potentially unrepresentative. Therefore, Pearson coefficients of correlation are not to be fully trusted. Linear regressions aren't suitable either. When plotted in a scatter plot (as shown below), the variables seem to not be correlated.

I want to prove they aren't correlated. I can not use the Pearson coefficient (which is r = -0.196) because of the reason explained before. What can I use to show their lack of correlation?

I'm fairly new to statistics, as you may notice. Thank you in advance. • Please tell us what a "variation coefficient" is. It sounds like you might intend to write "coefficient of variation" (CV) but that wouldn't make any sense in this context, because it is purely a property of a univariate variable or dataset and supplies no information about the association or relationship between two variables. Please know, too, that you cannot prove a lack of correlation: you can only assess it. It is crucial to distinguish the correlation in the data (which you have measured at -0.196) from any correlation you might be trying to infer: which is your question about?
– whuber
Feb 7, 2022 at 16:10
• The use of "variation coefficient" instead of "coefficient of variation" is a consequence of the fact that I'm not an English native speaker. I studied statistics in my native language, it was just poor translation on my part. I did mean the coefficient of variation. I don't understand why CV doesn't affect the measures of association of two variables. The Pearson coefficient is calculated using the Z scores of two variables and is (to my understanding) affected by the existence of outliers. Therefore, very disperse distributions should produce less reliable Pearson coefficients. Feb 7, 2022 at 20:18
• I apologize again for my poor English. Feb 7, 2022 at 20:19
• Z scores are affected, purely and simply, by all the data on which they are based. Calculating a Z score assumes nothing about how the data were generated or their distribution. The Pearson correlation tells you nothing about the CV of the variables and the CVs of the variables tell you nothing about the correlation.
– whuber
Feb 7, 2022 at 20:26
• I know the CV doesn't say anything about the correlation, I'm saying it affects the measures of correlation. I looked this up in Wikipedia because I thought maybe I was being crazy. It says "Like many commonly used statistics, the sample statistic r is not robust, so its value can be misleading if outliers are present". Therefore, it still is not clear to me how to measure the correlations among variables that have outliers or that vary too much, since the Pearson coefficient is unreliable on this cases. Feb 7, 2022 at 21:04