The Limited Usefulness of Population Correlation Coefficients on Inferring the Magnitude Relationship of Two Y Values on the Basis of Their Paired X Values

The population correlation coefficients are the indices of the correlational strength of two variables, X and Y. However, when asked what it means, people often suggest that a positive correlation means a higher value on variable X is coupled with a higher value on variable Y. Then, they implicitly and even explicitly expect that if sample A is higher on variable X than sample B, sample A's value on variable Y is also higher than that of sample B. But this inference is not always correct unless the correlation coefficient is one. The present study analyzed the triad relationship between the probability of such inferences' accuracy, the strength of the correlation, and the standardized distance between two X values of samples A and B. The conclusion is that such practice has very limited usefulness in realistic situations. The article proposes using a 95% range of variability of Y when one needs to expect the Y value based on the paired X value, or reporting the probability of dominance when comparing two samples is necessary.