The trap of Pearson's product-moment correlation: How the instability and mathematical indetermination of this coefficient have made work inaccurate o

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Do you truly believe that Pearson correlation coefficients of r=0.70 and above are strong and reliable? Perhaps not! Widely utilized in scientific and commercial contexts, Pearson’s product-moment correlation coefficient assesses variable relationships but lacks inferential properties. While a clever tool, it faces criticism for its sensitivity to outliers in small samples and its limitations in assessing nonlinear relationships. To identify linearity, one must plot data dispersion using Recursive Least Squares (RLS) estimation, yet regression lines fail to separate residuals from explanatory components. I propose that Pearson’s coefficient is mathematically indeterminate—akin to zero-by-zero division—when one covariate's variance is predominantly made up of residuals. This indeterminacy is challenging to demonstrate, as it relies on deviant data points that obscure the issue. When a variable's relationship is largely influenced by residuals, it reflects Heisenberg’s uncertainty principle: increased knowledge of one variable reduces knowledge of the other. Here, the only representative of a variable x is its mean, nullifying the explanatory power of its covariates and rendering Pearson’s coefficient and regression analyses unfeasible. I offer a method to identify proximity to this mathematical indeterminacy. Through numerous examples, I illustrate the instability and lack of scientific support for results derived from Pearso