Zrelated

Zrelated is a term used in statistics and data science to describe a type of association between two variables after standardization to z-scores. The term is not widely standardized in formal literature but appears in software documentation and methodological discussions to emphasize analysis in standardized units.

Definition: Given variables X and Y with means mu_X, mu_Y and standard deviations sigma_X, sigma_Y, define Z_X = (X - mu_X) / sigma_X and Z_Y = (Y - mu_Y) / sigma_Y. X and Y are said to be Zrelated if the standardized pair (Z_X, Z_Y) exhibits a strong linear or monotone relationship, as quantified by a correlation measure such as Pearson's r or Spearman's rho applied to the z-scores. In practice, because standardization is a linear transformation, Pearson correlation between Z_X and Z_Y equals that between X and Y; thus Zrelated emphasizes comparability across variables on different scales rather than altering the fundamental association.

Origin and usage: The term is encountered in methodological notes and some software workflows that preprocess data by standardization before modeling or visualization. It is often used informally to indicate that relationships observed are robust to scale and centering.

Properties: If X and Y are perfectly linearly related, Zrelated is achieved with r = ±1 in z-space, regardless of the original units. If one variable is transformed nonlinearly, Zrelated may not hold; nonparametric measures on z-scores can capture monotone associations.

Applications: Preprocessing for machine learning, cross-variable comparisons, outlier-resilient visualization, and exploratory data analysis.

See also: Standardization, Z-score, Correlation, Spearman's rho, Principal component analysis.