Adaptive Multi-Interval Scale (AMIS): Construction of a Unified Metric Space for Heterogeneous Data

Context and relevance: Comparison and integration of heterogeneous metrics with different units of measurement and statistical distributions is a fundamental problem in interdisciplinary research and applied analytics, especially in fields such as psychology, sociology, and educational studies. Existing methods (linear scaling, z-standardization) have critical limitations: ignoring the distribution shape, loss of interpretability, and the lack of a unified metric space for correct arithmetic operations. Objective: Development and presentation of a new normalization method – the Adaptive Multi-Interval Scale (AMIS) – for constructing a unified metric space that enables correct comparison and analysis of heterogeneous data. Methodology: AMIS implements a novel approach to normalization by constructing an adaptive scale through the iterative calculation of control points—the centers of mass of successively subdivided regions within the source data. A piecewise-linear transformation based on these points projects the data onto a 0–100 interval scale, where equal intervals correspond to statistically equivalent segments of the data. This ensures natural adaptability to any distribution shape, including skewed distributions and those with outliers. Results: The practical efficacy of AMIS is proven with two examples. In educational analytics, the method eliminates systematic errors when aggregating academic grades across different subjects, ensuring correct calculation of integral indicators. When applied to world GDP data, AMIS adequately scales distributions with extreme outliers, unlike linear normalization and z-standardization, which proved inadequate. Conclusions: The proposed AMIS method solves the problem of creating a unified interval space for initially incomparable indicators. It combines interpretability, adaptability to distribution, and the metric rigor necessary for mathematically correct operations of comparison, averaging, and weighting. The method has broad application prospects, primarily in psychological, sociological, and educational research, as well as a data preprocessing tool for machine learning.

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