Assessment of product quality risks by qualimetric methods using functionally dependent statistics

In modern production systems, ensuring high product quality while minimising risk is a critical challenge. Traditional quality assessment methods often rely on expert judgment or complex models, which may introduce subjectivity or require large datasets. This study aims to develop a universal methodology for assessing product quality risks using a mathematically grounded approach that eliminates the need for expert-based evaluations and can be easily implemented in various industrial contexts. A qualimetric method based on nonlinear mathematical dependence using the error function “erf” is proposed. The method transforms measured quality indicators into a dimensionless scale and derives functionally dependent statistics under the assumption of a uniform distribution. The model is validated through analytical derivations and numerical experiments on piston components in precision mechanical engineering. A new mathematical model was established to calculate the probability density function of transformed quality indicators. The methodology enables the estimation of the probability that a quality indicator will fall within a risky range near tolerance limits. Numerical experiments confirmed the validity of the model, demonstrating its applicability to real-world production scenarios and its alignment with known principles of qualimetry. The proposed method provides a universal, objective, and practical tool for risk-based quality assessment. It can be applied across different industries, integrated into existing quality management systems, and used to support decision-making in production control. Future research should expand the model to accommodate nonuniform distributions and explore its integration with real-time quality monitoring systems.