Probability distributions and calculations for Hake's ratio statistics in measuring effect size

Ratio statistics and distributions play a crucial role in various fields, including linear regression, metrology, nuclear physics, operations research, econometrics, biostatistics, genetics, and engineering. In this work, we examine the statistical properties and probability calculations of the Hake normalized gain as a measure of effect size and educational effectiveness in physics education. Leveraging existing knowledge about the Hake ratio as a ratio of normal variables and utilizing open data science tools, we developed two novel computational approaches for computing ratio distributions. Our pilot numerical study demonstrates the speed, accuracy, and reliability of calculating ratio distributions through (1) DE quadrature with/without barycentric interpolation, a very quick and efficient quadrature method, and (2) a 2D vectorized numerical inversion of characteristic functions, which offers broader applicability by not requiring knowledge of PDFs or the independence of ratio constituents. These numerical explorations not only deepen the understanding of the Hake ratio's distribution but also showcase the efficiency, precision, and versatility of our proposed methods, making them highly suitable for fast data analysis based on exact probability ratio distributions. This capability has potential applications in multidimensional statistics and uncertainty analysis in metrology, where precise and reliable data handling is essential.