Quantifying and analyzing nonlinear relationships with a fresh look at a classical dataset of student scores

Student past and present performances are analyzed, compared, and reflected upon by teachers, curriculum developers, and educational researchers. For the tasks, methods and techniques of traditional statistics are frequently employed. Recent advances in statistical theory and practice, although not yet covered by widely accessible statistics textbooks, shed additional light on the area and facilitate the improvement of old, and the development of new, curricula that are better aligned with learning goals and outcomes. In the present paper we discuss and illustrate the use of an index of increase that has been designed to quantify, and thus compare, relationships between dependent random variables (e.g., student scores in different study subjects) that rarely follow linear relationships; hence, the use and interpretations of the celebrated Pearson correlation coefficient become problematic. The aforementioned index of increase is the proportion of upward movements among all the movements in the scatterplot arising from paired observations. To appreciate the index from both graphical and mathematical points of view, we have illustrated its performance using a classical and easily accessible educational dataset. We have provided examples of how the index values can aid teachers and educational researchers in determining relationships between student performances in different study subjects, and thus in turn help them in, for example, developing curricula.