A Least Squares Hyperbolic Transformation to Moderate School Assessments

Abstract

This research develops a new method to mathematically moderate school assessments to make them comparable across schools in a large-scale public examination system. In many systems the school assessment is partly used to determine the final mark in a course. A hyperbola was chosen for this method as it has two important features. First, if chosen from the second or fourth quadrants of the Cartesian plane it is always monotonic increasing. Second, it gives a curvilinear transformation which helps to overcome differences in skew between a school’s raw assessments and the school’s exam marks. If the skew difference is large it can disadvantage the top students in a linear moderation, giving them moderated assessments below their exam marks. A hyperbola is closely fitted to the moderation criterion (the sorted exam marks) using a least squares process. The moderation equation itself is simple, with three constants, but their calculation is more complex. However a central education authority should have the computing resources to calculate these constants for each school group. This new method provides another option for school systems to consider in a high-stakes large-scale public examinations environment.