A Comparative Analysis of Gaussian, Triangular, and Trapezoidal Membership Functions in a Mamdani Fuzzy Inference System for Evaluating Mathematics Learning Achievement

Abstract

Educational assessment systems conventionally rely on crisp numerical scores that inadequately capture the gradual and uncertain nature of student learning, and while fuzzy inference systems (FIS) have been applied to address this limitation, prior studies rarely compare alternative membership function designs under identical experimental conditions, leaving open the practical question of which membership function type best supports fuzzy-based educational assessment. This study addresses that gap by presenting a comparative evaluation of Gaussian, Triangular, and Trapezoidal membership functions, which were selected because they are among the most widely used membership functions in fuzzy educational assessment and differ in their smoothness, computational simplicity, and plateau characteristics within a Mamdani Fuzzy Inference System (FIS) for assessing student learning achievement. Three fuzzy models were developed using examination score, assignment completion, and class participation as input variables, while learning achievement served as the output variable. A dataset comprising 360 secondary school students was used for model development and validation. Teacher evaluation scores were employed as benchmark values for assessing model performance. The three models were implemented using identical rule bases, inference mechanisms, and defuzzification procedures, differing only in the type of membership function employed. Model performance was evaluated using the Pearson correlation coefficient, root mean square error (RMSE), and mean absolute error (MAE). The results revealed that the Gaussian model achieved the highest correlation with teacher evaluations (r = 0.9623), indicating superior agreement with teacher assessment patterns, whereas the Triangular model produced the lowest RMSE (7.194) and MAE (4.867), demonstrating greater numerical prediction accuracy.