Use of the Triangular Fuzzy Numbers for Student Assessment

In an earlier work we have used the Triangular Fuzzy Numbers (TFNs)as an assessment tool of student skills.This approach led to an approximate linguistic characterization of the students’ overall performance, but it was not proved to be sufficient in all cases for comparing the performance of two different student groups, since tywo TFNs are not always comparable. In the present paper we complete the above fuzzy assessment approach by presenting a defuzzification method of TFNS based on the Center of Gravity (COG) technique, which enables the required comparison. In addition we extend our results by using the Trapezoidal Fuzzy Numbers (TpFNs) too, which are a generalization of the TFNs, for student assessment and we present suitable examples illustrating our new results in practice.

💡 Research Summary

The paper revisits the use of fuzzy numbers for assessing student performance, addressing a fundamental limitation of earlier work that employed Triangular Fuzzy Numbers (TFNs). In the original approach, each student’s achievement was represented by a TFN (a, b, c), where a denotes the lowest possible score, b the most plausible (modal) score, and c the highest possible score. While this representation captured linguistic labels such as “low,” “average,” and “high,” it suffered from a comparability problem: two TFNs are only partially ordered. When the intervals of two TFNs overlap, it is impossible to say definitively which one reflects a higher level of achievement. This non‑comparability hampers any attempt to rank or compare groups of students, such as different classes, cohorts, or instructional interventions.

To overcome this obstacle, the authors introduce a defuzzification technique based on the Center of Gravity (COG) method. The COG treats the fuzzy number as a shape with uniform density and computes the x‑coordinate of its mass center. For a TFN, the COG reduces to the simple arithmetic mean (a + b + c)/3. By converting each TFN into a single real number, the COG restores a total order: any two students (or groups) can now be compared directly by their COG values. Importantly, the COG retains the essence of the underlying fuzzy representation, because it is derived from the entire shape rather than a single point estimate.

The paper further extends the methodology by incorporating Trapezoidal Fuzzy Numbers (TpFNs), a generalization of TFNs defined by four parameters (a, b, c, d). In a TpFN, the interval