## 3 Results
### Assessing the Distribution and Spread of Simulated Grades
The Summative method output (\(\chi^{2}=1.0663\), \(p=0.5868\)) suggests a good fit to a normal distribution, as the \(\chi^{2}\) statistics resulted in the lowest of all methods, while its \(p\)-value is relatively high. The ST method, as indicated by the \(\chi^{2}\) value (\(\chi^{2}=1.3905\), \(p=0.7078\)), still yielded a good fit to a normal distribution due to its relatively low \(\chi^{2}\) value, which, along with the highest \(p\)-value, exhibited the best fit among the four methods. Correspondingly, the Rubric method was associated with a \(\chi^{2}\) value of 3.1181 (\(p=0.3738\)), which, despite a relatively high \(p\)-value, yielded a higher Chi-squared statistic, indicating a weaker fit to normality compared to the Summative and ST methods. Finally, the ST-FIS method yielded \(\chi^{2}=3.5090\), \(p=0.1730\), thereby holding the highest \(\chi^{2}\) value along with the lowest \(p\)-value, suggesting that its grade distribution deviates more from normality compared to the other methods but still does not show strong evidence of deviation from normality. To summarize, the Rubric resulted in the highest average student grades, but its fit to the normal distribution was relatively less consistent than those of other methods. Compared to the Rubric method, the Summative and ST methods produced similar average grades with moderate variability and a better fit to normality. The ST-FIS, while having a reasonable mean score, exhibited the widest variation in grades and a relatively weak fit to a normal distribution. Despite the commonly held assumption of normality of a grade distribution, we acknowledge that grade distributions in real-world educational settings may deviate from normality due to several factors (such as grade inflation, institutional policies, or specific assessment practices). Our use of normal distribution fitting here is not intended to imply that real grades always follow a normal law; rather, it serves as a diagnostic tool with which to assess whether the simulated methods produce distributions that are balanced and unbiased under controlled conditions. This provides an additional layer of internal validation for the proposed simulation methods. To fulfill the need for detailed statistical visualizations, we performed 100 simulation runs for each method to obtain distributions of the Consistency indicator (\(CI_{M}\)) and the Efficiency indicator (\(\beta_{M}\)). Figure 3 presents boxplots of the simulated \(CI_{M}\) values across methods. These plots provide a detailed view of variability, medians, and overall distributional behavior. The ST-FIS method exhibits the highest median consistency and a relatively narrow spread, indicating strong central performance and good stability. Although the Summative method shows a slightly narrower interquartile range, its median value is notably lower, and the overall dispersion is comparable. Similarly, Figure 4 displays the distribution of simulated \(\beta_{M}\) values. The ST-FIS method again stands out by achieving both the highest median efficiency and one of the smallest ranges, suggesting better and more consistent performance. These figures offer a comprehensive view of each method's behavior under uncertainty and help identify those with the most robust and reliable outcomes. Figure 4: Boxplot of simulated values of the Efficiency indicator \(\beta_{M}\) across grading methods \(M\). Panel (**a**) Summative. Panel (**b**) Rubric. Panel (**c**) ST. Panel (**d**) ST-FIS. The ST-FIS method stands out by achieving both the highest median efficiency and one of the smallest ranges, suggesting better and more consistent performance. Figure 3: Boxplot of simulated values of the Consistency indicator \(CI_{M}\) across grading methods \(M\). Panel (**a**) Summative. Panel (**b**) Rubric. Panel (**c**) ST. Panel (**d**) ST-FIS. Each boxplot summarizes 100 simulation runs per method and displays the median, interquartile range, and potential outliers.