# Tracing Student Activity Patterns in E-Learning Environments: Insights into Academic Performance
## 3 Methodology and Results
### Correlation between Final Grades and Other Metrics  
Table 4 presents the values of the metrics presented above for the five indicatively selected students. More analytically, Table 4 presents each student's final grade, Entropy, and Frobenius Norm. It also includes the Euclidean Distance between each student and the mean student profile, as well as the Cosine Similarity of pairs of vectors (student's profile vector and mean student profile vector). Table 5 presents the mean averages of students' final grades, as well as the mean averages of the metrics presented above. From the data provided in Tables 4 and 5, we can draw some indicative conclusions:  
1. Students achieving higher grades (e.g., ID 23 or ID 39) exhibit elevated Entropy, potentially indicating a natural inclination towards curiosity and exploration. This tendency may lead them to explore a broader array of learning materials and engage in diverse discussions or activities within Moodle. These actively involved students, who participate in discussions, complete quizzes, and access supplementary materials, may contribute to increased Entropy values. However, their active engagement in the learning process could also enhance comprehension and retention of the material. On the other hand, the student with ID 142, despite having low performance (final grade 5.8 out of 10), exhibits a similar entropy value to that of high-performing students with ID 23 or 39. This observation could be attributed to the fact that students with low performance may navigate Moodle with a sense of exploration but without a clear direction or purpose. They may explore various resources, engage in discussions, or attempt quizzes without a focused approach, leading to higher entropy. 2. Student ID 153 exhibits the highest Frobenius Norm value. This suggests that the student has been significantly active within Moodle, involving numerous transitions between different states or engaging extensively with various resources and activities. Despite the high level of activity, the low grade may appear contradictory, but it could imply that the student is investing considerable effort or time without attaining satisfactory academic success. This randomness in behavior might potentially contribute to the lower grade if it results in a lack of focus or ineffective study habits. \begin{table}
\begin{tabular}{c c c c c c} \hline
**ID** & **Final Grade** & **Entropy** & **Frobenius Norm** & **Euclidean Distance** & **Cosine Similarity** \\ \hline
23 & 9.5 & 1.32456 & 0.56163 & 0.04302 & 0.99713 \\
39 & 8.8 & 1.51869 & 0.50020 & 0.15418 & 0.96576 \\
95 & 7.5 & 1.29223 & 0.57918 & 0.06050 & 0.99467  
3. Although some students achieve high final grades (e.g., students with ID 23 or 39), they do not exhibit similar Euclidean Distance values. This discrepancy could stem from students attaining comparable grades through diverse learning strategies or approaches to utilizing Moodle. Certain students may heavily rely on specific features or resources within Moodle that closely align with average student behavior, resulting in low Euclidean Distances. Conversely, other students may adopt unconventional or personalized learning strategies, leading to profiles significantly divergent from the mean and with consequently high Euclidean Distances, despite achieving similarly high grades. 4. Upon reviewing the provided data, it becomes apparent that all students display relatively high Cosine Similarity values, ranging from approximately 0.96576 to 0.99713. While there are slight variations among the Cosine Similarity values of different students, they all remain close to 1, indicating a strong resemblance between each student's profile vector and the mean student profile vector in the 7-dimensional space. Similar values were calculated for all students. Contradictory and unexpected data reflect the complex and diverse ways in which students engage with Moodle and achieve academic success.