## 3 Experimental Design
### Cluster Analysis
#### 3.3.1 Selection of the Optimal K Value  
1. K-means Clustering and Determination of the Optimal K Value  
In this study, K-means clustering is employed to uncover potential grouping patterns in the physical health test data of the 2020 cohort of students from Yunnan Agricultural University and Southwest Forestry University. Considering that the effectiveness of the K-means algorithm is highly sensitive to the number of clusters (K) and the initial centroid selection, this study integrates the canopy algorithm as a preprocessing step to enhance clustering performance and robustness. The canopy algorithm rapidly divides the dataset into overlapping subsets (canopies) using a loose similarity measure defined by two distance thresholds (T\({}_{1}\)\(>\) T\({}_{2}\)). A canopy refers to a preliminary cluster or region that helps roughly group similar data points. This step effectively reduces the computational complexity of K-means by narrowing the search space for centroid initialization and providing a more informed basis for selecting the optimal K value. Following the canopy-based partitioning, fine-grained clustering is conducted within each canopy using the K-means algorithm. This hierarchical approach not only improves clustering accuracy but also enhances computational efficiency, particularly for handling the high-dimensional and large-volume dataset collected in this study. In our study, the canopy algorithm served not only as a coarse clustering method to group similar data points, but also played a critical role in narrowing the range of candidate values for K in the subsequent K-means clustering. Specifically, based on the initial canopy partitioning, we limited our search for the optimal number of clusters to the range of K = 3 to 10 for male students, and K = 4 to 10 for female students. This reflects the data characteristics revealed by the canopy structure and helped reduce computational overhead while maintaining clustering quality. To further ensure the scientific validity of the clustering results, this study combines the elbow method with the Calinski-Harabasz index to determine the optimal number of clusters. Additionally, the silhouette score is employed to assess the overall quality of the clustering outcomes. Figure 8: Boxplot of each characteristic attribute of females in the dataset. Note: “The red line in each box represents the median value of the corresponding characteristic attribute”. Application of the Elbow Method  
The elbow method is a commonly used technique for determining the optimal K value [36]. The core idea is that as the number of clusters (K) increases, the intra-cluster cohesion improves, and the sum of squared errors (SSE) decreases gradually. When K reaches the true number of clusters, the rate of decline in SSE slows down significantly, forming an \"elbow\" point in the graph. The K value corresponding to this \"elbow\" is considered the optimal number of clusters. However, the elbow method can be somewhat subjective in practice, especially when the data distribution is complex or when the \"elbow\" is not distinct. Relying solely on the elbow method may lead to bias. Therefore, this study introduces the Calinski-Harabasz index [37] as a supplementary verification tool. Introduction and Validation of the Calinski-Harabasz Index  
The Calinski-Harabasz index (CH index) is an important metric for evaluating the effectiveness of clustering. The calculation formula is as follows:  
\[CH=\frac{Inter-cluster\ dispersion}{Cluster\ density}\times\frac{n-k}{k-1} \tag{5}\]  
\(n\) is the total number of data points and \(k\) is the number of clusters. The greater the between-cluster dispersion, the more distinct the differences between clusters. The smaller the within-cluster compactness, the closer the data points within the same cluster are to each other. Thus, the optimal clustering solution corresponds to the value of \(K\) that maximizes the \(CH\) index, as it represents the best clustering outcome [38; 39].