## 3 Experimental Design
### Variable Analysis
#### 3.2.2 Multivariate Analysis  
To investigate the interrelationships among physical fitness indicators, multivariate analysis [32] was employed. Figure 3 presents the Pearson correlation heatmap of selected physical fitness indicators. Pearson's correlation coefficient was employed because it quantifies linear associations between continuous variables, which is suitable given that most physical fitness test scores approximate a normal distribution [33]. This type of multivariate analysis enables the simultaneous evaluation of interrelationships among multiple variables, allowing for the detection of interaction effects and multicollinearity that are not discernible through univariate methods alone. Such an approach provides a more comprehensive understanding of how different fitness components contribute to overall physical health and lays the foundation for predictive modeling and targeted intervention design. Figure 3: Results of multivariate analysis of the dataset. The heatmap reveals several noteworthy correlations. The 800 m running score shows a moderate positive correlation with the total fitness score (r = 0.74), while sit-up performance also demonstrates a moderate association (r = 0.53). The 50 m sprint score is positively correlated with the 800 m score (r = 0.41), indicating that both reflect aspects of cardiovascular endurance. The standing long jump rating exhibits a weak-to-moderate correlation with sprint performance (r = 0.38), suggesting a shared component of lower-body power. In contrast, the 'year' variable shows negative correlations with several performance indicators, including the 50 m sprint score (r = \(-\)0.38) and total score (r = \(-\)0.39), possibly reflecting a temporal decline in students' fitness levels. Correlation strengths were interpreted according to [PERSON]'s conventional thresholds: \(|\,\)r\(\mid\) \(<\) 0.1 (negligible), 0.1 \(\leq\)\(|\,\)r\(\mid\) \(<\) 0.4 (weak), 0.4 \(\leq\)\(|\,\)r\(\mid\) \(<\) 0.7 (moderate), and \(|\,\)r\(\mid\) \(\geq\) 0.7 (strong). This objective classification clarifies the significance of observed relationships and ensures consistent interpretation across variables [34; 35]. The analysis was extended by employing a Pearson correlation heatmap to quantify linear associations between variables, such as the moderate to strong relationships observed between 800 m running scores and total fitness score (r = 0.74) or between sit-up performance and total score (r = 0.53). It also reveals potentially redundant variables and inverse trends over time, with the 'year' variable negatively correlated with multiple performance metrics, suggesting a temporal decline in physical outcomes. This multivariate approach deepens our understanding of the interconnected nature of physical fitness components, beyond what univariate methods alone can reveal. In Figure 4, the pair plot serves as an integrative tool that validates and visually reinforces insights from both Figures 3 and 4. The diagonal histograms in Figure 4 confirm earlier distributional observations, such as skewness in BMI and sit-up scores, while the off-diagonal scatter plots visualize the correlations outlined in Figure 4, such as the alignment between the sit-and-reach and total score, or the weak association between pull/sit-ups and overall performance. Additionally, it enables the rapid identification of outliers and nonlinear patterns not easily captured by correlation matrices, while also informing the refinement or exclusion of non-informative features based on visual redundancy or distributional anomalies. In summary, Figures 2-4 collectively provide a layered and complementary framework for exploratory data analysis (EDA), facilitating both distributional assessment and inter-variable relationship exploration.