## 4 Discussion  
We studied the relationships between a variety of measures of core strength tests and countermovement jump (CMJ) heights with/without arm-swing. Significant correlations were detected between all 12 core strength measurements and CMJ heights with/without arm-swing. The results of the linear regression analyses showed that NPDF\({}_{180}\) and NPSF\({}_{120}\) shared 72% variance with CMJ height with arm-swing (H\({}_{\text{AS}}\)), where NPSF\({}_{120}\) shared 57.0% variance with CMJ height without arm-swing (H\({}_{\text{NAS}}\)). Therefore, NPDF\({}_{180}\) and NPSF\({}_{120}\) combined were good predictors for H\({}_{\text{AS}}\), and NPSF\({}_{120}\) alone was a good predictor for H\({}_{\text{NAS}}\). Although some differences exist in how previous research explains the impact of arm-swing on vertical jump, it stresses the importance of core strength during takeoff [1; 3; 5]. Individuals with a stronger core could benefit more from arm-swing during the jump's takeoff, compared to people with a weaker core [6; 8; 9]. In the present study, significant correlations between CMJ height with arm-swing and all 12 core strength measurements were detected and supported the hypothesis. Moreover, the 12 core strength measurements were also significantly correlated with CMJ height without arm-swing, Dynamic and static core strength play different roles in exercise performance. Static core strength, especially core flexion strength, can keep the body stable and provide a good foundation for CMJ performance. Again, dynamic core strength, especially the core flexion strength, can facilitate force and power transfer to improve CMJ performance [6; 8]. Coaches and professional and recreational athletes could try and see if improving both dynamic core strength and static core strength could improve CMJ height. The relationship between core strength and performance reported in previous studies was inconsistent. Some studies have observed positive results [14; 15; 16], but others disagreed [17; 18; 22; 25]. [PERSON] et al. [15] found that spike jump and block jump improved significantly after nine weeks of core strength training. [PERSON] et al. [14] observed sprint and kicking performance improvement after core strength training on stable and unstable surfaces. [PERSON] et al. [26] found that dynamic core extensor strength correlated with drop jump height without arm-swing significantly but showed only a weak correlation between dynamic core flexor strength and drop jump height. The standing position was chosen to test the [PERSON] study's participant's core strength, but we tested the core strength with a sitting position in the present study. The sitting position could decrease the core extension strength but is an excellent position to evaluate the maximum core flexion strength. That may be the reason for strong correlation between CMJ height and core flexion strength in the present study. However, the kinetic and kinematic structures of the core strength test in sitting position were inconsistent with the CMJ in upright positon, in which the feet need to load the body weight. That may affect the degree of correlation between core extension strength with CMJ performance in the present study. We have attempted to approximate the sporting movement in the strength test. The sitting position used in  
Figure 5: Results of regression analysis of CMJ height without arm-swing (H\({}_{\text{NAS}}\)) with 12 core strength measurements (n = 20) are the same as in Figure 4.  
our study and the subsequent range of movement, approximates the position the participants would be at when they start to generate forces in the jump and the range of movement. [PERSON] et al. [17] investigated the effect of core strength and endurance training on performance. They reported that neither training protocol showed superiority, and both were ineffective in improving sprint, vertical jump, and agility performance. They suggested that the large sample size and core musculature power training should be considered for future projects.