# Effect of Formation Size on Flocking Formation Performance for the Goal Reach Problem
## 3 Results
### Experimental Results
#### 3.2.1 Experimental Results in an Environment without Obstacles
The \(p\)-value obtained for each of these models was less than 0.001, which means that the sample data provides enough evidence to reject the null hypothesis, and that changes in the flock size are significantly associated with shifts in the distance to center of mass, on the population level. The \(R^{2}\) value associated with the inverse function for the distance to the center of mass indicates that 69.7% of the variation in the distance to center of mass is explained by the flock size, while the \(R^{2}\) value associated with the cube function was somewhat higher than that, at 71.1%. Speed matching time: A similar observation of distance to center of mass can be drawn for the speed matching time measure. Figure 5h shows curve estimates for each of the six functions relating the flock size to the speed matching time. The \(p\)-value obtained for each of these models was less than 0.001, which means that the sample data provides enough evidence to reject the null hypothesis, and that changes in the flock size are significantly associated with shifts in the speed matching time on the population level. For the speed matching time, the highest \(R^{2}\) value is that associated with the inverse function and it accounted for only 12.3% of the variation in speed matching time due to changes in the flock size. Collisions per member: The average collisions per member tend to increase along with the increase in the flock size as shown in Table 3. The number of collision avoidance steps needed for a larger flock size is larger than the number needed for a smaller flock size. The scatter plot in Figure 5c also shows that the standard deviation is small (2.5 collisions). The \(p\)-value obtained in each of the six curve estimation models was less than 0.001, which means that the sample data provides enough evidence to reject the null hypothesis, and that changes in the flock size are significantly associated with shifts in the number of collisions on the population level. However, the \(R^{2}\) values associated with these models are a bit lower than those obtained for the models of the distance to center of mass variable. The \(R^{2}\) value associated with the quadratic function indicates that 31.5% of the variation in the number of collisions is explained by the flock size, while the \(R^{2}\) value associated with the cube function was closely higher than that at 33.7%. Separations per member: Table 3 shows a steady increase in the average number of applied separation steps along with the increase in the flock size. The scatter plot in Figure 5d) clearly supports that. The shape of the scatter is close to a quadratic or a cubic function. In fact, the \(p\)-value obtained in each of the six curve estimation models was less than 0.001, which means that the sample data provides enough evidence to reject the null hypothesis, and that changes in flock size are significantly associated with shifts in the number of applied separation steps on the population level. The highest \(R^{2}\) value is that associated with the cubic function (81.4%), closely followed by the \(R^{2}\) value associated with the quadratic function at 79.5%. Alignments per member: Similar to goal localization time, Table 3 shows that the average number of alignment steps does not exhibit any monotonic relation along with an increase in the flock size. The distribution of data points in the scatter plot shown in Figure 5e supports this observation. It appears that the average number of alignment steps per member is not affected by the flock size. The scatter of data points appears in every flock size. Figure 5e also shows curve estimates for each of the six functions relating the flock size to the average number of alignment steps per member. The \(p\)-value obtained in each of these models was larger than \(\alpha=0.05\) (Appendix B); thus, the null hypothesis cannot be rejected, meaning that there exists no association between changes in flock size and shifts in the number of alignment steps.