## 5 Analysis
### Parameter Configuration Summary
In addition, all MAE and MSE values in Tables 3 and 4 are also accompanied by an indication of the reduction achieved in MAE and MSE in percentage terms when using our \(\mu_{p}\) (in Table 3) or \(\sigma_{p}\) (in Table 4) estimates, compared to using \(PD\) or \(\sigma_{K}\), respectively. These values are included within parentheses. Multiple readings can be made from Tables 3 and 4. For the sake of clarity, we limit ourselves to the most essential. A quick inspection of the first columns of Table 3 shows that values of \(R>0.90\) are achieved for both duration variabilities. \(R^{2}\) values are also very close to 90%. Overall, this means that there is a high degree of correlation between our estimates and the actual values. Also, in Table 3, inspection of the MAE confirms that our expressions manage to reduce the variability from the PERT estimates by around two-thirds (63% and 69%). However, the reduction in the MSE is even higher (90% and 87%). This means that, when using our estimates, values can be expected to be around two-thirds closer to the actual values compared to the deterministic project duration estimates, but also that the amount of large deviations (measured by squared errors) would be significantly reduced. Results in Table 4 are still good, but clearly not as good as those in Table 3. In this table, our project duration standard deviation estimates are not indisputably superior when compared to \(\sigma_{K}\) (which, it must be borne in mind, are already better estimates than PERT's). Still, in the case of normal variability (central block), we can observe \(R>0.80\) and \(R^{2}\) values again around two-thirds. The absolute errors are reduced by 40% on average and the squared errors by around 70%. As \(\sigma_{K}\) is already better than the PERT estimate, we can expect our estimates to double (at the very least) PERT's accuracy in absolute terms concerning its project duration standard deviation estimate. In squared terms, we could expect our estimates to improve PERT's estimate by around 75% at least. Results on the lognormal variability (rightmost block in Table 4) also improve the \(\sigma_{K}\) estimates, but not as much as hoped. Indeed, there are a couple of datasets (II and X) which worsen the \(\sigma_{K}\) estimates (negative reduction values) in the MAE, MSE, or both. These datasets coincide with the configurations with lower activity duration variability (\(CV_{i}\) = 0.10). However, configurations with higher variability (datasets III and XIII, with \(CV_{i}\) = 0.3) do not report much better results compared to the case with average variability. Overall, this means that the expression for estimating \(\sigma_{p}\), although promising because of its decent correlation values, still needs some work. This is something we already disclosed when introducing it; however, this will probably come at the expense of complicating it numerically. This is an option for future research. Tables 3 and 4 also contain two columns each reporting the values of coefficients \(K_{1}\) and \(K_{2}\). During the design stage, we experimented by letting these coefficients vary freely in order to minimize the MSE in every dataset. Results of these alternative scenarios can be found in Appendix D of the Supplementary Materials (Tables S3 and S4). It can be seen, however, that the reduction from making coefficients \(K_{1}\) and \(K_{2}\) constant did not have a substantial impact on the expressions' performance. Finally, for those visual readers, Figures 2 and 3 represent the regression graphs of the first dataset (dataset I). Dataset I can be considered the most representative of all datasets, as it approximately presents the average parameters and distribution configurations. In Figures 2 and 3, the top graphs represent the regression results in ratio scale. On observing the \(\mu_{p}\) estimates (in Figure 2), it is evident that they are superior to the \(\sigma_{p}\) estimates (in Figure 3) as the datapoint clouds of the former follow a clearer bisector pattern.