## 7 Discussion  
In this paper, we proposed several expressions that allow calculating the overall project duration average (\(\mu_{p}\)) and standard deviation (\(\sigma_{p}\)) from purely deterministic schedule information. Calculations are quite straightforward and were exemplified step by step by means of a small project network in Section 4. These calculations can be implemented either manually or in a simple spreadsheet. The only requirement is that the scheduler is familiar with the critical path algorithm. Basically, identifying the critical activities of a project involves the calculation of the activities' earliest and latest start dates. The difference between earliest and latest start dates is equal to the activities' total slacks (\(s_{i}\)). Eventually, by comparing the size of \(s_{i}\) against the slack admissibility threshold \(\tau\) from Equation (6), we can discriminate which activities are critical or subcritical. Hence, any basic spreadsheet that sequences activities departing from the activity durations and precedencies can implement the formulae proposed here. The only requirement is to follow the calculations in the same exact order used in the small project example. For those project schedulers who are not familiar with how to calculate the critical of bigger projects, it is always possible to resort to a commercial scheduling software (e.g., Microsoft Project, Oracle Primavera) to extract the values of the activity slacks (\(s_{i}\)). Then, the rest of the calculations can be implemented in a spreadsheet, even manually. However, beyond obtaining a more accurate estimate of the project duration average, there are other potentially interesting practical applications of the estimates proposed in this paper. Obviously, the first is the possibility of calculating a more accurate average project duration estimate (\(\mu_{p}\)) from basic deterministic schedule information with hardly any extra effort. This improved estimate will allow project managers, for instance, to plan execution and resource allocation better with lower margins of error. Also, we anticipated that calculating both \(\mu_{p}\) and \(\sigma_{p}\) values could allow us to make probabilistic inferences about any likely project duration (\(X\)). More precisely, we could say that \(z=\left(X-\mu_{p}\right)/\sigma_{p}\)  
Figure 3: Dataset I \(\sigma_{p}\) regressions with normal (**a**) and lognormal (**b**) activity duration variability. Then, assuming \(X\) follows a normal distribution, the value of \(z\) could be looked up in a standard normal distribution table and an (approximated) project completion date inferred. There are other project management tools and techniques that can also benefit from more accurate \(\mu_{p}\) and \(\sigma_{p}\) values. Earned value management (EVM), for example, is a deterministic project monitoring and control technique that also produces project duration forecasts, named \(EAC(t)\) (see [PERSON] et al. [31] for a recent and comprehensive comparison of the most relevant \(EAC(t)\) expressions). All possible parametrizations of \(EAC(t)\) include a deterministic estimate of the planned duration \(PD\) which, after this research, could be replaced by \(\mu_{p}\), enhancing their overall accuracy. Similarly, schedule risk analysis (SRA) is a proactive planning and monitoring technique which ranks project activities and determines their relative importance and exerts different levels of project control on them. SRA uses Monte Carlo simulations to calculate its metrics because, so far, there is no other way of calculating their parameters from deterministic schedule information. For example, the significance index (\(SI\)) metric uses \(E(PD)\) (the expectation of the project duration) which is our \(\mu_{p}\) here. Also, the schedule sensitivity index (\(SSI\)) and the criticality slack sensitivity index (CSS) both use \(\sigma_{p}\).