## Statistical sums such as PERT

The team can assume that the numbers fall under a certain distribution curve, such as a Beta distribution. Then values can be calculated using standard formulae for the distribution. For example, average cost, and standard deviation of the cost can be calculated. Obviously the results are only as good as the selection of the appropriate distribution.

simulation

Using a standard simulation technique, such a Monte Carlo (which implies using a computer program to avoid the need for intense manual calculations), the risk of certain costs, overall project cost, final completion dates, and interim dates can be calculated. The team needs to decide on some parameters to input to the program, since the results will be only as good as the inputs. The idea here is the instead of using a single number for each activity cost or duration, and running a project program such as MS Project to calculate the project completion date or budget, the program is run many times, and each time it selects a value for each activity parameter. The values are selected by selecting values for the parameter from a given distribution. The program can be told, for each activity cost, or for each activity time, or for some of these, what distribution applies to that parameter for that activity. The user also inputs other relevant values such as the mean for that activity and the standard deviation. This allows the program to construct the curve for each parameter. The program then runs, selecting values for each parameter on each run, in such a way that the large set of values for any activity falls under the provided curve. In other words, the program does a 'what if'analysis. It asks, for every variable parameter, what if it were this value, or that value, selecting the values from the distribution expected for that item.

For each run the program will calculate the overall cost, and the project completion date. The multiple runs (the number of runs must be high enough to guarantee statistical stability) provide a distribution of completion dates or costs. These distributions show the probabilities that the project will complete within timeframes or budget windows. Hence the risk is quantified. This technique can also calculate the probability that a given activity is on the critical path, which gives information to the PM about the criticality of monitoring that item.

decision trees

Decision trees can be used to show the paths resulting from project risk events. Decision points can also be included. Probabilities and impacts are followed through each path.

Consider an example:

We want to upgrade local facilities and to purchase racks of DSL data sets to offer service in a new town - our competitor will offer high speed internet via cable. The local facilities are old, so until each is upgraded, performance will be poor. Our cost to upgrade and purchase the DSL data sets would be \$5M

Consider a decision tree, with 2 risk events:

1) Our competitor announces before us - 20% probability

2) Their new cable modem technology performs better than our DSL offering - 40% probability

When there are multiple events in a path, the second event on a path has certain probabilities these occur given that we get to that point in the path. The same applies for subsequent events. Figure 2