# General statistical techniques

Since risks are associated with probability techniques, it seems natural that we use probability distribution functions to help describe the impact and probability of the various risks in the project.

Figure 8-3 shows a skewed probability distribution. This is typical for project cost and schedule risks. The distribution shows the possible occurrences of cost or schedule completion for a particular task along

Figure 8-3: **Typical probability distribution for cost and schedule risk**

the X axis and relates them to the probability of that possibility occurring along the Y axis. The various possibilities are due to the risk associated with the task. For example, a task in a project has a most likely date. This date is plotted along the X axis. The probability associated with this date has the highest probability of any other date and is plotted at the corresponding point on the Y axis.

In a probability distribution the most likely date will always be at the peak of the probability distribution curve. This is not necessarily the average date for the task, which in a skewed distribution can be higher or lower, earlier or later, than the most likely date. Notice that the optimistic and pessimistic dates are the earliest and latest dates on the X axis and correspond to the lowest probability.

There are many distributions that can be applied. They can be symmetrical or skewed. Figure 8-4 shows a few that might be used in risk analysis—triangular, even, normal, and skewed distributions. The triangular distribution shows that probabilities increase uniformly from the optimistic point to a certain point where the highest probability is reached and then decrease uniformly until the pessimistic point is reached.

The even distribution has the characteristic that any value on the X axis has exactly the same probability of occurring. There is no optimistic, pessimistic, average, or most likely point.

The normal distribution is one that most of us have seen many times. It is a convenient distribution because calculations associated with it are simple to make and are generally close enough for most phenom-

Figure 8-4: **Probability distributions**

ena that we need to estimate. In the normal distribution the mean value and the most likely value are the same because the distribution is symmetrical. A special measurement, the standard deviation, relates specific ranges of values along the X axis with the probability that the actual value will be between the high and low value. This is particularly useful in project management because it allows us to predict a range of values along with a probability that the actual value will occur when we do the project. A skewed distribution is one where the most likely value on the X axis is different from the mean value. The skewed distribution is frequently encountered in cost estimating and schedule estimating for projects.

# Computer simulations

Today computer simulations are quite simple and inexpensive to use. Not many years ago simulations were done on analog computers, which made them expensive and not very accurate. The digital computers that most people have on their desks now are able to run simulations quite easily. Simulations use a model to simulate the real phenomena that we are trying to find out something about. There are two reasons to use simulations. One is that solving the problem mathematically is very difficult and expensive or even impossible. The second is that studying the actual phenomena is impossible or impractical in full scale. In either case simulation or modeling can be practical.

The most popular simulation for project management is the Monte Carlo simulation. This technique was discussed in Chapter 5. Monte Carlo analysis is important because it completes the PERT analysis of schedule estimation. In the PERT analysis we predict schedules based on ranges of values and probability for the durations of the project tasks. Since the durations of the tasks can be a range of values, it is possible that the actual duration values will determine a critical path that is not the one that is predicted by the most likely values. The Monte Carlo analysis evaluates these possibilities and gives us statistical guidelines for the project schedule.

Other computer simulations can be used to analyze the risks associated with the engineering, manufacturing, sales, marketing, quality, and reliability of the project deliverables.