The notion of risk permeates every major PM process; and during my preparation for the PMP exam I have learned that Risk Management probably carries more weight than I had usually conceded.

• If we use PERT Analysis to solve a three-point estimate problem correctly, should we automatically use PERT as well in other risk-related questions of the PMP exam?
• Would a Triangular Analysis be perhaps a better choice in some cases?
• What is the real difference between PERT and Triangular analysis and why has PMI named the former as the standard procedure for three-point estimates?

Consider the following practice question borrowed from Crowe's book and slightly modified to serve the purpose of this post:

You have asked a team member to estimate the duration of a specific activity, and she has reported back to you with three estimates. The best-case scenario is that the activity could be completed in 3 days; however, because of some contingencies in the horizon, her most likely estimate for the task is 7.5 days. She has also indicated that there is the possibility the task could take as long as 10 days. Which of the following is closest to the three-point estimate for this activity?

• A. 10.5 days.
• B. 7.5 days.
• C. 7.1 days.
• D. 6.8 days.

The answer to this question is fairly straightforward if one has memorized the appropriate formula from the PMBOK, 4th Edition, Three-Point Estimate, p. 150, PERT analysis calculation of expected activity duration using a weighted average of the three estimates:

Plugging-in the numbers one gets:

The correct answer to the practice question would therefore be C.

• Option A is above the optimistic estimate so it could not be correct.
• Option B is equal to the most likely estimate and, in some cases, this estimate might be equal to the expected value. Option B, however, would only be correct if we were using, for example, a Normal Distribution, which is symmetrical, instead of a PERT Distribution.
• It is important to notice that even though the wording of the question does not explicitly require that a PERT analysis be applied, this requirement must be assumed as a PMI set standard for three-point estimates.
• Option D would also be a correct answer, but not the best one as to PMI standards.
• Option D assumes a Triangular Distribution. The PMBOK, 4th Edition, Probability Distributions, p. 297, states that this distribution is widely used and is compatible with the data typically produced in quantitative risk analysis.

The formula for the expected value in a Triangular Distribution is:

Plugging-in the numbers from the practice question one gets the following result for the expected value:

This would then be an acceptable answer but not the best if we keep PMI standards.

In order to compare PERT and Triangular distributions, I ran two simulations consisting each one of 60,000 random measurements of the activity duration. I applied separately PERT and Triangular distributions to the case at hand in order to obtain those measures. The results are described in the following table.

The first column of the table shows different intervals for the activity duration. The range goes from 3.0 to 10.0 days, and each row represents a bin containing a number of measures within the indicated limits. The next three columns register for each distribution the number of measures per bin, the probability of a given measure to fall into a given bin, and the corresponding cumulative probability.

The following graph illustrates the same data in a form that is more amenable to comment. From the results, I believe that, at least in the case at hand, using PERT or Triangular is indistinct. Both work very well in the example. The Triangular distribution has, of course, the advantage that it is easier to work with mathematically. (Please, click on the graph to render it in a new, resizable window so you can inspect its contents comfortably). Thanks a lot for reading this post. Any comments or precisions to what I have stated are very welcome.