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## Time Management—Formulas relating to PERT analysis

In preparing for the PMP exam, it is of course a good idea to get a PMP exam prep text that has sample test questions at the end of each chapter covering one of the knowledge areas on the exam, but to really make sure you are ready for the exam, I strongly recommend getting text prep software that will allow you to test yourself on a larger bank of exam questions.

When reviewing the Time Management chapter, it seemed that questions involving the formulas involved with PERT analysis came up with high frequency on the practice mini-exams. The purpose of this post is to review these formulas and to give an example of the kind of question you would get on the examination that would involve them.

1. Introduction

Well, what is PERT? It is a variation on the three-point estimate, which can be used for estimating costs or time durations. A one-point estimate would be a “best guess” of what the time duration would be, let’s say. However, a more accurate estimate would be a three-point estimate, with one of the points being the most likely duration, then a pessimistic estimate of the duration, and an optimistic estimate of the duration.

Anybody who drives in LA is intuitively familiar with the concept, even if they are not a project manager, quality engineer, or have any interest in statistics whatsoever. If Mapquest tells you that to go from point A to point B it takes 30 minutes, then you assume that it is the most likely estimate. However, if there is a risk to your being late, let’s say, if it is an appointment with a client or a job interview, then you would probably do a more pessimistic estimate of, say, 45 minutes, to account for the risk of there being a traffic accident or something that will slow you down. And if you are fortunate to have a GPS system or even a smartphone that tells you about any traffic accidents that have occurred on your route, you have the possibility of reducing the risk by taking an alternate route. An optimistic estimate might be 25 minutes, if you just happen to hit all the green lights on the way to the freeway. So this concept of optimistic and pessimistic estimates is something people do all the time, but the PERT analysis formalizes it in the context of project management.

2. PERT formulas

The regular three-point estimate is simply the average of the pessimistic (P), most likely (M), and optimistic (O) estimates or

(P + M + O)/3

PERT analysis takes the weighted average, which gives 4 times as much weight to the most likely or M estimate. Here’s the formula for the PERT or weighted average:

Formula 1: Weighted average

(P+ 4M + O)/6

Here’s how to remember the six in the denominator: instead of the regular average, with the weighted average, it is like you are adding M four different times to the sum, but since you have now six terms to cover in the average, and hence you divide by six.

The standard deviation is the measure of the difference between the values in the distribution on either end and the value in the middle (the average). The bigger the standard deviation, the wider the spread of the values. The formula for the standard deviation for PERT analysis is

Formula 2: Standard deviation

(P – O)/6

NOTE: This is really an approximation of the standard deviation for the purpose of statistically challenged project managers. Any person with a Six Sigma background would take a look at this formula and laugh. However, in the spirit of professional harmony, a project manager can safely ignore any sound of derision coming from the quality department and use the formula safely in the knowledge that it’s what’s recommended by PMBOK.

The variance is just the standard deviation squared, or

Formula 3: Variance

[(P – O)/6]2

This is used if you trying to get the estimate for the whole project. What you do is get the total of the estimates for all activities A through Z along the critical path. Then you take the variance for each activity, sum them up across the whole project, and take the square root to get the standard deviation for the project estimate. However, for questions only involving one activity, you will only need formulas 1 and 2 for the weighted average and standard deviation, respectively.

There’s one more thing you need to know for PERT analysis question is the following three figures for the standard deviations:

±1σ = 68.27%

±2σ = 95.45%

±3σ = 99.73%

If you have a “normal” distribution, and you try to calculate those values which are plus or minus 1 sigma or standard deviation from the average (aka the mean), you can be assured that 68% of the values fall within that range. Now on the exam, it will say “95.45%” and you are expected to know that that is 2 standard deviations, similarly with “68.27%” for 1 standard deviation and “99.73%” for 3 standard deviations or 3 sigma.

NOTE: That is, by the way, the origin of the term “six sigma”, meaning three sigma or standard deviations above and below the mean, meaning you are striving for quality such that 99.73% of your production is without defect.

3. Sample question

Let’s take a three-point estimate example from the world of IT. If programming a certain module will take an average programmer 25 hours to do as the most likely estimate, the pessimistic estimate of 35 hours may be based on the assumption that a newbie is assigned to the task, whereas the optimistic estimate of 20 hours may be if they can get the veteran programmer Jones, who is rumored to dream in code. Using the weighted average technique, you determine that there is a 95.5% probability that the module will be completed in:

1. Between 25.83 and 28.33 hours
2. Between 23.33 and 28.33 hours
3. Between 25.83 and 30.33 hours
4. Between 20.83 and 30.33 hours

Step 1: What is the PERT weighted average?

Weighted average = (P + 4M + O)/6 = (20 + 4*25 + 35)/6 = 25.83 hours

Step 2: What is the standard deviation?

Standard deviation = (P – O)/6 = 2.5 hours

Step 3: What is the range?

This last step means that the upper end of the range equals the weighted average + (1, 2, or 3 standard deviations), and the lower end of the range equals the weighted average – (1, 2, or 3 standard deviations). How do you know how many standard deviations you add or subtract? You have to infer that from the 95.5% probability figure, which is a confidence level corresponding to 2 standard deviations away from the weighted average.

So the upper end of the range = weighted average + 2σ = 25.83 + 2*2.5 hours = 30.83 hours

And the lower end of the range = weighted average – 2σ = 25.83 – 2*2.5 hours = 20.83 hours

The answer therefore is D. The other answers are for people that assume that it is only one standard deviation and/or those who remember you have to get the range by adding AND subtracting from the weighted average.

In the next post, I will move onto the question of the critical path and how you can make it work for you as a project manager.

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### 6 Responses

1. This is a very clean breakdown of how to solve these types of questions. Better than what was presented in my PMP Exam Prep courses.
Thank you.

Cody W., San Diego CA

2. The “NOTE” about 6 sigma is incorrect. 6 sigma has 6 sigmas on either side of the mean, with a built in 1.5 sigma shift to give 4.5 sigma which equates to 3.4 defects per million (DPMO), or 99.99966%

• Thanks, Bob! I appreciate any and all who correct my mistakes on the blog.

3. […] The busiest day of the year was November 3rd with 1,427 views. The most popular post that day was Time Management—Formulas relating to PERT analysis. […]

4. wonderful information

5. It was a very good read… Thank You…