Category: Project Management Professional (PMP)® Exam using A Guide to the Project Management Body of Knowledge (PMBOK® Guide)

PMP® Formulas and Calculations - The Complete Guide

When I speak to Project Management Professional (PMP)® students, there's often one thing on their mind: How are they going to learn all the PMP® formulas they need for their Project Management Institute (PMI)® exam?

There's no getting away from it. There are a lot of these formulas and calculations that you have to learn for the exam.

Difficult math formulas for the PMP exam
PMP® Formulas - The Complete Guide.

In this (almost) complete guide we'll go through the PMP formulas with examples. Stick with me and by the end you'll see that learning the PMI® formulas isn't going to be that bad. Read on and you'll learn a lot about how to crack formula-based PMP questions to help pass your PMP exam.

What Formula Question Types are on the PMP Exam?

Tip

You must know the formulas well enough to be able to write them out on your "brain dump" quickly and correctly once you started the exam. Read time management and brain dump tips from students...

For the PMP exam, you must know how to correctly answer questions with formulas about earned value, communications, procurement, probability, network diagrams, project selection, depreciation, and some mathematical basics. You also have to know a lot of acronyms.

Here is a list with the types of questions you have to expect:

  • Apply a formula: These are straightforward questions where you are given values and are expected to apply the correct formula.
  • Apply two formulas: In these questions you get a set of values and asked to calculate a result. At first these look as if you can simply apply one formula. But as you are applying this first formula you suddenly realize that one value is missing. This missing value must then first be calculated via a second formula.
  • Invert a formula: These questions test your ability to take a basic formula and invert it. For instance instead of asking “4+6=?” the question would be “4+?=10” and it is your job to invert the formula and calculate “10-4=6”.
  • Result interpretation: In these types of questions you are given a result and asked, “What does this result mean for the project?”.
  • Find the correct formula: For these types of questions you are given a scenario and various formulas as options. Your task is to select the formula which best applies to the given scenario.
  • Use a formula based on keywords: There is more than one way to calculate earned value results. Which formula to use depends on the progression (or health) of your project. These are scenario-based questions that contain certain keywords. You must recognize these keywords and apply the correct formula

So you can easily see that being "mathematically ready" for your PMP exam means more than simply knowing your earned value (EV) formulas. Yes, you'll definitely see at least one EV question on your exam but that's simply not all there is.

Formula Question Example

Here is a formula-based sample PMP exam question for you:

  • There are 20 stakeholders on a project. What is the total number of potential communication channels in this project?

    A. 10
    B. 45
    C. 190
    D. 100

  • Correct Answer: C. 190

    Explanation: The total number of communication channels can be calculated using the formula n(n – 1)/2, where n = number of stakeholders. So in this case the total number of communication channels is (20*19)/2 = 190.

More questions like this one...

The Earned Value Formulas

Student Success Story

Know earned value formulas and their application inside out. I probably had at least 10 EV questions on my exam, and knowing the formulas meant that I could answer these with confidence and move on quickly. Read more...

Warwick Kowalczyk, PMP

Many PMP aspirants find the concepts behind earned value management (EVM) hard to understand and the formula even harder, so that's where we are going to start.

First, let's define the term:

Earned Value (EV)1. The measure of work completed expressed in terms of the budget authorized for that work.

A Guide to the Project Management Body of Knowledge (PMBOK® Guide)

 

The thing that complicates it for many people is the question: "What on earth is value?" For the earned value calculations you just have to remember that value equals money. The total value of a project (in EVM terms) is equal to the budget of the project. As you work through the project you spend the budget in order to achieve the project's objectives, which in turn deliver business value. You can assess how much value you have 'earned' (read: 'achieved') by knowing how far through the project you are and how much you have spent.

The earned value management formulas are simply the calculations that give you the data to work out the EV position on your project. There are 12 earned value calculations in total.

Still not clear? You'll see what I mean with an example.

The Earned Value Calculation

Let's say we're working on a project to design an app for a smartwatch. Overall we are trying to answer the following two questions:

  • Are we over, on, or below budget?
  • Are we ahead, on, or behind schedule?

The earned value management formulas give us the information we need to determine that, and we'll get further into how we can calculate cost and schedule performance a little later.

The formula:
 EV = % complete * BAC 
What you get:
A monetary value.

(BAC = Budget at Completion)

Let's assume the following situation: Our smartwatch app project is going to take six months and cost $60,000. We're two months into the project and we've spent $20,000. Our project sponsor wants to know if the project is doing "OK".

Generally, when sponsors ask a question like that, then they want to know if you are burning through the budget too quickly and if you are going to hit the end date which they have published to the Board. The EVM formulas can tell you exactly that.

Let's continue with the example and assume that that project was supposed to deliver 36 work packages, and that each package has the same planned value (PV) and takes the same amount of effort to complete. Now, let's say that the project team has completed 12 work packages so far during the first two months of the project. We can see that 12/36 shows that the team is a third of the way through the project. As a percentage, 33.33% of the work has been completed. That's the % complete figure in the simple formula.

The budget at completion (BAC) is the total amount budgeted for the project, in this case $60,000.

Plugging those figures into the formula we get: 33% * $60,000 = $20,000. The earned value (EV) of the project is $20,000.

The formula:
 EV = Sum of PVs of all completed activities 
What you get:
Project's earned value. Why? If each work package requires a different level of effort to complete, we cannot use the number of completed work packages as an indicator of project’s percentage of completion. In such cases, we need to go one level deeper, for example, breaking the work packages into person-hours of effort. When determining project’s percentage of completion is difficult, grab the latest project schedule. Each project activity should have its own planned value (PV) assigned to it. Add the PVs for all the completed activities (the emphasis is on 'completed'), and this value will be your project’s earned value.

Is There A Formula for Planned Value?

Earned value alone isn't that useful a figure. You need to have something to compare it to. Planned value (PV) is great for that.

Planned Value (PV)1. The authorized budget assigned to scheduled work.

There is no formula for PV. It's simply the approved budget for a task. You can use PV for the budget of a phase, stage or work package. It's normally measured over a particular time period.

How to Calculate Schedule Variance

Schedule variance tells us whether our smartwatch app project is ahead, on, or behind schedule.

Schedule Variance (SV)1. A measure of schedule performance on the project, expressed as the difference between project's earned value and planned value.

One argument that I often get from my students is something along the lines of "I don't see how SV is possible! I mean we are using numbers from the budget and you expect me to believe that they help predict how I'm doing on the schedule??" It's not an unreasonable argument. But it is indeed possible. For now, I'm going to ask you to take a leap of faith (I did when I originally came across SV) and I'll explain how it all works. Let us begin with the formula:

The formula:
 SV = EV - PV 
What you get:
A monetary amount. A negative number means you are behind schedule (that's bad). A positive number means you are ahead of schedule (generally that's good).

Although we generally see a negative SV as something bad and a positive SV as something good, contemporary project management considers all variances, positive or negative, as potentially harmful to the project and the performing organization. Variance tolerances are defined, such as +/- 5% or +/-15%, and whenever the variances are found outside this range, an investigation is recommended. One may argue saying, "I don’t get it, how can be a positive 20% SV bad for the project or the organization? A positive 20% SV means I am 20% ahead of schedule and I should be getting a bonus for that!" Well, we didn’t say that a positive 20% SV is always bad. We said it is potentially harmful to the project and the organization. If the 20% faster progress happened due to the brilliance of the project team, they need to be rewarded. However, in most cases, it has been found that positive schedule variance is a result of conservative estimation during project planning. If a project is conservatively estimated or scheduled, it will consume organizational resources for a more extended period than it should have been, which is not good for the organization.

Earlier we showed you that the project team had completed 12 out of 36 planned work packages meaning the project is 33.33% complete. Further, the team has just finished the 2nd month in the project, and with the total project duration of 6 months, the team has consumed 33.33% of the time planned.

It seems that we are on track!

But wait a minute, what if the project was front-loaded? A front-loaded project requires more resources and work to be completed at the beginning of the project in comparison to the end. Let us now assume that in this project we planned to complete 66.67% of the work in the first two months, and very few resources are involved in the last three months of the project. Therefore, now, looking again at 33.33% of the time consumed and 33.33% of work packages completed, we understand that these values are not good for the front-loaded project since according to the plan we should have completed 66.67% of the work.

The EV of our fantastic new app development project is $20,000, as we saw earlier. However, the Planed Value (PV) is $40,000 (66.67% of $60,000). In other words this means according to our plan we should have completed $40,000 worth of work. Using those numbers in the schedule variance formula we get $20,000 - $40,000 = -$20,000.

These are really simplified figures to help you understand the math more easily and you can clearly see that this is bad news. Now, when we are starting to dig into the numbers, we can see that the project is running behind schedule.

One of the emerging trends in project management is the new approach to schedule variance. The development of the new SV formula was a result of criticism of the traditional SV formula that expresses schedule variance in monetary terms. The new SV formula expresses the schedule variance in time units. This formula uses two components: earned schedule (ES) and actual time (AT). However, the calculation of these components is relatively complicated. Therefore, it is unlikely that you will be required to make any calculations on the exam that involves ES and AT. Anyway, if you want to see the formula, here it is:

The formula:
 SV = Earned Schedule (ES) – Actual Time (AT) 
What you get:
Time units. The interpretation of SV calculated by this formula is the same as the traditional SV formula, i.e., positive SV = project ahead of schedule; negative SV = project behind schedule.

What is the Cost Variance (CV) Formula?

Schedule variance is one way to get a view on how your project is performing. Another way to look at it is to use the cost variance formula.

The CV formula is also used to work out if the project is over, on, or under budget.

The formula:
 CV = EV – AC 
What you get:
A monetary amount. A negative number means you are over budget (that's bad). A positive number means you are under budget (hurrah!). It's most useful to report CV alongside the project budget so that you can easily see the magnitude of any variance. As mentioned earlier, contemporary project management sees all project variances, positive or negative, as potentially harmful to the project and the performing organization. Similarly to schedule variance, the cost variance outside the defined threshold limit (e.g., +/-15%) should be investigated, and a root cause identified. For example, positive cost variance could be a result of conservative estimation during project planning. If a project was conservatively estimated, extra funds allocated to it could have been spent elsewhere by the organization.

(AC = Actual Cost)

If we take the figures from our smartwatch project we can see what this means to the example. The EV is $20,000. Let's assume that the actual cost of the work done so far – the money we have spent already – is $20,000. Putting these in the formula gives us:

$20,000 - $20,000 = $0. Perfect! We are exactly on budget! (For the purposes of this example, anyway...)

So, what do we know about the project so far? It's neither over nor under budget, but it is running behind schedule. We thought we would have completed more activities as per the plan by now, but we did not. This is an early warning sign that the timescales for the project might slip.

It's important to consider the context behind the numbers. Our company has never delivered this kind of smartwatch app before, so we're learning as we go. Maybe that's why we aren't making the progress we expected. Maybe one of the big mobile platforms has taken longer than expected to get back to us about how to get our product in their store. There could be lots of reasons that explain the numbers, so when you're reporting the project's status to the sponsor, don't just rely on the numbers to tell the story.

How Do You Calculate the Cost Performance Index (CPI)?

We report to the project sponsor. She understands the reasons for the numbers but wants to know if the situation can be recovered.

A useful piece of information to help paint the whole picture is the cost performance index (CPI) calculation. This is an alternative way of looking at the cost performance of a project and is often preferable to CV because the answer you get is a ratio. Ratios are self-explanatory and don't need further information to highlight how far off-track performance is.

CPI in project management measures the cost efficiency of a project.

The formula:
 CPI = EV / AC
What you get:
A number. You're aiming for 1. That means that you are getting $1 of value for every $1 spent. You are using your project budget as planned. If it's more than 1, you are getting more than $1 for every $1 spent. This could mean that your initial budget was not put together in a robust way and your estimates were too conservative.

If the number is less than 1 then that's bad news. You are getting less than $1 of value for the project for every $1 spent. This normally happens when you are spending your budget in ways you hadn't expected, such as dealing with unforeseen risks or your initial estimates were just too tight.

Using the figures from our app development project we get:

$20,000 / $20,000 = 1. That looks good. We're spending money in the right way and on the right things, but there's still more to uncover about how this project is performing.

Challenge Yourself With a Really Hard CPI Sample PMP Exam Question

Click the image below to watch a video with a really hard CPI question. This one not only requires you to identify the correct formula, but then you also have to adjust the formula, or you'll get the wrong result - tell us in the comments if you got it right or wrong:

Using Estimate at Completion (EAC)

Next, we look at the EAC formula. That's Estimate at Completion and it works out the expected final and total cost of the project, based on project performance.

The EAC formula PMPs need to know most often is:

The formula:
 EAC = BAC / CPI 
What you get:
A monetary value.

(BAC = budget at completion. CPI = cost performance index)

There is EAC approach (a thorough bottom-up re-estimate of remaining work) to use if the original project estimates are now thought to be invalid, but this one is the one that is most likely to turn up in the PMP exam.

For this project we decide to re-estimate the costs of the remaining works (ETC). We use bottom-up estimation and find out that we need further $50,000 to complete the rest of the project. Since we have already spent $20,000 on the project already, the new estimate at completion (EAC) becomes $70,000.

Cost-wise, it's still not that bad. Our project sponsor can now be confident that we're going to hit the new budget target, i.e. EAC, that we have determined for the project. However, remember SV? That didn't look so good. She asks us to dig into that a bit further.

What is Schedule Performance Index (SPI)?

The SPI formula is used to work out if the project is:

  • Ahead, on, or behind schedule.
  • Going to finish when predicted.
The formula:
 SPI = EV / PV 
What you get:
A number. Again, as this one is a ratio too you are aiming for 1 as that means you are working through the project at the rate you had expected. If the number is greater than one, it means you are racing through your tasks faster than you had planned (but it doesn't comment on the quality of the work done – just something to think about!). If the number is less than 1 it means you are progressing more slowly than planned and the tasks are taking longer.

The SPI calculation, when applied to our app project, shows this:

$20,000 / $40,000 = 0.5

Ouch! We are working at half the speed that we planned. This is where the project is struggling! We had better stop doing project management formulas and start helping our team get the work done!

As we mentioned earlier, one of the emerging trends in project management is the new approach to schedule variance. Same goes to the schedule performance index (SPI) that uses earned schedule (ES) and actual time (AT). Since the calculation of these components is relatively complicated, we believe it is unlikely that you will be required to make any calculations on the exam that involves ES and AT. Anyway, if you want to see the formula, here it is:

The formula:
 SPI = Earned Schedule (ES) / Actual Time (AT) 
What you get:
The interpretation of SPI calculated by this formula is the same as the traditional SPI formula, i.e., greater than “1” = the project is ahead of schedule; less than “1” = the project is behind schedule.

But before we leap into the To Do list, let's just check a couple of other formulas.

Now you know that there is a CPI and an SPI. But what exactly is the difference? Find the answer here...

How to Use the Variance at Completion (VAC) Formula

Variance at completion (VAC) is another useful PMP earned value management formula. It helps you look forward and anticipate the difference between your original BAC and the newly calculated EAC. In other words, it's the total cost we originally planned minus the total cost that we now expect.

The formula:
 VAC = BAC - EAC 
What you get:
A monetary value. This shows you how much over or under budget (the variance) we will be at the end of the project. A value of $0 means you'll hit budget. Less than zero means you'll be over budget so ideally you're looking for a number near $0.

For our project, right now we're looking at:

$60,000 - $70,000 = -$10,000

That tells us there will be a negative variance of $10,000 by the time this smartwatch app is developed. Our sponsor is going to want to know about that too.

What is the Estimate to Complete Formula (ETC)?

The Estimate to complete (ETC) formula is essentially an inversion of the EAC calculations, so if you can do that one, you can work this one out easily.

There are several ways to calculate ETC in project management but this is the simplest and easiest to use in straightforward situations. It's also the one from A Guide to the Project Management Body of Knowledge (PMBOK® Guide).

The formula:
 ETC = EAC - AC 
What you get:
A monetary value that tells you how much more the project will cost.

Our app development project is pretty straightforward so the calculation looks like:

$70000 - $20,000 = $50,000 In other words, we need another $50,000 to see the project through to the end.

What if my situation isn't straightforward? There are 4 other ways to define and calculate ETC. They are all covered in detail in study guides like The PMP Exam Formula Study Guide.

Understanding the To Complete Performance Index (TCPI)

The last bit of PMP mathematics we are going to do is to work out the TCPI. It's worth a mention because it's one of the PMP exam formulas that people find most difficult. TCPI is used to calculate the cost performance that must be achieved to hit your cost target (either BAC or EAC).

You can work out TCPI against your BAC or EAC. We know that the smartphone project has already had the budget re-estimated once so it's best to use the latest figures. We're going to use the EAC to work it out.

The formula:
 TCPI = (BAC - EV) / (EAC - AC) 
What you get:
A figure.

Let's do this is in stages.

BAC – EV = $60,000 - $20,000 (remember? That's our EV) = $40,000 

EAC – AC = $70,000 - $20,000 = $50,000

$40,000 / $50,000 = 0.8

TCPI on our development project is 0.8. In order to get the intuition behind TCPI, let's break the formula in two parts. The first part, BAC – EV, indicates how much project work is remaining, i.e., how much value remains that need to be achieved. The second part, EAC – AC, indicates the money available to finish the project. Hence, the TCPI gives you the ratio of work that needs to be completed and the money available to complete the project.

The smartwatch app project started off looking quite good but we had to re-estimate the budget and we uncovered that we aren't getting through the tasks as quickly as we need to. In fact, the amount of work to do is unlikely to get done in time if we only apply the same amount of effort as we are now.

Thanks to the PMP cost management formulas we know a lot more about our project now.

Phew! We made it through all the earned value management formulas. Here's a table that summarizes all of that.

Formulas for Earned Value from PMBOK Guide

If you're creating a PMP formula sheet for your exam revision, there are a few more calculations that you'll want to include.

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PMP Formulas PDF Free Download

Why don't you practice on some free questions from The PM Exam Simulator?

The PDF download document has two sections:

Section 1 - Self-Assessment Questions: The difficulty level of these questions is easy to medium. They are intended to measure general understanding of formula concepts.

Section 2 - Answers and Explanations: In this section we repeat the question and then give you the answer, explanation and a reference where you can read up on the topic.

Five Formulas Explained

The Formula of Standard Deviation

Ah, standard deviation! Doesn't this take you back to high school math. Standard deviation is simply a reflection of the uncertainty in the estimates. It just highlights the variability of an activity in statistical terms and you use it when an activity has different estimates: optimistic, most likely and pessimistic.

The formula:
σ  = (Pessimistic - Optimistic) / 6
What you get:
A figure that represents the variation in the estimates, in the same units of measurement as your estimate.

The σ is the sign that represents standard deviation and is read as "sigma".

There's a task on the smartwatch app project to build the notification tools, so that when something new happens the user gets an alert. Given that we haven't done this task before we want a good understanding of the risk and we've used subject matter experts from an agency to help estimate the work.

The pessimistic estimate is 12 hours. The optimistic estimate, if we had an expert developer in-house, is 5 hours. Pop those into the formula and we get:

(12 – 5) / 6 = 1.16

The larger the standard deviation, the greater the risk. The risk level on this task seems OK but we can always take it to our sponsor for a second opinion.

The Advantage of PERT Formula

You have another tool to help you estimate: PERT. The most likely PERT formula PMP aspirants are going to come across in the exam gives you a weighted average for your estimates. It's a three point estimate for the expected duration of a schedule activity using pessimistic, optimistic and most likely durations.

The formula:
Beta = 
(Pessimistic + (4 * Most Likely) + Optimistic) / 6
What you get:
The estimated duration of the activity as a weighted average.

Let's see what the weighted average is for the task to build the notification module of our smartwatch app. First, we need another piece of data: the most likely duration for the task, which our lead developer tells us is 8 hours. That gives us:

(12 + (4 * 8) + 5) / 6 = 8.2 hours. Now there's a number our sponsor can understand.

How to Use the Communication Channels Formula

The sponsor isn't the only person on the project team who needs to know this information. The team is only 6 people but there are still multiple communication channels. How many exactly? Well, there's a communication channels formula to work that out!

The formula:
 n * (n-1) / 2 
What you get:
The number of communication channels in the team.

In the formula, 'n' stands for the number of people. On our project, the communication channel math looks like this:

6 * (6-1) / 2 = 15

That's 15 different routes for messages!

How to Practice PMP Formulas

This article has covered the most common PMP formulas but the best way to really get to grips with them is to practice using them frequently.

Download our free redacted PMP Formulas PDF Guide so that you've got all the data and formulae to hand.

Another great way to practice is to do sample exam questions. This free batch of sample PMP math questions will help you see where you need to build your skills.

Learn More about PMP Exam Formulas

Conclusion and Recommendation: Practice. Practice. Practice!

There are 49 PMP Exam Formulas that you have to master for the PMP Exam.

Mastering these formulas is more than simply knowing them! You have to know how to apply them to a given question scenario and you may even have to apply TWO formulas. You may have to invert a formula, you might be asked to interpret a formula (CPI = 0.8 means?), or you may be asked to determine which formula is most appropriate in a given scenario based on certain keywords in the question.

Being able to do this takes time and practice. Don't expect miracles overnight. Here is my recommended approach:

  • First of all, study the right formulas. There are unfortunately still some websites out there that list incorrect or outdated PMP exam formulas!
  • Review the EVM formulas that are listed in the PMBOK® Guide every other day for two weeks. Pause for one week, then do it again. The more you repeat, the more they will sink in.
  • Make sure that you know all the earned value formulas in and out. PMI loves earned value!
  • Answer at least one formula-based question on every day that you are studying for the exam. Since most people take about 3 months for their preparation that means you will answer about 90 formula-based questions.

Knowing and understanding all the formulas and their variations, concepts, keywords, value and acronyms may seem daunting at first. But if you plan on taking just one small step forward every day and practice, practice, practice, then you'll soon find that the calculations start to come naturally.

© 2018 OSP International LLC. All rights reserved. This copyrighted article may not be reproduced without express written consent of OSP International LLC.

*1 -- These definitions are taken from the Glossary of Project Management Institute, A Guide to the Project Management Body of Knowledge, (PMBOK® Guide) – Sixth Edition, Project Management Institute Inc., 2017.
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