Solving a PERT (Program Evaluation and Review Technique) problem involves estimating task durations using three time estimates, mapping those tasks into a network diagram, finding the critical path, and then calculating the probability of finishing by a given deadline. The core formula is simple, but a full PERT solution chains several calculations together. Here’s how to work through each step.
The Core PERT Formula
Every PERT calculation starts with three time estimates for each task:
- Optimistic (O): How long the task takes if everything goes perfectly.
- Most Likely (M): The realistic duration based on experience, accounting for normal delays.
- Pessimistic (P): How long it takes if major problems hit.
You combine these into a single expected duration using a weighted average:
Expected Duration (Te) = (O + 4M + P) / 6
The “4” in front of M gives the most likely estimate four times the weight of the other two. This pulls the result toward reality while still accounting for best-case and worst-case scenarios. Calculate Te for every task in your project before moving on.
Step 1: List Tasks and Dependencies
Before any math, identify every task in the project and determine which tasks depend on others. A dependency means one task cannot start until another finishes. For example, in a software project, you can’t begin testing until coding is complete. Write out each task with its predecessors. This dependency list is the foundation for your network diagram and ultimately determines which path through the project is the longest.
Step 2: Build the Network Diagram
A PERT network diagram is a visual map of your project’s workflow. Nodes (circles or squares) represent milestones or events, and arrows between them represent tasks. Draw the starting node on the left, then connect tasks as arrows flowing left to right based on your dependency list. Each arrow gets labeled with the task name and its expected duration (Te). When multiple tasks feed into one milestone, draw all their arrows converging on that single node. When a milestone triggers several parallel tasks, draw arrows branching out from it.
The diagram lets you see every possible path from start to finish, which you need for the next step.
Step 3: Find the Critical Path
The critical path is the longest sequence of dependent tasks from start to finish. Its total duration equals your project’s expected completion time, because every other path is shorter and will finish while the critical path is still running. A delay on any critical-path task delays the entire project.
To find it, trace every possible route through your network diagram and add up the Te values along each route. The path with the highest total is your critical path. In a simple project with five or six tasks, you can do this by inspection. For larger projects, use a forward pass and backward pass:
- Forward pass: Starting from the first task, calculate the earliest start and earliest finish for each task moving left to right. When two tasks feed into the same node, take the larger finish time.
- Backward pass: Starting from the last task, calculate the latest start and latest finish moving right to left. When a node branches into two tasks, take the smaller start time.
Any task where the earliest start equals the latest start has zero float, meaning there’s no scheduling wiggle room. The continuous chain of zero-float tasks is your critical path.
Step 4: Calculate Project Variance
PERT’s real power is quantifying uncertainty. Each task has a standard deviation and variance that measure how spread out the three estimates are:
Standard Deviation (σ) = (P − O) / 6
Variance (V) = σ² = [(P − O) / 6]²
A task with optimistic and pessimistic estimates close together has low variance, meaning you’re fairly confident in the estimate. A task with a wide gap has high variance and more risk.
To get the project-level variance, add up the variances of only the tasks on the critical path. Non-critical tasks don’t factor in because they aren’t driving the project’s timeline. The project standard deviation is the square root of that summed variance.
Step 5: Calculate Completion Probability
This is where PERT answers the question managers actually care about: “What’s the chance we finish by the deadline?” You use a z-score to translate your estimates into a probability.
Z = (Due Date − Expected Completion Date) / Project Standard Deviation
Once you have the z-score, look it up in a standard normal distribution table (or use a calculator) to find the corresponding probability.
A Worked Example
Suppose your critical path has an expected completion time of 28 weeks and a project variance of 9.11. Your deadline is 29 weeks.
First, find the project standard deviation: √9.11 = 3.01 weeks.
Then calculate Z: (29 − 28) / 3.01 = 0.33.
Looking up 0.33 in a normal distribution table gives roughly 0.6293, or about 63%. That means there’s a 63% chance the project finishes by week 29. If your stakeholders want 90% confidence, you’d need a later deadline or a way to reduce variance on critical-path tasks.
A negative z-score means the deadline falls before your expected completion date, giving you less than a 50% chance of finishing on time. A z-score of zero means the deadline exactly matches the expected date, which corresponds to a 50% probability.
Putting It All Together
Here’s the full sequence in summary form:
- Gather three estimates (O, M, P) for every task.
- Calculate Te for each task using (O + 4M + P) / 6.
- Map dependencies into a network diagram.
- Identify the critical path by finding the longest route through the diagram.
- Sum the variances of critical-path tasks to get project variance.
- Use the z-score formula to find the probability of meeting a specific deadline.
How PERT Differs From CPM
If you’ve encountered the Critical Path Method (CPM), PERT can feel similar since both use network diagrams and identify critical paths. The key difference is how they handle time. CPM uses a single, fixed duration for each task, making it deterministic. It works well for repetitive projects like construction or manufacturing where durations are well known from past experience.
PERT is probabilistic. It uses three time estimates to account for uncertainty, making it better suited for one-time projects like research initiatives, product launches, or any project without reliable historical data. PERT focuses on managing time risk, while CPM is also used for time-cost tradeoffs, where managers spend extra money on resources to compress the schedule.
Software That Handles PERT
For classroom problems, you’ll solve PERT by hand. For real projects, several tools automate the process. Microsoft Project supports scheduling, task dependencies, and critical path visualization. Lucidchart offers diagramming with increasingly automated chart creation. SmartDraw provides PERT-style diagramming tools, and Microsoft Visio handles network diagram creation. These tools won’t replace understanding the math, but they eliminate manual errors when you’re managing dozens or hundreds of tasks.