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Estimating Under Uncertainty
Quantitative Analysis · Chapter 8: Project Timelines and the Critical PathSection 8.2 · lesson 36 of 5512 min · free with a free account
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Next lesson: The Critical Path Method →← Sources of Uncertainty in Project TimelinesNext lesson: The Critical Path Method →
In this lesson
Statistical methods for predicting project completion times.
- Build three-point estimates for activity durations
- Compute expected duration, variance and standard deviation
- Roll activity statistics up to a project distribution
- Turn any deadline into a probability with a Z-score
Transcript
Everything said in the video, section by section (1,749 words). Read along, or search it.
Project Timelines and the Critical Path · Section 8.2
Estimating Under Uncertainty
Welcome to Section eight point two, Estimating Under Uncertainty. Every project schedule you have ever seen was a guess dressed as a fact. Someone wrote down a completion date, and everyone downstream treated it as certain. This section takes the honesty back. You will learn to build three-point estimates for each activity's duration. From those, you will compute an expected duration, a variance, and a standard deviation. Then you will roll those activity statistics up into a distribution for the whole project. And finally, you will turn any deadline you like into a probability. The technique is called PERT, the Program Evaluation and Review Technique. It is one of the few places in project management where statistics does real work. By the end, a deadline stops being a promise and becomes a bet with known odds.
The Honest Approach
CPM Collapses Uncertainty. PERT Keeps It.
In real-world projects, activity durations are rarely known with certainty. A task that normally takes two weeks may finish in one, if conditions are favourable. Or it may stretch to five weeks, if a key supplier is late. The Critical Path Method handles this by collapsing each activity to a single deterministic estimate. One number in, one number out, and no stated confidence anywhere. PERT takes a more honest approach. It treats activity durations as random variables, not as fixed quantities. It asks for three time estimates per activity to capture the uncertainty. Then it applies the laws of probability to predict the chance of meeting any deadline. So instead of saying the project takes thirty weeks, you can say something far more useful. There is a seventy-five per cent chance we are done within thirty-two weeks.
The Route Ahead
How PERT Reaches a Probability
Before the formulas, hold the shape of the whole method in your head. PERT runs in one direction only, and there are five moves. First, three estimates per activity, which bracket the plausible range. Second, those three numbers become activity statistics, an expected duration and a variance. Third, the roll-up. Both quantities are summed along the critical path. That gives you a project mean and a project variance. Fourth, the normal approximation, which is where the Central Limit Theorem earns its keep. Total project duration turns out to be roughly normal, whatever the pieces looked like. Fifth and last, the deadline probability. You compute a Z-score against your target date and read the normal table. Every step from here is one of those five. Keep the map in view.
Step One
Three-Point Duration Estimation
Step one. Three-point duration estimation. Instead of asking an expert for a single best guess, PERT asks for three. The optimistic time, written as a, is the shortest the activity could take. Everything goes right in that scenario. It is the lower bound. There is roughly a one per cent chance the activity finishes faster than a. The most likely time, m, is what the work would take most often if repeated. Statistically, m is the mode of the duration distribution, and the single best guess. The pessimistic time, b, is the longest the activity could take if serious problems occur. That is the upper bound, and again about a one per cent chance of exceeding it. Behind those three numbers sits a beta distribution, bounded below by a and above by b. Its peak, the mode, sits at m, and the expected duration is pulled toward that peak.
Activity Statistics
The Three PERT Formulas
From those three estimates, PERT computes three statistics. The first is the expected duration, written t sub e. It equals a, plus four times m, plus b, all divided by six. Notice this is a weighted average, not a plain one. The weight of four says the most likely time is the strongest signal you have. But the optimistic and pessimistic estimates still pull the expected value. That matters most when the uncertainty is asymmetric. If the downside is far worse than the upside is good, the expected duration shifts upward. The second statistic is the variance, b minus a over six, all squared. A wide gap between optimistic and pessimistic means higher variance, and greater uncertainty. The third is the standard deviation, simply b minus a over six. It is in the same units as the duration itself, weeks or days, so it reads more easily.
Where The 6 Comes From
Six Standard Deviations, End to End
Where does that six in the denominators come from? It comes from an approximation of the beta distribution. The range from a to b is assumed to span about six standard deviations. So dividing that range by six recovers one standard deviation. It is the same reasoning behind the plus or minus three sigma rule from quality control. This is also why the one per cent tails on a and b matter so much. If your experts give you a comfortable range rather than a true extreme, that span breaks. The expected duration and the standard deviation are computed independently of each other. Take one activity estimated at four, five, six, and another at one, five, nine. Both have an expected duration of exactly five. But their standard deviations are zero point three three three and one point three three three. Only the second one threatens your schedule.
Rolling Up
From Activities to a Project Distribution
Now we roll the activity statistics up to the project level. The critical path is the chain whose total duration equals the project's total duration. It therefore controls the schedule, and only its activities enter the roll-up. The expected project duration is the sum of the expected durations along that path. The project variance is the sum of the activity variances along that same path. The project standard deviation is the square root of that sum. Here is the classic mistake. Variances add, standard deviations do not. You must square, then sum, then take the root, in that order. One condition is attached to that variance sum, and it is a real one. Summing variances assumes the durations are independent of one another. That is, a delay on one activity does not change the duration of another. In practice that is a simplification, but it works well enough for most planning purposes.
The Enabling Theorem
The Central Limit Theorem and the Z-Score
The Central Limit Theorem is the key statistical result that makes PERT useful. When a large number of independent random variables are summed, the sum is approximately normal. And crucially, that holds regardless of the shape of the individual distributions. So the beta shape stops mattering once you reach the project level. Project duration is a sum of many activity durations, so we can treat it as normal. Its mean is the expected project duration, and its spread is the project standard deviation. Once a quantity is normal, deadline questions become table lookups. Take your target date, subtract the expected duration, and divide by the project standard deviation. That is the Z-score. It says how many project-level standard deviations your target sits from the expected completion. Look that Z up in a standard normal table for the probability of finishing by the target.
Worked Example
Five Activities on the Critical Path
A project has five activities on its critical path, labelled A through E. For each we have an optimistic, a most likely, and a pessimistic estimate, in weeks. The job is to find the expected duration and variance for each, then the project totals. Work through activity C, because it is the interesting one. Its estimates are five, eight, and thirteen weeks. Expected duration is five, plus four times eight, plus thirteen, all over six. That is fifty over six, which is eight point three three three weeks. Its standard deviation is thirteen minus five, over six, or one point three three three. Squaring that gives a variance of one point seven seven seven eight. That is the largest variance in the set, tied with activity E. Do the same for all five, and the expected durations sum to thirty point three three three. The variances sum to five point nine four four four.
Step One
Compute the Project Statistics
Step one of the solution. Compute the project statistics. The expected project duration is just the column sum of the expected durations. That comes to thirty point three three three weeks, which we round to thirty point three three. The project variance is the column sum of the variances, five point nine four four four. Take its square root to get the project standard deviation. That gives about two point four three eight weeks. Both sums run over the five critical-path activities only. Now read what that spread is actually telling you. Two point four three eight weeks on a thirty-week project is about eight per cent of the schedule. That is moderate risk. Not comfortable, but not alarming either. It also tells you where to act, because activities C and E carry the largest variances. Narrow their optimistic to pessimistic ranges and the project's spread falls fastest.
Steps Two And Three
Two Deadlines, Two Probabilities
Steps two and three. Turn two candidate deadlines into probabilities. Start with twenty-eight weeks. Subtract thirty point three three three, then divide by two point four three eight. That gives a Z-score of minus zero point nine five seven. The cumulative probability there is zero point one six nine three. So roughly a seventeen per cent chance of finishing within twenty-eight weeks. That sits below the fifty per cent mark, because twenty-eight is shorter than the expected duration. Now thirty-two weeks. Thirty-two minus thirty point three three three, over the same spread, gives zero point six eight four. The cumulative probability is zero point seven five two nine, or about seventy-five per cent. The four weeks between those deadlines are worth about fifty-eight percentage points of confidence. One caution on the lookup. Round the Z first and the table returns different values. Carry the full Z into the table, and round only the probability at the end.
What It Means
From a Point Estimate to a Promise
The twenty-eight week deadline is aggressive. There are less than even odds of meeting it. The thirty-two week deadline is realistic, at roughly a seventy-five per cent chance. And if the sponsor demands ninety-five per cent confidence or better? Quote the expected duration plus one point six four five standard deviations. That works out to about thirty-four point three four weeks. The expected duration weights the most likely time most heavily, through the factor of four. Variance grows with the gap between the optimistic and pessimistic estimates. Project statistics sum over the critical path, assuming the activities are independent. By the Central Limit Theorem, completion time is approximately normal for large projects. A Z-score and a standard normal table give the probability of meeting any deadline. So PERT stops saying it will take twenty-eight weeks. It says instead, there is a seventy-five per cent chance we finish within thirty-two.
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