PMGT1865 Chap.9 Probability of Completion and Target Dates
Probability of Completion and Target Dates
An expected project duration on its own is a single number with the uncertainty stripped back out of it, which is precisely what three point estimating existed to avoid. This chapter puts the uncertainty back and answers two questions in the language a sponsor uses. Given a scheduled completion time, how likely are we to meet it? And given a required level of confidence, what date should we promise?
Both run through the same three quantities: the project standard deviation, obtained by adding the variances of the critical activities and taking the root; the Z statistic, which measures the distance between the target and the expected duration in standard deviations rather than in days; and the normal probability table, which converts that distance into an area.
The chapter is deliberate about one fact that a duration alone conceals: the expected duration is the fifty per cent point, so a project reporting its critical path length and stopping there has promised a coin flip without choosing to.
What this chapter covers
- 01
Steps four and five: variances on the critical activities, and the root of their sum
- 02
Why variances add and standard deviations do not, and what adding deviations costs
- 03
Why only the critical path enters the sum, and where that simplification is unsafe
- 04
The Z statistic, and reading its sign before reading the table
- 05
The probability as an area under the curve to the left of a target date
- 06
Reading a two decimal Z, and using symmetry for a negative one
- 07
Rounding Z rather than the inputs, and why the order matters
- 08
The inverse question, and rearranging to schedule for a chosen confidence
- 09
The accelerating price of certainty, and the fifty per cent point most promises land on
- 10
Two levers on a probability: moving the target date, and narrowing the spread
Two probabilities and a confident date
- +2Sum the variances of the critical activities: 0.111 plus 1.362 plus 0.445 plus 2.779 plus 0.111 is 4.808. The square root gives a project standard deviation of 2.193 days.
- +2For the 35 day target, Z is 35 minus 31.5, all over 2.193, which is 1.596 and rounds to 1.60. The table value at 1.60 is 0.9452, so the probability of completing within 35 days is 94.52 per cent.
- +1For the 30 day target, Z is 30 minus 31.5, all over 2.193, which is minus 0.684 and rounds to minus 0.68. The sign is negative because the target is earlier than the expected duration, so expect an answer below fifty per cent.
- +1Printed tables start at zero, so use symmetry. The value at plus 0.68 is 0.7517, and one minus 0.7517 is 0.2483, so the probability of completing within 30 days is about 24.8 per cent.
- +1For the inverse question, read the table backwards: a probability of 0.90 corresponds to a Z of 1.28. Then the scheduled time is 1.28 times 2.193 plus 31.5, which is 34.3 working days.
Key terms
- Project standard deviation
- The spread of the expected project duration, obtained by adding the variances of the critical activities and rooting the total. It is the unit in which the Z statistic measures distance.
- Z statistic
- The distance between a target date and the expected duration, expressed in project standard deviations. Changing the unit from days to deviations is what allows a normal table to price the gap.
- Scheduled completion time
- The date being tested or promised, written Ts. It is an input to the probability question and an output of the inverse question.
- Normal probability table
- The look up that converts a two decimal Z into the probability of completing within the scheduled time. Values below zero are obtained by symmetry rather than from the table itself.
Probability of Completion and Target Dates FAQ
What happens if I add the standard deviations instead of the variances?
The spread roughly doubles and every probability is dragged towards fifty per cent. In the worked example the five deviations sum to 4.167 while the correct project deviation is 2.193, barely half of that. It is the single most expensive error in this material because it produces a number that looks entirely reasonable, and the only defence is the habit of squaring first, adding second and taking the root last.
How do I handle a Z that comes out negative?
Use the symmetry of the normal curve. Look up the positive value and subtract the result from one. A Z of minus 0.68 gives a table value of 0.7517 at plus 0.68, so the answer is about 24.8 per cent. The trap is reporting the table value itself, which is the probability of finishing after the target rather than before it and is three times too large in that example.
Is it safe to use only the critical path variances?
It is the method as taught, and it carries a known direction of error. A near critical path with high variance can overtake the critical path on a bad run, and the calculation does not see it. Where the second path is comfortably shorter, as in the worked example, the simplification is safe.
Where two paths sit within a day or two of each other, say so in your answer rather than reporting the probability as though it were exact.
What should I do with a probability that is too low?
Two levers exist. Moving the target date to the right buys confidence directly, and narrowing the spread buys it more cheaply where one activity dominates the variance. In the worked example the build contributes 2.779 of the 4.808 total, more than the other four combined, so anything that narrows that one range, a prototype, a fixed price, a clearer specification, is worth more than a week of padding elsewhere.
What is never available is adjusting the estimates until the answer improves.
Assessment move
Write this calculation as four labelled lines in a fixed order: sum of variances, project standard deviation, Z value, probability. Each on its own line with its own label, before any prose. A marker awarding four marks can find four lines, whereas a paragraph containing the same four numbers has to be hunted through, and hunting is where part marks disappear.
Drill the five recurring errors until they feel wrong: adding deviations instead of variances, reporting the complement for a negative Z, including non critical activities in the sum, rounding the deviation before computing Z, and answering the wrong direction by giving a probability when a duration was asked for. Then practise the reporting habit the chapter is really teaching, which is pairing a date with a confidence.
Thirty five working days at 94.5 per cent, or 31.5 days at 50 per cent. Two numbers in one sentence, and the sponsor now knows both what you are promising and how firm the promise is.
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