PERT Notes

Last reviewed 16 Sept 2026 · 5 min read

Programme Evaluation and Review Technique (PERT)

PERT was developed in 1958 for the US Navy's Polaris missile programme (with Booz Allen Hamilton and Lockheed). It is used for projects with uncertain activity durations — research, development and first-of-a-kind projects.

Features

Probabilistic — each activity has three time estimates.
Event-oriented — emphasis on events (milestones).
Estimates the probability of completing the project by a scheduled date.
Uses slack (for events) and the critical path.

Three time estimates

Estimate	Meaning
Optimistic time $t_{o}$	Shortest possible time if everything goes exceptionally well
Most likely time $t_{m}$	Time required under normal conditions (mode)
Pessimistic time $t_{p}$	Longest time if things go badly (excluding major catastrophes)

Activity durations are assumed to follow a beta (β) distribution, which can be skewed either way and has finite end points.

FormulaExpected time and variance of an activity

t_{e} = \frac{t _{o} + 4 t _{m} + t _{p}}{6}

σ = \frac{t _{p} - t _{o}}{6}, σ^{2} = (\frac{t _{p} - t _{o}}{6})^{2}

If $t_{p} - t_{m} > t_{m} - t_{o}$ , the distribution is skewed to the right and $t_{e} > t_{m}$ .
A larger range $t_{p} - t_{o}$ means more uncertainty.

Project duration and probability

FormulaProject completion (central limit theorem)

Expected project duration $T_{E} = \sum t_{e}$ along the critical path.
Project variance $σ_{T}^{2} = \sum σ^{2}$ of critical activities; standard deviation $σ_{T} = \sum σ^{2}$ (standard deviations are not added).
The project duration is assumed normally distributed.

Probability of completion by scheduled time $T_{S}$ :

Z = \frac{T _{S} - T _{E}}{σ _{T}}, P (T \leq T_{S}) = Φ (Z)

Time for a desired probability:

T_{S} = T_{E} + Z σ_{T}

Useful normal distribution values

Z	−2	−1	0	0.5	1.0	1.28	1.5	1.645	2.0	3.0
Φ(Z)	0.0228	0.1587	0.500	0.6915	0.8413	0.90	0.9332	0.95	0.9772	0.9987

$T_{S} = T_{E}$ → probability 50%.
$T_{E} \pm 1 σ$ covers ≈ 68%, $\pm 2 σ$ ≈ 95.4%, $\pm 3 σ$ ≈ 99.7% of outcomes.

Slack

Event slack $= T_{L} - T_{E}$ (latest allowable minus earliest expected time).
Positive slack — ahead of schedule/resources to spare; zero slack — critical; negative slack — behind schedule (when the scheduled date is earlier than the expected date).

When two paths have close durations, the path with higher variance may govern the probability; standard practice uses the critical path, but near-critical paths should be checked.

Worked example

Worked ExamplePERT network

Activity	Events	$t_{o}$	$t_{m}$	$t_{p}$
A	1–2	2	4	6
B	2–3	3	5	13
C	2–4	2	3	4
D	3–5	4	6	8
E	4–5	5	8	11

(Durations in weeks.)

Step 1 — expected times and variances

Activity	$t_{e}$	$σ^{2}$
A	(2 + 16 + 6)/6 = 4	(4/6)² = 0.444
B	(3 + 20 + 13)/6 = 6	(10/6)² = 2.778
C	(2 + 12 + 4)/6 = 3	(2/6)² = 0.111
D	(4 + 24 + 8)/6 = 6	(4/6)² = 0.444
E	(5 + 32 + 11)/6 = 8	(6/6)² = 1.000

Step 2 — paths

A–B–D: 4 + 6 + 6 = 16 weeks; variance = 0.444 + 2.778 + 0.444 = 3.667
A–C–E: 4 + 3 + 8 = 15 weeks; variance = 1.556

Critical path A–B–D, $T_{E} = 16$ weeks, $σ_{T} = 3.667 = 1.915$ weeks.

Step 3 — probability of completion in 18 weeks $Z = \frac{18 - 16}{1.915} = 1.04$ → $P \approx 0.85$ (about 85%)

Step 4 — probability of completion in 14 weeks $Z = \frac{14 - 16}{1.915} = - 1.04$ → $P \approx 0.15$ (about 15%)

Step 5 — duration for 95% confidence $T_{S} = 16 + 1.645 \times 1.915 =$ 19.15 weeks

Worked ExampleSingle activity

An activity has $t_{o} = 4$ , $t_{m} = 7$ , $t_{p} = 16$ days. Find $t_{e}$ and $σ$ .

Solution. $t_{e} = \frac{4 + 28 + 16}{6} = 8$ days; $σ = \frac{16 - 4}{6} = 2$ days (variance 4)

PERT vs CPM

Aspect	PERT	CPM
Nature	Probabilistic	Deterministic
Time estimates	Three ( $t_{o}$ , $t_{m}$ , $t_{p}$ )	One
Orientation	Event-oriented	Activity-oriented
Distribution	Beta (activities), normal (project)	None
Used for	R&D, new, uncertain projects	Construction, repetitive projects with experience
Cost	Time is the main concern	Time–cost trade-off (crashing)
Critical measure	Slack (events)	Float (activities)
Origin	US Navy, Polaris (1958)	DuPont and Remington Rand (late 1950s)

Frequently tested points

PERT: probabilistic, event-oriented, three time estimates, beta distribution.
$t_{e} = (t_{o} + 4 t_{m} + t_{p}) /6$ ; $σ = (t_{p} - t_{o}) /6$ .
Project variance = sum of variances on the critical path; $σ_{T} = \sum σ^{2}$ .
$Z = (T_{S} - T_{E}) / σ_{T}$ ; $T_{S} = T_{E}$ gives 50% probability.
95% → Z = 1.645; 84% → Z ≈ 1; 97.7% → Z = 2.
Slack (PERT) vs float (CPM); negative slack → behind schedule.
CPM for construction and crashing; PERT for R&D.

Common MistakeCommon mistakes

Adding standard deviations instead of variances.
Using the most likely time instead of the expected time for the critical path.
Forgetting that probability is 50% when the scheduled time equals the expected time.

Revision SummaryChapter summary

PERT handles uncertain durations using optimistic, most likely and pessimistic estimates with a beta distribution.
Expected time is $(t_{o} + 4 t_{m} + t_{p}) /6$ and standard deviation $(t_{p} - t_{o}) /6$ .
Project expected duration and variance are sums along the critical path, with a normal distribution assumed.
$Z = (T_{S} - T_{E}) / σ_{T}$ gives completion probabilities and required durations.
PERT (probabilistic, event-oriented, slack) complements CPM (deterministic, activity-oriented, float, crashing).

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Programme Evaluation and Review Technique (PERT)

Features

Three time estimates

Project duration and probability

Useful normal distribution values

Slack

Worked example

PERT vs CPM

Frequently tested points

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