PERT 3-point Estimate Calculator
The PERT (Program Evaluation and Review Technique) three-point estimate is a standard project-management technique: for any task you give three numbers — optimistic (O), most-likely (M) and pessimistic (P) — and the tool returns the weighted expected value E = (O + 4M + P)/6, the standard deviation σ = (P − O)/6, and the 1σ / 2σ / 3σ confidence ranges. Useful for project duration, cost or engineering effort. PMBOK 7th edition and PRINCE2 both list it as a core estimation method, and it works equally well for personal scheduling.
All three estimates must be valid numbers.
Expected E
7
(O + 4M + P) / 6
Std. dev. σ
1.667
(P − O) / 6
Variance σ²
2.778
σ²
Triangular mean
8
(O + M + P) / 3
1σ range (≈ 68%)
5.33 – 8.67
E ± σ — about two-thirds of outcomes
2σ range (≈ 95%)
3.67 – 10.33
E ± 2σ — common planning buffer
3σ range (≈ 99.7%)
2.00 – 12.00
E ± 3σ — risk worst-case envelope
Outputs share the unit of the inputs (days / weeks / dollars). The σ bands assume a normal approximation.
Formula
E = (O + 4M + P) / 6 σ = (P − O) / 6 σ² = ((P − O) / 6)² 1σ ≈ 68% : [E − σ , E + σ] 2σ ≈ 95% : [E − 2σ, E + 2σ] 3σ ≈ 99.7%: [E − 3σ, E + 3σ] (Triangular-distribution mean = (O + M + P) / 3, shown for comparison.)
- · O is the shortest / cheapest value assuming everything goes right; P is the worst plausible case (still realistic, not catastrophic); M is the single most-likely value. The three must satisfy O ≤ M ≤ P — the calculator surfaces an error otherwise.
- · The 4× weight on M comes from PERT's beta-distribution assumption that the mode is more probable than the endpoints. When M is skewed away from (O + P) / 2, the PERT mean E diverges from the simple triangular mean — the calculator shows both for sensitivity-checking.
- · The ÷6 in σ comes from the convention "P − O ≈ 6 standard deviations" (covering ~99.7% of beta-distribution mass). For one task this can mildly under-estimate tail risk, but for N independent tasks the project σ_total = √Σ σᵢ² is reliable.
- · The 1σ / 2σ / 3σ bands use a normal approximation. For a single skewed beta-distributed task the error can be material, but project managers traditionally use this shortcut for buffer planning.
- · Common mistake: setting O to "the typical fast case" or P to "the historical slowest". PERT requires O and P to represent the ~1% and ~99% tails, not sample extremes — picking sample extremes severely under-estimates σ.
- · Typical uses: critical-path analysis (CPM + PERT σ), R&D cost forecasts, baseline distributions for Monte Carlo sampling, sensitivity analysis (widen or shrink P − O and watch σ).
- · Origin: Malcolm, Roseboom, Clark & Fazar, Operations Research 7 (5), 1959 (the U.S. Navy Polaris missile program). See also MacCrimmon & Ryavec 1964 and PMI, PMBOK Guide 7th ed., §4.4.2.
Frequently asked
What is the difference between the PERT expected value and the triangular-distribution mean — which should I use?
The PERT expected value E = (O + 4M + P) / 6 corresponds to a beta distribution and gives the most-likely value M four times the weight of the endpoints — appropriate when the task's actual duration tends to cluster around M and O / P are rare tails. The triangular-distribution mean = (O + M + P) / 3 corresponds to a piecewise-linear density and weights the three points equally — appropriate when you are unsure whether M really is the most-likely value. In practice: (1) for a single task PERT is usually more conservative (if M is below the midpoint, E > M); (2) the closer the two values are, the more "centred" M is; (3) the calculator shows both, so use PERT E as your baseline and the triangular mean as a sensitivity cross-check.
I have 20 independent tasks — can I just add up every E and σ?
You can add expected values directly: E_total = Σ Eᵢ. But you cannot add standard deviations — for *independent* tasks, the variances add: σ²_total = Σ σᵢ², so σ_total = √Σ σᵢ². Example: five tasks each with σ = 2 days → σ_total = √(5 × 4) = √20 ≈ 4.47 days, not 10. This is the central limit theorem in action — for N independent random variables, σ grows as √N, not N, so a portfolio of parallel tasks has much tighter aggregate variance than a naïve sum would suggest. The key assumption is independence: if the tasks share a resource (same engineer, same vendor) or correlated risk, the formula breaks down and you need Monte Carlo or to add covariance terms.
When picking O and P, should I use the historical fastest and slowest?
Usually no — PERT's O and P correspond to the ~1% and ~99% tails of the beta distribution ("only a 1% chance the task finishes faster than O or slower than P"), not sample extremes. Historical samples (especially N < 30) rarely cover the true tails and consistently under-estimate σ. Better practice: (1) ask experienced practitioners "if everything went right, how long?" (O) and "if foreseeable worst-case risks occur, how long?" (P); (2) derive P from a known-risk checklist (vendor delays, staff turnover); (3) PMI guidance suggests P − O ≥ 2× typical experience to avoid compressing σ. If you must use samples, take the 5th and 95th percentile rather than min and max.
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