Schedule Probability, Variance, and Risk Interpretation
From durations to a confidence date
PERT treats project duration as a random variable, approximately normal, with mean TE (the critical path length built from three-point estimates) and standard deviation σpath = √(sum of variances of activities on the critical path). Then:
Z = (T − TE) / σpath
where T is the target date. Convert Z with a standard normal table: Z = 0 gives 50%; 0.67 gives 75%; 1.00 gives 84%; 1.28 gives 90%; 1.645 gives 95%; 2.00 gives 97.7%; 3.00 gives 99.87%. Invert for the date at a chosen confidence: T = TE + Z × σpath, so a P90 date is TE + 1.28σ.
Interpretation is where marks are lost. The output is a probability of finishing by a date, not a commitment. PERT systematically understates risk because it ignores merge bias (path convergence makes expected project duration longer than the expected critical path when several paths compete to be critical), because near-critical path variance is omitted, and because activity durations are rarely independent. Skewed distributions and real-world correlation break normality further.
Traps. Dividing by an activity's σ instead of the path σ; adding standard deviations instead of variances; quoting 95% when Z is negative; treating the P50 date as a commitment; and forgetting that a question asking for the probability of finishing later than T needs 1 minus the table value.
Worked example. Critical path TE = 40 days, variances sum to 16, so σpath = 4. Target 46 days: Z = (46 − 40)/4 = 1.5, about 93.3% on-time probability. Target 36 days: Z = −1, about 15.9%, meaning roughly an 84% chance of overrunning.
Rule of thumb. Each additional σ above the mean adds roughly 34 percentage points from 50%, then gains shrink quickly.