PERT, the three-point method of estimating how long work will take, turns an optimistic, a most likely and a pessimistic time into an expected time of (O + 4M + P) ÷ 6. We took a chain of three tasks estimated at (2, 3, 6), (4, 6, 10) and (1, 2, 5). It totals 12.00, standard deviation 1.37, and a normal model gives a 92.7% chance of finishing by 14. Add one equal, independent path that must also finish, and the same model gives 86.0%. Our simulations of two beta readings gave 81.3% to 87.2%; triangular tasks gave about 52%. The US Navy's July 1958 PERT report had already described this bias, four years before a RAND memorandum of December 1962 analysed it.
The Navy's appendix weighs the likely time four times over
The project began on 27 January 1958 with people from the Navy's Special Projects Office, Booz, Allen & Hamilton and Lockheed's Missile Systems Division. Its Summary Report, Phase 1 appeared that July. Malcolm, Roseboom, Clark and Fazar described the work in Operations Research in 1959, according to its abstract.
In Appendix B, for unimodal distributions, the standard deviation "can be estimated roughly as one-sixth of the range". The authors declined to use M as the mean because "an activity time will more often exceed than be less than an estimated likely time". The pessimistic time was one "which might be required in one out of a hundred similar activities", a long time rather than a ceiling.
Each activity became a beta distribution peaking at M, running from O to P, its spread fixed at a sixth of the range. That beta's exact mean needs a cubic equation, so the report tabulated it and kept a straight line "because it is accurate enough for our purposes". The choice of the beta itself, it admitted, "was dictated by intuition because empirical evidence is lacking".
Skew leans the exact mean at most 3.86% of the range off the shortcut
We recomputed the appendix's Table 1 and matched each printed value to within 0.0001. The table already shows the widest gap: with M at O, the report puts the beta's mean 0.2053 of the way along the range and the shortcut at 0.1667. Our sweep of every mode position found nothing wider than that 3.86%.
The sign flips on the way: near either end the shortcut sits too close to M, and with the mode near a quarter of the range it overshoots by 1.05% of the range. MacCrimmon and Ryavec's RAND memorandum asked how far PERT can sit from any unimodal distribution on the same range, a wider question than the one beta the Navy assumed.
Another beta balances the mean exactly and unsettles the spread
A second beta, Beta(1 + 4r, 1 + 4(1 − r)) with r the position of M along the range, matches (O + 4M + P) ÷ 6 exactly at every mode. In our calculation its standard deviation is 0.845 times a sixth of the range with M at an end and 1.134 times for a symmetric task.
Triangular tasks load up to a sixth of the range onto the tail
A triangular distribution on the same numbers has mean (O + M + P) ÷ 3. It exceeds the PERT mean, on our algebra, by (1 − 2r) ÷ 6 of the range, a full sixth with M at O and zero for a symmetric task. Its spread is wider too, by a factor of 1.225 to 1.414. On the chain above the triangular total is 13.00 against PERT's 12.00. RAND treats the triangular as an equally reasonable, easier reading, since the beta had no prior justification either.
Parallel paths of equal length tilt the default deadline odds to 86.0%
The 1958 report warned that giving an event the time of its longest path "gives biased estimates and the bias is such that the estimated expected times of events are too small". It gave an example of two parallel activities, each a year long with a standard deviation of a month. Using a result it cites from Kendall, it puts the later of the two 0.56 standard deviations beyond either, so "the half month is lost". The exact figure, 1/√π, is 0.5642.
The opening chain is the default in our PERT project time estimator. An update dated 7 November 2026 gives it a control for equal parallel paths. Enter the three default tasks, a target of 14 and two paths, and it gives 86.0%, an expected finish of 12.78 and a P95 of 14.69, up from 14.26. Under this normal model the 12.00 total becomes a 25% date with two paths. Before the update the calculator's FAQ called the total "the critical-path sum" without that warning, and a single task of (0, 0, 10) produced a 95% range starting at −1.60.
The beta and triangular figures are our own simulation, a million draws per case; the calculator does not give them. Compared like with like, two paths take the 1958 beta from 93.4% to 87.2%, a loss of 6.2 points. The second beta falls from 90.1% to 81.3%, or 8.8 points, against the normal model's 6.75. So "about seven points" holds for the normal model and both betas; triangular tasks lose about 20.
RAND reached the same shape analytically. Two equal beta paths with mean 0.50 finish at 0.63, which we recomputed as 0.6286. In one example network PERT gave a mean of 4 where the true mean was 4.69. The memorandum's summary puts the largest errors in networks with many independent parallel paths of about equal length; the study appeared as a journal paper in 1964.
Splitting a task lightens the spread through independence alone
Project spread is the square root of the summed variances, so independent delays partly cancel. One task entered as (0, 10, 30) has a standard deviation of 5.00 and a P95 of 19.89. The same work entered as eight tasks of (0, 1.25, 3.75) keeps the expected total at 11.67, but its standard deviation falls to 1.77 and its P95 to 14.57.
The spread shrank by √8 only because the model assumes the eight pieces cannot go wrong together. Tasks sharing people or equipment are not independent, and their spread is wider than the arithmetic shows.
Correlation and unequal paths counterweigh part of the bias
The seven-point drop is for two equal, independent paths. If the paths are correlated, or if the second path is shorter, the effect is smaller. Under the normal model, two paths correlated at 0.5 give 87.5% on the target of 14, and at 0.8 they give 89.3%; perfectly correlated paths return 92.7%. A second path nine-tenths as long gives about 92.3%.
The 1958 authors still called the longest-path method "appealing", and offered "a very rough guess" that the simplification then in use might be biased "to the extent of a couple of months". By December 1962 Fulkerson's paper on expected critical path lengths gave an estimate its abstract calls "usually much better, and never worse" than using each time's mean, and Grubbs's abstract found the conditions for validating PERT's means and variances "rather restrictive".
View author profile → · Editorial policy
