What low, most-likely and high must mean
A Monte Carlo QRA is only as good as the three-point estimates feeding it. The simulation logic is well understood and fairly consistent across tools. The input ranges are where models differ, and where most of them are weakest.
Before any numbers are collected, fix the definitions and write them on the input sheet. Most likely is the mode: the single most probable outcome, which is not the mean and need not be the plan figure. Low and high need one of two meanings, and the model must be parameterised to match.
Absolute bounds: low is the least the activity could take and high is the most, with nothing outside. Triangular, PERT and BetaPERT inputs are absolute bounds. The simulation never samples outside them. Estimators rarely believe in true bounds, so they tend to give values that are too tight.
Percentiles: low is the P5 or P10 and high is the P95 or P90, so there is a 5% or 10% chance of falling outside each end. This is easier for experts to judge and is the better elicitation basis. It needs a distribution that takes percentile inputs and extrapolates the bounds, such as a Trigen-type feature in the tool, or a lognormal. Feeding P10/P90 values into a triangular or PERT silently treats them as absolute bounds and understates the tails.
Pick one meaning for the whole model, state it in the model basis, and check that the tool parameterisation matches. AACE International Recommended Practice 66R-11, Selecting Probability Distribution Functions for Use in Cost and Schedule Risk Simulation Models, is the reference for choosing the distribution.
The PERT formula and what its weighting does
PERT (Program Evaluation and Review Technique) was developed for the US Navy Polaris programme in the late 1950s. The Beta-distribution approximation in Malcolm et al. (1959) gives a mean of (O + 4M + P) / 6 and a standard deviation of about (P - O) / 6, where O, M and P are the optimistic, most-likely and pessimistic values.
The 4 is not a statement that the most-likely outcome is four times as probable as either extreme. It comes from fitting a Beta distribution to the three points. In the modern BetaPERT form it is a shape parameter, lambda. With a = 1 + lambda (M - O) / (P - O) and b = 1 + lambda (P - M) / (P - O), the Beta runs from O to P and has mean (O + lambda M + P) / (lambda + 2). Lambda = 4 is classic PERT. Lambda = 3 gives the (O + 3M + P) / 5 variant. A higher lambda concentrates the distribution more tightly around the mode.
Worked example: O = 8, M = 12, P = 22 days. Classic PERT mean = (8 + 48 + 22) / 6 = 13.0 days. The lambda = 3 mean = (8 + 36 + 22) / 5 = 13.2 days. The triangular mean = (8 + 12 + 22) / 3 = 14.0 days. The Beta standard deviations are 2.54 days (lambda = 4) and 2.76 days (lambda = 3). The (P - O) / 6 shortcut gives 2.33 days, which is only an approximation.
The (P - O) / 6 shortcut assumes the range spans about six standard deviations, which fits near-absolute bounds. If low and high are P5 and P95, the range spans about 3.3 standard deviations, and dividing by 6 understates the spread by nearly half. For percentile inputs use percentile-based approximations. Swanson's rule takes the mean as 0.3 x P10 + 0.4 x P50 + 0.3 x P90. The Pearson-Tukey approximation takes it as 0.185 x P5 + 0.63 x P50 + 0.185 x P95. With P10/P50/P90 of 10/14/24 days, Swanson gives 0.3 x 10 + 0.4 x 14 + 0.3 x 24 = 15.8 days. With P5/P50/P95 of 9/14/27 days, Pearson-Tukey gives 0.185 x 9 + 0.63 x 14 + 0.185 x 27 = 15.5 days.
These formulas give a point estimate. They are useful for sense-checking an input, for a deterministic expected value, or for a small cost line that does not justify full simulation. In a live QRA the simulation samples the whole distribution, so the shape you assign matters more than the formula.
Which distribution, and what it does to the output
Triangular, PERT and BetaPERT are all bounded. PERT is BetaPERT with lambda = 4, not a separate family. None of them samples outside low and high, and the Beta forms have thinner tails than the triangular between the same bounds. Only unbounded forms such as the lognormal, or tools that extrapolate the end points from percentile inputs, can produce values beyond the stated range.
Shape changes the answer when the inputs are skewed. For 8/12/22 days, the triangular mean is 14.0 days and its P80 is 16.7 days. Classic PERT gives 13.0 days and 15.3 days. That is a 1.4-day gap at P80 on one activity, and the gap compounds across a long chain of right-skewed activities. For near-symmetric inputs the difference is small.
The practical rule is to choose the distribution before eliciting, tell the experts what it is, and check the choice against 66R-11. If the P80 of the model moves materially when you swap distribution, the inputs are doing too much work and need more evidence. Triangular is a reasonable default for early-stage work provided low and high are treated honestly as bounds or as percentiles with a tool that supports them.
Why estimates come out too narrow
The most common failure is a range that is too tight. Asked for an optimistic value, people start from the base estimate and subtract a little. Asked for a pessimistic value, they add a little. This is the anchoring effect described by Tversky and Kahneman in Judgment under Uncertainty: Heuristics and Biases (Science, 1974). The result is a narrow, symmetric distribution centred on the plan, and a simulation output that barely differs from the deterministic estimate.
A second effect is overconfidence. In calibration studies, reviewed by Lichtenstein, Fischhoff and Phillips (1982), people asked for ranges they are 90% sure contain the true value capture it far less often than 90% of the time. The consistent finding is that stated ranges are too narrow. The HM Treasury Green Book (2026 edition) makes the same point at programme level, which is why it requires an explicit adjustment for optimism bias.
Reference class forecasting, associated with Bent Flyvbjerg, is the structural correction. Compare the project or activity with the outcomes of a class of similar ones, and anchor on the distribution of those outcomes, not on the plan.
Calibration practice
Calibration means that when an assessor says 90%, the true value falls inside the range about 90% of the time. It can be trained, and the training is simple.
Train on known quantities. Give assessors ten questions with answers they can later check, for example the length of a well-known structure or a published unit rate. Ask for 90% ranges. Score them: if the answer sits inside the range fewer than nine times in ten, they are overconfident. Calibration training of this kind is described by Douglas Hubbard in How to Measure Anything.
Give feedback straight away. Show the results, discuss why ranges were too tight, and repeat with a fresh set. Repeat until the hit rate approaches the stated confidence.
Use an equivalent-bet test on each real estimate. Ask: would you rather win a prize if the true value falls inside your range, or win the same prize if a spun wheel lands in a 90% sector? If you prefer the wheel, the range is too narrow. If you prefer the range, it is too wide. Adjust until you are indifferent.
Keep a track record. Log each three-point estimate against the actual outcome and score the hit rate by assessor and by activity type. On a live programme this is the only real test of whether the estimates were calibrated.
Calibrating against evidence
The strongest estimates rest on data: historic outturn, benchmarks and analogous activities. Bring the evidence to the workshop and challenge ranges that sit outside it. If every comparable programme ran 12 to 16 weeks and a participant claims a 6-week pessimistic value, ask what is different about this one.
Convert rates to durations before fitting a range. Suppose 200 m of piling has been achieved at between 8 m/day and 25 m/day, with a mean of 16 m/day. The durations are 200 / 25 = 8 days to 200 / 8 = 25 days. The mean rate gives 200 / 16 = 12.5 days, which is not the mean duration, because the reciprocal skews the range to the right. Model the duration, not the rate.
Where there is no data - first-of-a-kind work, or early-stage scope - expert judgement is unavoidable, and the basis should be written down. For example: low 30 days, most likely 45 days, high 80 days, reflecting uncertainty about equipment availability and thin analogous data. A bare set of numbers without a reason cannot be reviewed.
Structured elicitation protocols, such as SHELF and Cooke's classical method, formalise this for high-stakes inputs. They are rarely justified for every activity but are worth considering for the few that dominate the result.
Breaking the anchor in the workshop
Ask for conditions before numbers. Instead of what the optimistic value is, ask under what specific conditions the activity would finish early, and how early. Instead of what the pessimistic value is, ask what would have to go wrong for it to overrun the plan significantly, and by how much. Conditional questions get participants working with scenarios rather than adjustments to the anchor.
Where the base estimate is not yet fixed, collect low and high from each participant independently before showing the plan figure. Where the base is already agreed, asking for conditions still produces wider and better-reasoned ranges.
Prioritise. A model with hundreds of activities cannot give each five to ten minutes. Run a quick first pass with coarse ranges, run the simulation, and spend the full time on the activities and costs with the highest sensitivity or criticality. The tail of low-influence items can stay on the coarse values, with the reason recorded.
Variability and event risk
Keep variability and event risk separate. Variability is the inherent uncertainty in an activity that will definitely happen: piling will take place, and the question is whether it takes 40, 50 or 70 days. Event risk is a discrete occurrence that may not happen, such as unexpected ground conditions with a 30% probability and a 5 to 10 day impact.
Variability belongs in the three-point estimate. Event risk belongs in the register, and in the model it is sampled as an event: it occurs or not in each iteration with its stated probability, and when it occurs its own three-point impact is applied. It is not applied as a fixed probability-weighted expected value.
The test is to ask whether the high value captures things that might go wrong, or how the activity varies when nothing goes wrong. If it is effectively the worst realistic case including identifiable risks, event risk has been bundled into variability. Pull the events out into the register and recalibrate the range to inherent variability only. Otherwise the model cannot show which risks drive the result, and cannot support targeted mitigation.
Elicitation record and correlation
For every three-point estimate that matters, record five things: the scope and assumptions, the conditions behind low and high, the evidence considered, the definition used for low and high (absolute or percentile), and any dissenting view. This is what lets an independent assurance reviewer see the thinking behind the inputs, not just the numbers.
Discuss correlation in the same session. If two activities have high values driven by the same severe weather, they are not independent. Capturing that during elicitation is more coherent than adding correlation afterwards, and it avoids a zero-correlation model that understates the tails.
Red flags when reviewing a QRA
Symmetric ranges on most activities are suspect. Real activities usually have a longer right tail than left, meaning more overrun exposure than saving. Symmetry everywhere suggests anchoring.
Narrow ranges on long activities are a second flag. An activity with a most-likely of 180 days and low and high of 170 and 195 days has a coefficient of variation of about 2.8% when modelled as a triangular. Compare that with the outturn of similar activities before accepting it.
A similar percentage spread on every activity is a third flag. Excavation, M&E installation, software integration and commissioning have different risk profiles, and their ranges should show it.
Other checks: no stated definition of low and high, a high value that mixes in event risks, no evidence column, and no link between the range and any outturn data.
How SOMA approaches this
On a QSRA or QCRA we agree the low and high definition first, run a coarse pass to find the drivers, and spend elicitation time on those. We record evidence and reasoning against each driver.