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Triangular Distribution

A triangular distribution is a probability distribution defined by three values: a minimum, a most likely value (the peak) and a maximum. It is the most widely used input shape in project Monte Carlo simulation because it is simple to explain, needs only a three-point estimate, and gives weight to the tails when data is scarce.

Min 20Most likely 25Max 40Mean 28.3P80 32.380% of outcomesDuration (days)
Triangular distribution. Min 20, most likely 25, max 40 days. The shaded area holds 80% of outcomes, ending at the P80 of 32.3 days.

Key formulas

Mean = (a + c + b) / 3
For x ≥ c: P(X ≤ x) = 1 − (b − x)² / ((b − a)(b − c))
where a = minimum, c = most likely, b = maximum

Worked example

Duration: minimum 20 days, most likely 25 days, maximum 40 days.

MeasureValue
Chance of finishing within the most likely 25 days25%
Mean28.3 days
P5027.8 days
P8032.3 days

Only one outcome in four meets the most likely duration. That single fact explains why schedules built on most likely values overrun.

Triangular versus PERT

A PERT (beta) distribution uses the same three points but concentrates more probability near the most likely value, giving thinner tails and a lower P80. Triangular is the more cautious choice when ranges come from judgement rather than data. PERT can suit well-understood, repeatable work where outcomes cluster tightly.

When to use it

  • Early-stage models with little historical data.
  • Workshop-derived ranges, where its simplicity makes challenge easier.
  • Skewed uncertainty: set the most likely value off-centre to show more room to overrun than underrun.

Common mistakes

  • Symmetric triangles for work that can overrun far more than it can underrun.
  • Hard limits taken literally. Real outcomes can fall outside a judgement-based maximum; consider a P10/P90 based input instead.
  • Same shape everywhere. Choose the distribution to fit the evidence for each input.

Related terms

Three-point estimateP80P50QSRASafran RiskFull glossary
Go deeper
Probability distributions for risk analysis: how to choose

Frequently asked questions

What is a triangular distribution?
A probability distribution defined by a minimum, most likely and maximum value. It is the most common input shape for project Monte Carlo simulation.
What is the mean of a triangular distribution?
The mean is the minimum plus the most likely value plus the maximum, divided by three.
Why use a triangular distribution in risk analysis?
It needs only a three-point estimate, is easy to explain and challenge, and gives reasonable weight to the tails when there is little data.

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