Selecting Probability Distribution Functions

What a probability distribution is

When you say a cost item is "$80k to $180k, most likely $100k," you've given three points. A distribution fills in everything between them: is it far more likely to land near $100k than near $180k? Is the high tail long (a small chance of a very bad outcome)? Those questions matter, because a Monte Carlo simulation samples from the distribution thousands of times — so the shape you pick directly determines the contingency you get out.

The four shapes you'll use most

You don't need exotic statistics. A handful of distributions cover almost all project risk work:

• Triangular — Defined by low, most likely, high. Simple, intuitive, no special knowledge needed. The workhorse for expert-judgment ranges. Mean = (low + most likely + high) ÷ 3.
• PERT (BetaPERT) — Same three points, but a smoother bell that weights the most-likely value more heavily. Often more realistic than triangular. Mean = (low + 4×most likely + high) ÷ 6.
• Uniform — Every value between low and high is equally likely — no "most likely" point. Use when you genuinely have no idea where within the range it'll land. Mean = (low + high) ÷ 2.
• Normal — The symmetric bell — defined by a mean and standard deviation. Good for quantities that cluster around a center with equal spread either way (and where extreme values are rare).

Distribution explorer

Pick a shape and set the low / most-likely / high points. Watch how the shape — and the resulting mean — change for the same three numbers:

Choosing the right distribution

Match the shape to what you actually know. If you have only expert low/likely/high judgment, triangular or PERT is right — PERT when you trust the most-likely value, triangular when you don't. Use uniform only when you truly have no central tendency. Reserve normal for quantities with genuine symmetric spread. Above all, respect skew: most cost and schedule risk has a longer tail on the bad side, so don't force a symmetric shape onto an asymmetric reality.

Ten things to remember

1. A distribution describes how likely each value of an uncertain quantity is.
2. A range gives the ends; the distribution fills in the middle.
3. Triangular: low/likely/high, simple. Mean = (lo + likely + hi) ÷ 3.
4. PERT: smoother bell, trusts most-likely. Mean = (lo + 4×likely + hi) ÷ 6.
5. Uniform: all values equally likely. Mean = (lo + hi) ÷ 2.
6. Normal: symmetric bell from mean and standard deviation.
7. Same 80/100/180 → triangular $120k, PERT $110k, uniform $130k.
8. Respect skew — cost/schedule risk usually has a long bad-side tail.
9. Symmetric shapes can understate big overruns — don't force them.
10. The distribution is the raw material Monte Carlo simulations run on.