Three shapes for three kinds of estimate.

#1. Triangular

Three numbers: minimum, most likely (the mode), maximum. Caladia draws a triangular probability density that rises linearly from min up to mode and falls linearly from mode down to max. The probability of any duration is proportional to the height of the triangle at that point.

Easiest to explain to non-stats people: “What’s the floor? What’s the ceiling? What’s your best guess in between?” It captures asymmetry happily — 3 / 5 / 12 says “5 is most likely but it could blow out long” without committing to a specific shape.

Where it falls short: a triangular puts more probability into the tails than most real-world durations do. If you have historical data showing a smoother shape, PERT-beta will fit it better. Use triangular when you genuinely don’t know more than three points.

#2. PERT-beta

Same three inputs — min, mode, max — but a beta distribution underneath. The shape is smoother than triangular and concentrates more probability near the mode. The classic PERT formulas baked in:

• Expected duration ≈ (min + 4 × mode + max) / 6
• Standard deviation ≈ (max − min) / 6

The 4× weight on the mode is what makes PERT-beta different from triangular: trust the mode more than the endpoints. Use this when the mode is grounded in past experience but the bounds are softer guesses. PERT-beta has been the project-management default since 1958.

Worth knowing: the expected duration is usually higher than the mode alone, because the asymmetric tail to the right pulls the mean up. For 3 / 5 / 12 the PERT mean is (3 + 20 + 12) / 6 ≈ 5.83 — almost a day above the “most likely” number. Surfacing that to the estimator is a calibration check: if they don’t buy 5.83, their max is probably too high.

#3. Normal

Two inputs: mean and standard deviation. Symmetric bell curve. Use it for activities where you have actual historical data — past run times of similar work — and the spread is roughly symmetric around the mean.

Two pitfalls worth flagging:

• Durations aren’t symmetric. Most activities have a hard floor (a minimum below which they cannot finish) and an open ceiling (delays can compound). A normal allows arbitrary values either side of the mean, including negatives. If your “minimum reasonable” is the same distance below the mean as the “maximum reasonable” is above, you’re probably looking at the wrong distribution.
• Familiarity isn’t a fit. Prefer triangular or PERT-beta unless you have empirical justification — ideally a histogram of past run times that actually looks bell-shaped.

When normal is the right call: machine-paced tasks, well-instrumented operations with logged completion times, anything where you’ve already seen the distribution in the data.

#4. Eliciting parameters from experts

Most estimates come from a person in a room, not a histogram. A few techniques make the resulting numbers more honest:

• Ask for endpoints before the mode. “How fast could this finish if everything went right?” “How slow if everything went wrong?” Then “what’s most likely?” Asking for the mode first anchors the estimator — they’ll cluster their min and max tightly around it and the resulting spread will be too narrow.
• Widen the range you’re given. Domain experts are reliably overconfident: minimums are too high and maximums are too low. A range elicited as “95% confidence” typically covers more like 50% of the cases the expert can imagine. Push back and look for surprises.
• Show the PERT mean. For triangular and PERT-beta, compute (min + 4×mode + max) / 6 and put it next to the mode. If the expert disagrees with that mean, the inputs need revising — usually the max is the optimistic one to revisit.
• Worry about the long right tail more than the left. Activities rarely finish dramatically earlier than expected; they often finish dramatically later. If you’re unsure whether to widen min or max, widen max.

For a project of any size, the handful of activities that dominate the schedule and cost tornadoes (see Concepts · Sensitivity) are the ones to spend elicitation effort on. Everything else can be triangular or PERT-beta with rough bounds — the simulation absorbs noise.