Check what a triangular cost estimate assumes about its low and high values

A triangular distribution uses a lower bound, a peak and an upper bound. Choosing it asserts that outcomes lie within those endpoints and that probability density rises and falls linearly around the peak. It is a specific modeling assumption, even when the three inputs came from an informal conversation.

Before using the result in a spending decision, ask what each estimate means. A supplier's best case, expected quote and difficult case may be scenarios rather than the minimum, mode and maximum needed by the distribution. A value described as a 90th percentile is especially different from an absolute upper bound.

MathWorks' distribution documentation defines the triangular parameters and their formulas. The numerical examples below use those mathematical definitions with invented cost assumptions. They do not establish that a triangular distribution fits any particular company's project experience.

Suppose a cost model uses a minimum of $100,000, a mode of $120,000 and a maximum of $200,000. The mean of a triangular distribution is the sum of those three values divided by three, giving $140,000. Calling the $120,000 mode the expected cost would understate that modeled mean by $20,000.

The long range above the peak explains the difference. In this example, only 20% of the modeled probability lies at or below $120,000. The peak marks the greatest density, not the midpoint of the probability distribution. For a continuous distribution, no single exact dollar value carries a positive point probability.

The median is approximately $136,754 and the 90th percentile approximately $171,716 under the assumed shape. These numbers are mathematical consequences of the three inputs. Their apparent precision should not be mistaken for evidence that the inputs themselves are known to the nearest dollar.

If an estimator intended $100,000 to mean a low but possible case and $200,000 to mean a high case with some chance of exceedance, using them as bounds removes those outside outcomes. The model then assigns zero probability below the first and above the second by construction.

Ask the estimator whether an outcome above $200,000 is physically or contractually impossible, merely unlikely, or excluded from the quoted scope. Those answers require different treatments. A contractual cap may still exclude change orders or delay costs, so check what the bound actually covers.

Where experts can supply only percentile judgments, use a method consistent with those judgments or retain named scenarios. Do not silently reinterpret their statements to satisfy a software input form. Record the elicited meaning and any additional assumptions needed to convert it into a distribution.

A single triangle may combine routine uncertainty with a different event such as a supplier failure. Consider whether the business process is better described by a routine distribution plus a separate disruption scenario. That choice should follow operating evidence, not a preference for a smoother chart.

For an invented illustration, suppose routine cost follows the $100,000/$120,000/$200,000 triangle and a separate 10% disruption case has total cost of $500,000. If routine outcomes have 90% weight, the mixture mean is 90% of $140,000 plus 10% of $500,000, or $176,000.

Specify whether the $500,000 is total cost in the disruption case or an additional cost on top of routine spending. Adding $500,000 to the triangle when it was intended as a total would count the routine component twice. Preserve the event definition beside its probability and cost treatment.

A distribution for the total project is different from one for the work still uncommitted. If $70,000 is already spent, do not add that amount to a total-cost triangle that already includes it. Alternatively, if the triangle covers only remaining work, identify the historical amount separately in the full project estimate.

Keep price date, currency, quantities and exclusions consistent across the three inputs. A low estimate based on current prices and a high estimate including several years of escalation can embed an unstated time assumption. Show the mechanism rather than treating the values as interchangeable expert opinions.

Review relationships between cost components. Several independently sampled triangles may miss a common delay that moves them together. Equally, an explicit shared schedule driver should not receive an additional blanket dependency adjustment for the same event. The combined model needs one coherent account of the operating cause.

Software packages do not always order or name triangular parameters identically. Some use minimum, mode and maximum directly; others express the peak relative to the lower bound and total range. Verify the documentation for the tool being used and check a simple known case before loading a large cost register.

For the example, sampled values should remain between $100,000 and $200,000, and a sufficiently large independent sample should have a mean near $140,000, allowing for sampling error. A sample centered near $120,000 suggests that the parameters or distribution may have been interpreted differently.

Keep all cost units consistent. Entering 100 for one parameter and 120,000 for another can satisfy a basic numeric check while describing the wrong scale. Review the input table, distribution summaries and a few generated observations before trusting a total-cost percentile.

As quotes, design decisions or completed work narrow uncertainty, update the relevant parameters and preserve the earlier version. Explain whether a change reflects better evidence, a scope change or a different distribution choice. Those are different reasons for a movement in the reported budget range.

Present the three inputs with their meanings and the summary management is using, such as mean cost or a selected percentile. A triangular model is most useful when it makes the estimate's assumptions visible. It should not turn three loosely described numbers into a stronger claim about certainty than the evidence supports.