The basic concept
The triangular distribution is a continuous model described by a minimum a, a most likely value c (the mode), and a maximum b. This guide uses a < c < b. Its density rises linearly to the mode and then falls linearly to zero.
It is useful for a transparent first model when experts can supply plausible bounds and a most likely value but data are limited. The shape remains an assumption, rather than something those three estimates prove.
Support, density and cumulative probability
The probability density function (PDF) is zero outside [a,b]. The cumulative distribution function (CDF) is zero below a and one above b. At c, both branches agree.
Worked example: project completion time
Assume a project takes at least 4 weeks, most likely 6 weeks and at most 12 weeks, with a triangular distribution.
| Question | Calculation | Answer |
|---|---|---|
| Mean time | (4 + 6 + 12)/3 | 7.333 weeks |
| Variance | (4²+6²+12²−4×6−4×12−6×12)/18 | 2.889 weeks² |
| Standard deviation | √2.889 | 1.700 weeks |
| P(completion ≤ 8 weeks) | 1 − (12−8)²/[(12−4)(12−6)] | 66.67% |
| P(completion > 10 weeks) | (12−10)²/[(12−4)(12−6)] | 8.33% |
The mean exceeds the most likely value because the upper side extends farther from the mode.
Excel cumulative probability
Enter the minimum in B1, mode in B2, maximum in B3 and threshold in B4.
With 4, 6, 12 and 8 respectively, the result is 66.67%. First validate B1 < B2 < B3.
Simulation by inverse transformation
Place one uniform draw =RAND() in B5. Use that same draw throughout the following formula.
Repeated recalculation produces different simulated completion times. A large simulation should approach the theoretical mean, but any finite simulation has sampling variation.
Triangular versus three-point beta models
The triangular mean is (a+c+b)/3. A commonly used project-estimation weighted mean, (a+4c+b)/6, belongs to a different modelling convention. Do not use that weighted mean while claiming to use the triangular distribution.
Hard bounds mean outcomes beyond them have zero model probability. If delays can exceed 12 weeks, the chosen bound is too restrictive and may conceal risk.
Check your understanding
Does the most likely value have positive point probability?
No. The density peaks at the mode, but the model is continuous. A probability requires an interval.
Interactive distribution laboratory
Change the parameters and threshold, then switch between cumulative probability and density or mass. Values are numerical approximations for learning.
Horizontal axis: the variable’s value. Vertical axis: density, probability mass, or cumulative probability as labelled. Shading highlights values at or below the threshold. The plot shows a finite viewing window.
Sources and further reading
- National Institute of Standards and Technology: Gallery of Distributions
- Microsoft: Statistical functions reference
All numerical finance and project examples are hypothetical teaching illustrations. Probability assumptions require validation before practical use.