Triangular Distribution

The triangular distribution is one of the simplest and most widely used probability distributions in quantitative risk analysis. It is defined entirely by three parameters: the minimum possible value, the most likely value (mode), and the maximum possible value. The probability density forms a triangle: it rises linearly from zero at the minimum to its peak at the most likely, then falls linearly back to zero at the maximum. The area under the triangle is 1.0 (as with any probability distribution). The triangular distribution is easy to explain to subject matter experts, easy to elicit estimates for, and requires no special statistical knowledge to use.

The main advantage of the triangular distribution over more complex alternatives is its transparency. An expert can directly specify minimum, most likely, and maximum values and immediately understand what those values mean in terms of the shape of the distribution. The main limitation is that it assumes a perfectly linear relationship between the mode and both extremes — the probability rises and falls symmetrically or asymmetrically but always linearly. Real-world uncertainties are often better represented by distributions that taper more gently toward the tails (like the BetaPERT distribution), but triangular is a reasonable approximation when precise tail behaviour is not critical.

Choosing between a triangular and a PERT or BetaPERT distribution depends on how important the tail behaviour is for the risk item in question. For most routine activity durations and cost elements, triangular is entirely adequate. For risks where the extreme values are genuinely important — because they are what drive the P90 or P95 outcome — the BetaPERT or lognormal distribution may give more realistic tail behaviour. In practice, the choice of distribution rarely changes the overall QRA output as much as the quality of the input estimates. Getting the right three-point values from well-calibrated subject matter experts matters far more than whether you use triangular or BetaPERT.