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Triangular Distribution Calculator

Enter min, max, mode and the point x to evaluate.

How it is calculated

The distribution is set by a minimum a, a maximum b and a most likely value (mode) c, with a < b and a ≤ c ≤ b. At the point x the density is 2(x − a) ÷ ((b − a)(c − a)) for x below c, 2 ÷ (b − a) at x = c, and 2(b − x) ÷ ((b − a)(b − c)) for x above c, and 0 outside a to b. The cumulative probability P(X ≤ x) is (x − a)² ÷ ((b − a)(c − a)) up to c and 1 − (b − x)² ÷ ((b − a)(b − c)) above c, and it is 0 below a and 1 above b. The mean is (a + b + c) ÷ 3. All four inputs must use the same unit, and the density is per unit of x, not a probability itself. Invalid settings, such as c outside a to b, give no result.

Worked example

A task is estimated at a minimum of 10 days, a maximum of 30 and a most likely 14. Evaluate at x = 20 days. Since 20 is above the mode, density: 2 × (30 − 20) ÷ ((30 − 10) × (30 − 14)) = 20 ÷ 320 = 0.0625. Cumulative probability: 1 − (30 − 20)² ÷ (20 × 16) = 1 − 100 ÷ 320 = 0.6875. The mean is (10 + 30 + 14) ÷ 3 = 18.00. So there is about a 68.75% chance the task takes 20 days or less.

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Common mistakes

  • Reading the density as a probability. A value of 0.0625 is per day, not 6.25%; the probability of finishing by x is the cumulative figure.
  • Mixing up the mode with the mean. The mode here is 14 days, but the mean is 18, because the longer tail to the right pulls the average up.
  • Entering a mode outside the minimum and maximum, or a minimum that is not below the maximum. The triangle is then not defined and the tool returns nothing.
  • Mixing units, such as a and b in days with x in weeks. All four values have to share one unit.
  • Using it as the PERT weighted average. PERT uses (a + 4c + b) ÷ 6, while this mean is (a + b + c) ÷ 3, so the two figures differ.

About this calculator

The triangular distribution is a simple continuous probability distribution defined by just three numbers: a minimum a, a maximum b, and a most-likely value (mode) c, forming a triangle-shaped probability density that peaks at c. This calculator evaluates that distribution at any point x, returning the probability density (PDF) at x, the cumulative probability (CDF) of a value falling at or below x, and the distribution's overall mean, (a+b+c)/3.

Because it needs no historical data — only a best guess, worst case and most likely estimate — the triangular distribution is a go-to tool in project management, risk analysis and Monte Carlo simulation, especially in PERT-style estimating where a task's duration or cost is bounded by optimistic, pessimistic and most-likely figures supplied by subject-matter experts. Statistics students also meet it as an accessible first example of a non-uniform continuous distribution before moving on to more complex bell-shaped curves.

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