Calculators

Triangular Distribution Calculator

Calculate triangular distribution density, cumulative and upper-tail probabilities, mean and variance privately in your browser.

Model a continuous value using a minimum, most likely value, and maximum. Calculations stay in your browser.

Enter numbers between −1e12 and 1e12. Maximum must exceed minimum by at least 1e-9, and mode must lie between them. Evaluation values outside the range are allowed. Results use 12 significant digits.

Your triangular distribution results will appear here.

Model a bounded continuous quantity.

A triangular distribution models a continuous quantity from a minimum a, mode c, and maximum b, with a < b and a ≤ c ≤ b. Below the mode, density is 2(x − a)/[(b − a)(c − a)] and cumulative probability is (x − a)²/[(b − a)(c − a)]. At or above the mode, density is 2(b − x)/[(b − a)(b − c)] and upper-tail probability is (b − x)²/[(b − a)(b − c)]. Endpoint modes use the appropriate one-sided branch. Outside [a, b], density is zero; cumulative probability is zero below a and one above b. Mean = (a + b + c)/3 and variance = (a² + b² + c² − ab − ac − bc)/18. For a = 0, c = 5, b = 10 and x = 5, density is 0.2, cumulative probability is 0.5, mean is 5 and variance is approximately 4.16666666667. Density can exceed one and is not a point probability: P(X = x) = 0. Enter finite numbers with absolute value at most 1e12 and b − a at least 1e-9. Scientific notation is accepted. Results use floating-point arithmetic rounded to 12 significant digits; extreme tails may round to zero. This assumed shape is a model, not a fit to observed data. All calculations stay in your browser.

Frequently Asked Questions

Everything you need to know about this tool, how it works, and privacy.

Can the mode equal an endpoint?

Yes. Set c = a for a decreasing triangle, or c = b for an increasing triangle. Minimum and maximum must still differ.

Is density the probability of an exact value?

No. Density has inverse measurement units and may exceed one. A single exact value has probability zero in this continuous model; probabilities come from areas under the density.

What do values outside the bounds return?

Below the minimum, cumulative probability is zero and upper-tail probability is one. Above the maximum, these are one and zero. Density is zero outside the bounds.

When is a triangular model useful?

Use it as a simple bounded model when you have a minimum, maximum and most likely value but limited data. It assumes straight sides and does not establish that real observations follow this shape.