Triangular Distribution Generator (min/mode/max)

Q: What do a, c, and b mean?

a is the minimum, c is the mode or most likely value, and b is the maximum.

Q: Why does the shape become skewed?

If c is not centered between a and b, more density shifts toward the side closer to c.

Q: How is this different from Beta or PERT?

Triangular uses straight-line density between the bounds, while Beta or PERT smooth the curve and change the shape near the center and edges.

Q: Why might another tool show different numbers?

Different defaults, units, sample sizes, or rounding rules can change the displayed mean, variance, or sampled histogram.

Q: What baseline should I try first?

Start with a simple minimum, mode, and maximum, then change only the mode or one bound so the reason for each shape shift stays clear.

What is a triangular distribution?

The triangular distribution models uncertainty when you can specify min, most likely (mode), and max values. It’s commonly used for quick estimates and PERT-like inputs.

a: minimum (optimistic).
c: mode / most likely value.
b: maximum (pessimistic).
If c is closer to a or b, the distribution becomes skewed.

PDF: for a≤x<c, 2(x-a)/((b-a)(c-a)); for c≤x≤b, 2(b-x)/((b-a)(b-c)). Mean: (a+b+c)/3. Variance: (a²+b²+c²-ab-ac-bc)/18.

You don’t need to enter personal information to use this tool.

Presets

Quickly set common estimation shapes (you can tweak values after applying).

Generator

Set min/mode/max, sample size, bins, and RNG. Then generate samples and export results.

Distribution

Sample size (N) Bins

RNG

Show theoretical PDF

Sample stats

Samples (first 20)

How to use this tool

Use this page when you know a minimum, a most likely value, and a maximum and need a fast bounded model.

Use in 3 steps

Enter a, c, and b so that a ≤ c ≤ b.
Generate a sample and compare the histogram with the theoretical PDF and summary statistics.
Change only one input at a time to separate shifts in the center, skew, and total range.

How to read the result

The histogram shows the sampled shape. The theory panel shows the exact mean and variance implied by your inputs. If c moves toward a or b, the shape becomes visibly skewed.

Boundary checks

If c=a or c=b, almost all asymmetry points to one side.
Keep the same unit across scenarios before comparing outputs.
When comparing with Beta or PERT, match the same bounds and the same interpretation of the mode.

Frequently asked questions

What do a, c, and b mean?