Triangular Distribution (Symmetric)

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A triangular distribution (or triangle distribution) is a continuous probability distribution defined by three parameters:

  • a: the minimum or lower limit, (ac),
  • c: the mode (height or peak), (acb),
  • b: the maximum or upper limit (b ≥ c).

When a and b are equal but opposite in sign (e.g., -1, 1), the distribution is a symmetric triangular distribution, which is a special case of a triangular distribution.

A symmetric triangular distribution.

The parameters a, b, and c can be estimated from sample data:

  • a: Use the sample minimum,
  • b: Use the sample maximum.
  • c: Use the sample mean, mode or median.

The parameters can also be estimated by expert knowledge of likely values. The parameters, a, b and c change the triangle’s shape:

Like all probability distributions, the total probability (aka the area under the curve) equals 100% (1.0). This mean that wider ranges will have shorter peaks and more compact ranges will have higher peaks.

Properties of the Triangular Distribution

The probability density function is given by

PDF for the triangular distribution.

The mean for the triangular distribution is:
μ = 1/3 (a + b + c).

The standard deviation, s, is:
s = (1/√6) a.

  • The distribution is centered at zero,
  • Endpoints are known.

Support (range) = a  ≤ b


Modified from Stephanie Glen. “Triangular Distribution / Triangle Distribution: Definition” From Elementary Statistics for the rest of us!