# Triangular Distribution (Symmetric)

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.

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

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

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

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

Support (range) = a  ≤ b

## References:

Modified from Stephanie Glen. “Triangular Distribution / Triangle Distribution: Definition” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/triangular-distribution/