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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**, (*a*≤*c*),*c*: the**mode (height or peak)**, (*a*≤*c*≤*b*),*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/