Statistics How To

Triangular Distribution / Triangle Distribution: Definition

Probability Distributions > Triangular Distribution

What is a Triangular Distribution?

A triangular distribution (sometimes called a triangle distribution) is a continuous probability distribution shaped like a triangle. It is defined by:

  • a: the minimum value, where a ≤ c,
  • c: the peak value (the height of the triangle), where a ≤ c ≤ b,
  • b: the maximum value, where b ≥ c.

This makes it very easy to estimate the distribution’s parameters from sample data:

  • Use the sample minimum as an estimator for a,
  • Use the sample maximum as an estimator for b, and
  • Use any reasonable statistic (e.g. the sample mean, mode or median) as an estimator for c.

If you don’t have sample data, expert knowledge can be used to estimate a probable minimum, maximum and most likely value (i.e. the mode).

The three parameters, a b and c change the shape of the triangle:
triangular distribution 2

Like all probability distributions, the area under the curve is 1. Therefore, the wider the distance between a and c (i.e. the range), the shorter the height.

When the peak is centered at zero and a = b, it is called a symmetric triangular distribution. When this happens, a and b are equal but opposite in sign (e.g. -2, 2) and are sometimes referred to as -a and a instead of a and b.
triangular distribution

PDF, Mean and Standard Deviation

The probability density function, which is used to find the probability a random variable falls into a certain range, is given by:
pdf triangular

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

The standard deviation, s, is:
s = (1/√6) a.
This formula makes the assumption that the distribution is centered at zero and that the endpoints are known.

Samuel Kotz, S and van Dorp.J. (2004) Beyond Beta. Sample chapter on the Triangle Distribution available here from World Scientific.


Confused and have questions? Head over to Chegg and use code “CS5OFFBTS18” (exp. 11/30/2018) to get $5 off your first month of Chegg Study, so you can understand any concept by asking a subject expert and getting an in-depth explanation online 24/7.

Comments? Need to post a correction? Please post a comment on our Facebook page.

Check out our updated Privacy policy and Cookie Policy