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title: "Triangular distribution"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575431/Triangular%20distribution"
format: markdown
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The [Triangular distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575255/) is the most commonly used distribution for modeling expert opinion. It is defined by its minimum (*a*), most likely (*b*) and maximum (*c*) values. The figure below shows three [Triangular](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575255/) distributions: Triangular(0,10,20), Triangular(0,10,50), Triangular(0,50,50) which are symmetric, right-skewed and left skewed respectively.

 

![image](media://29773a1b-10db-46cb-9882-18fb01ca3d1c)

 

The Triangular distribution has a very obvious appeal because it is so easy to think about the three defining parameters and to envisage the effect of any changes.

 

The mean and standard deviation of the Triangular distribution are determined from its three parameters:

 

            mean = 

![image](media://dd81290b-955b-4fe0-b10b-51d2a11ea1fd)

 > Macro (mathinline)



            standard deviation = 

![image](media://62f6e123-ae5f-4bed-89cc-cf59181ac33c)

 > Macro (mathinline)



 

These formulas show that the mean and standard deviation are equally sensitive to all three parameters. Many models involve parameters for which it is fairly easy to estimate the minimum and most likely values, but for which the maximum is almost unbounded and could be very large.

 

<u>[Central Limit Theorem](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574953/)</u> tells us that, when adding up a large number of distributions (for example adding costs or tasks duration), it is the distributions' means and standard deviations that are most important because they determine the mean and standard deviation of the risk analysis result. In situations where the maximum is so difficult to determine, the Triangular distribution is not usually appropriate since the values generated from it will depend a great deal on how the estimation of the maximum is approached. For example, if the maximum is assumed to be the absolutely largest possible value, the risk analysis output will have a far larger mean and standard deviation than if the maximum is assumed to be a "practical" maximum by the estimating experts.

 

The Triangular distribution is often considered to be appropriate where little is known about the parameter outside an approximate estimate of its minimum, most likely and maximum values. On the other hand, its sharp, very localized peak and straight lines produce a very definite, unusual and very unnatural shape, which could be said to conflict with the assumption of little knowledge of the parameter.

 

There are two useful variations of the Triangular distribution that you can construct with simulation software. For these two variations of the Triangular distribution, you have to give the 5 and 95 percentile or the 10 and 90 percentiles. The way to construct both variations of the Triangular distribution is discussed [here](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575255/).

 

The figure below shows a Triangular distribution where the 5% percentile is 40, the Most Likely value is 50 and the 95% percentile is 80. You can see that this distribution (in which you specified the percentiles) of course extend beyond these "practical minimum and maximum" (40 and 80 here).

 

![image](media://5cd792b3-693b-4649-9b20-19a8698dbc85)

 

Both variations of the Triangular distribution are a useful way of avoiding asking the expert to estimate the absolute minimum and maximum of a parameter: questions that the expert often has difficulty in answering meaningfully since there may theoretically be no minimum or maximum. Instead, the analyst can discuss what values of *p (the left practical minimum)* and *q (the right* *practical minimum)* the expert would use to define "practical" minima and maxima respectively. Once this has been decided, the expert only has to give estimates for the practical minimum, most likely and practical maximum for each estimated parameter and the same *p* and *q* estimates are used for all estimates. One drawback is that the expert may not appreciate the final range to which the distribution may extend, so it is wise to plot the distribution and have it agreed by the expert before using it in the model.

 

 

 

 

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