Probability Distributions > PERT Distribution

## What is the PERT Distribution?

PERT stands for *Program Evaluation and Review Technique*. It was developed in the 1950s for the Polaris weapon system to calculate a probable time frame for completion of the project based on optimistic, pessimistic, and most likely time frames. Nowadays, it is used for project completion time analysis in Program Evaluation and Review Technique(PERT). PERT is a modeling technique to estimate completion time or other desired event, bases on best estimates for the minimum, maximum, and most likely values for the event.

This led to the development of the **PERT distribution** (also called the beta-PERT or three-point estimation technique) — a smooth version of the uniform distribution or triangular distribution.

## PERT Distribution properties

The PERT distribution is defined by:

- The
**minimum**: the smallest value in a set. Can be any real number. - The
**maximum**: the largest value in a set. Can be any real number. - The
**mode**(*c*), the most common number in a set; this tells you where the peak of the distribution lies on the horizontal (x) axis. Can be any real number, but must be larger than the minimum and smaller than the maximum. In more technical terms, that’s min < c < max. - An optional
**shape parameter**, λ. This must be a*non-negative*real number.

The quantities min, max and mode must share the same unit dimensions (e.g. length, weight, cost) while λ is a dimensionless quantity (in other words, it’s just a number). Sometimes the letters α, β and m are used, where:

- α is the minimum,
- β is the maximum,
- m is the mode.

The probability density function (PDF) is defined as [2]

where

*α*= 1 + 4((b – a) /(c – a)).- β = 1 + 4((c – b) /(c – a)).

The mean (average) for the PERT distribution is:

μ = (a + 4m + b) / 6.

This is a weighted mean, as it takes into account the maximum and minimum estimates for the project. This assumption about the mean was first proposed by Clark in 1962 [3]. Clark used the PERT technique to estimate the effect of uncertainty of task durations on the outcome of a project schedule in evaluation, hence the name PERT distribution. The standard deviation(σ) is given by:

σ = (β – α) / 6.

The PERT distribution produces a bell-shaped curve that is nearly normal. It is essentially a Beta Distribution that has been extended to the maximum and minimum and given strict definitions for the mean and variance (a technique called “reparameterization”).

## Advantages and Disadvantages

Unlike the triangular distribution, the PERT distribution PDF is a smooth curve which places more emphasis on values near the most likely value, favoring values around the edges; this resemblance to the normal distribution may lead to more accurate estimates.

*The difference between the triangular and PERT distribution.*

In general terms, PERT is beneficial because it gives you a better understanding of how long your project will take and what factors might affect its completion date. This can help you plan your resources more efficiently and make sure that your project stays on track. Additionally, it enables you to assess potential risks associated with your project before they become an issue. By tracking progress with PERT, potential issues can be flagged before they become serious.

PERT also takes into account multiple scenarios and outcomes instead of just the “most likely” scenario. This means it’s predictive capability can be more accurate than other methods such as Monte Carlo or Critical Path Method (CPM). This makes it easier to adjust or modify existing plans if needed without impacting the overall timeline of your project too much.

One major drawback with using PERT is that it’s a Utopian model that assumes you can place infinite resources where they will benefit completion time the most. In the real world, this is usually not possible. As such, relying on the PERT model alone can result in unrealistic expectations for management and workers.

## References

Top image: NASA. Retrieved July 15, 2016 from: http://www.nasa.gov/pdf/741989main_Analytic%20Method%20for%20Risk%20Analysis%20-%20Final%20Report.pdf. 1] David Vose, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[2] Rao, K. et al. PERT distribution and its properties. International Research Journal of Modernization in Engineering Technology and Science. Volume: 03/Issue:10/October-2021 [3] Clark CE (1962) The PERT model for the distribution of an activity. Operations Research 10, pp. 405406