# PERT distribution

< List of probability distributions < PERT distribution

The PERT distribution, also known as the beta-PERT or three-point estimation technique, is a smooth version of the uniform distribution or triangular distribution. It is widely used in project management and risk analysis. In this article, we will discuss why and how to use this useful tool.

## What is PERT?

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) — 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.

PERT is used to estimate project duration by providing an optimistic, pessimistic, and most likely time frame. It uses weighted averages to calculate these values by assigning weights that are based on the probability of each outcome occurring. For example, if an event has an 80% chance of happening in one week, it would be assigned a weight of 0.8. This means that it would be counted as 0.8 weeks when calculating the duration of the project.

## The PERT distribution

The PERT distribution (also called the beta-PERT distribution) is an almost normal, bell-shaped curve; It can be thought of as a transformation of the four-parameter beta distribution that includes a maximum and minimum and strict definitions for the mean and variance (a technique called “reparameterization”).

It is defined by:

• A real-valued minimum (a).
• A real-valued maximum (b).
• A mode (c): The “most likely” value, which can be any real number, but must be larger than the minimum and smaller than the maximum (min < c < max). Sometimes m is used instead of c.
• An optional non-negative, real-valued shape parameter, λ.

Min, max and mode must share the same unit dimensions (e.g., cost, length, or weight); λ is a dimensionless quantity.

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 (weighted) mean (μ) of the PERT distribution: μ = (a+ 4b + c) / 6.

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 = (b – a) / 6.

## Benefits of using a PERT distribution

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.

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.

## References

[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

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