# PERT distribution

Format: PERT(min, mode, max)

The PERT distribution (also known as the Beta-PERT distribution) gets its name because it uses the same assumption about the mean (see below) as PERT networks (used in the past for project planning). It is a version of the Beta distribution and requires the same three parameters as the Triangle distribution, namely minimum (a), mode (b) and maximum (c). The figure below shows three PERT distributions whose shape can be compared to triangle distributions:

##### Uses

The PERT distribution is used exclusively for modeling expert estimates, where one is given the expert's minimum, most likely and maximum guesses. It is a direct alternative to a Triangle distribution, so a discussion is warranted on comparing the two:

##### Comparison with the Triangle distribution

The equation of a PERT distribution is related to the Beta4 distribution as follows:

PERT (a, b, c) = Beta4(a1, a2, a, c)

where:

The last equation for the mean is a restriction that is assumed in order to be able to determine values for a1 and a2. It also shows how the mean for the PERT distribution is four times more sensitive to the most likely value than to the minimum and maximum values.

This should be compared with the Triangle distribution where the mean is equally sensitive to each parameter. The PERT distribution therefore does not suffer to the same extent the potential systematic bias problems of the Triangle distribution, that is in producing too great a value for the mean of the risk analysis results where the maximum for the distribution is very large.

The standard deviation of a PERT distribution is also less sensitive to the estimate of the extremes. Although the equation for the PERT standard deviation is rather complex, the point can be illustrated very well graphically. The figure below compares the standard deviations of the Triangle and PERT distributions with minimum a=0, maximum c= 1, and varying most likely value b.

The observed pattern extends to any {a,b,c} set of values. The graph shows that the PERT distribution produces a systematically lower standard deviation than the Triangle distribution, particularly where the distribution is highly skewed (i.e. b is close to the minimum or maximum). As a general rough rule of thumb, cost and duration distributions for project tasks often have a ratio of about 2:1 between the (maximum - most likely) and (most likely - minimum), equivalent to b = 0.3333 on the figure above. The standard deviation of the PERT distribution at this point is about 88% of that for the Triangle distribution. This implies that using PERT distributions throughout a cost or schedule model, or any other additive model with similar ratios, will display about 10% less uncertainty than the equivalent model using Triangle distributions.

You might argue that the increased uncertainty that occurs with Triangle distributions will compensate to some degree for the over-confidence that is often apparent in subjective estimating. The argument is quite appealing at first sight but is not conducive to the long term improvement of the organization's ability to estimate. We would rather see an expert's opinion modelled as precisely as is practical. Then, if the expert is consistently over-confident, this will become apparent with time and his/her estimating can be re-calibrated.

##### Limitations to using the PERT distribution

The PERT distribution came out of the need to describe the uncertainty in tasks during the development of the Polaris missile (Clark, 1962). The project had thousands of tasks and estimates needed to be made that were intuitive, quick and consistent in approach. The Four-Parameter Beta distribution (Beta4) was used just because it came to the author's mind (the Kumaraswamy distribution would also have been a good candidate, for example). The decision to constrain the distribution so that it's Mean = (Min + 4* Mode + Max)/6 was an approximation to their decision that the distribution should have a standard deviation of 1/6 of its range (i.e. Max - Min).

Farnum and Stanton (1987) demonstrated that, if one wishes to maintain this [standard deviation = range/6] idea then the PERT distribution should only be used with a certain range of values for the mode, namely:

Mode + 0.13(Max - Mode) < Mode < Mode + 0.13(Max - Mode)

i.e. that the mode should not lie less that 13% of the range from either the Min or Max values. In practice this is pretty good advice, and tends to occur when one has a very high Max value relative to the Min and Mode, since the distribution is very skewed and gives very small density in the extreme tail making the Max value estimate rather meaningless, for example:

Golenko-Ginzburg (1988) describes a study that analyzed many PERT networks and concluded that "the "most likely" activity-time estimate m [mode] is practically useless". They found that the location of the mode in most project tasks was approximately one third of the distance from the Min to the Max, i.e:

Mode = Min + (Max-Min)/3

Taking the Beta4(a1 ,a2,min, max) distribution again, this equates to a1 = 2, a2= 3. Thus, from Golenko-Ginzburg's viewpoint it is sufficient to use

Beta4(2, 3, min, max)

in place of

PERT(min, mode, max)

with the added advantage that one is only asking a subject matter expert for two values.

## The Modified PERT distribution

The PERT distribution, just like the Triangle, will produce just one shape from its three parameters. Thus, we are restricted to accepting this interpretation, or creating our own. The modified-PERT distribution is a quick alternative approach first proposed in Vose (2000) that modifies the weighting factor for the most likely value from 4 in a PERT to a user-defined value g:

The Modified PERT distribution is widely recognized as a very useful improvement over the Triangle and PERT and is offered by many simulation packages.

## Reference

Vose, David (2000). Risk Analysis - a Quantitative Guide.

## ModelRisk functions added to Microsoft Excel for the PERT distribution

VosePERT generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VosePERTObject constructs a distribution object for this distribution.

VosePERTProb returns the probability density or cumulative distribution function for this distribution.

VosePERTProb10 returns the log10 of the probability density or cumulative distribution function.

## PERT distribution equations

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