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The 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 Triangular distribution, namely minimum (a), most likely (b) and maximum (c). The figure below shows three PERT distributions whose shape can be compared to the Triangular distributions here.



Experts may not be able to easily estimate the real min and max parameters of a quantity. For example, if we were to ask you for the min and max working hours you have ever done, you are unlikely to have a precise recollection of those hours. However, you may be able to give us an approximated "feasible" min and max based on your recollection of your historical working hours, by providing percentiles rather than absolute min and max values.  This alternative way to model expert opinion is further explained in this section.  

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


            PERT (a, b, c) = Beta(a1, a2) * (c - a) + a



\alpha_1 = \frac {(\mu-a)*(2b-a-c)}{(b-\mu)*(c-a)} \textrm{, the first shape parameter of the Beta distribution}

\alpha_2 = \frac {\alpha_1*(c-\mu)}{(\mu-a)}\textrm{, the second shape parameter of the Beta distribution}

\ mu = \frac {a+4*b+c}{6} \textrm{, the mean of the Pert distribution}


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 Triangular 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 Triangular 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 Triangular and PERT distributions that have the same ab and c values.


To illustrate the point, the figure uses values of zero and one for a and c respectively and allows b (the most likely value; the x-as in the above figure) to vary between zero and one, although the observed pattern extends to any {a,b,c} set of values. It can be seen that the PERT distribution produces a systematically lower standard deviation than the Triangular distribution, particularly where the distribution is highly skewed (i.e. b is close to zero or one in this case). 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 Triangular distribution. This implies that using PERT distributions throughout a cost or schedule model, or any other additive model, will display about 10% less uncertainty than the equivalent model using Triangular distributions.


Some readers would perhaps argue that the increased uncertainty that occurs with Triangular distributions will compensate to some degree for the over-confidence that can occur when subjectively estimating. The argument is not conducive to the long term improvement of the organization's ability to estimate. We would rather see an expert's opinion modeled as precisely as is practicable. Then, if the expert is consistently over-confident, this will become apparent with time and the estimating can be corrected.




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