PERT APPROXIMATION FORMULAS (for mean and variance of activity durations)

The two statistics needed for the duration of each project activity are the mean time and the variance or standard deviation of activity duration. The mean value formula is a weighted average of the three given times where the weight on the minimum and maximum times is one and the weight on the modal time is 4, thus mean duration of activity j is given by
 

The variance formula is motivated by the fact that for the symmetric case, almost all of the probability distribution will be within 3 standard deviations of the mean, so that one-sixth of the range of the interval is a reasonable approximation for the standard deviation for the activity duration. Thus the variance is given by the equivalent formulas shown below:

It should be noted that these are empirical approximation formulas not derived from the beta distribution directly.  There is no theoretical argument showing that the relative weight of 4 on the modal time is better than a relative weight of 3 or 5, and the absence of the modal time in the variance formula runs counter to the properties of the beta distribution and seems to be based more on the symmetric normal distribution.  Presumably, some experimentation was done in the early days, and some empirical basis was found for these forms.  At this point, we merely accept the formulas as the "traditional" way of doing PERT, and note that the Monte Carlo simulation approach does not make use of these formulas, but rather works directly from the beta distributions assumed for the activity durations.