In Beta Distribution, (O 4M P)/6 . Why is M multiply with 4 and not any other number?

I saw this explanation in a search:

"4/6ths approximates 66.7%, which approximates 68%, which approximates the area under a normal distribution within one standard deviation of the mean."

"The reason why you multiply by 4 to get the Mean (and not 5, 6, 7, etc.) is because the number 4 is tied to the shape of the underlying probability curve."

"The cone of uncertainty uses the factor 4 for the beginning phase of the project."

I can't take credit, my friend Google helped me to find these answers :-)

You will find a discussion of the origins of the Beta PERT distribution here http://broadleaf.com.au/resource-material/beta-pert-origins/ . I got so fed up with people asserting that the distribution has special meaning and should be used in preference to others that I sought out the original papers and summarised them. It is very useful but it is just one of many you might choose.

Below are extracts from the paper, numbered 1 and 2. The formula for the standard deviation is an approximation based on an analogy with the Normal distribution. The coefficients of the formula used to estimate the mean (1, 4 and 1) are based on an approximate fit to the solutions of a cubic equation. It was all a perfectly valid expedient measure to make it possible to carry out calculations with slide rules and log tables in the days before electronic calculators let alone inexpensive ubiquitous computing power.

David Vose, in his book https://www.wiley.com/en-au/Risk+Analysis%...p-9780470512845 discusses the implications of choosing one distribution over another, especially BetaPERT versus Triangular. You just need to to think what it represents and how that relates to what you believe about the value it represents, hopefully getting the two in accord with one another. There is no simple answer that will always be valid. You either adopt a distribution blindly or understand the differences and make an informed choice.

My paper has references to the originals.

1. The choice of the Beta distribution is explained by the mathematician on the original PERT team (Clark, 1962) in the following way.

“The author has no information concerning distributions of activity times, in particular, it is not suggested that the beta or any other distribution is appropriate. But the analysis requires some model for the distribution of activity times, the parameters of the distribution being the mode and the extremes. The distribution that first comes to the author’s mind is the beta distribution.”

2. Derivation of mean and variance

The team knew that, to help the engineers assess the uncertainty in their activity’s durations, they were starting with a three point estimate. They needed to convert those three points into a mean and variance and, for convenience, Clark selected a Beta distribution to do this.

Clark further explains that three values, minimum, mode and maximum, is not enough to completely define a Beta distribution and a further assumption is required to permit the three point estimate to be converted to a mean and variance. By analogy with a Normal distribution, in which almost all outcomes (99.73%) fall within plus or minus three standard deviations of the mean or a range of six standard deviations (six sigma), he assumed that the range between the minimum and the maximum values of a forecast represents six standard deviations of the duration’s distribution. This defined the variance, the square of the standard deviation, and fixed the Beta distribution’s shape parameters.

Even once the Beta distribution was defined by the three point estimate and the six sigma assumption it was still a challenge to derive the mean analytically as it required the solution of a cubic equation, but a simple approximation was found to be a good fit to the exact solution. Using this approximation and the six sigma assumption, what we know as the Beta PERT distribution was defined. It is specified by three values, the minimum A, the mode or most likely value M and maximum B and has a mean and variance calculated as follows. [IMAGE NOT REPLICATED HERE]

The equation for the mean is an approximation to an exact solution and the equation for the variance is based on an approximate analogy with the Normal distribution.

Thank You Stephen for the references and such a nice explanation. Though lot of people use whatever they like for calculating the mean. But I think there should be a reason for every specific method over another for calculations.

The beta distribution is a weighted average in which more weight is given to the most likely estimate. This alteration to the formula and placing more weight on the most likely estimate is made to increase the accuracy of the estimate by making it follow the Normal Distribution shape. Hence, in most of the cases, the Beta (PERT) distribution has been proven to be more accurate than the 3-Point triangular estimation.

So I thought to choose this over another with better understanding. Thank you for your valuable response.