I’ve explained the mathematical derivations in brief within my Project Management Spreadsheet workbook; click here to go directly to the explanation page.  You may copy the file and use it yourself; please respect the open-source license (you may not sell it, and the attribution must be kept intact).

While I’ve been happy with the 1996 edition, the latest edition of the Guide to the Project Management Body of Knowledge and pertinent professional training opportunities can be found here, at the Project Management Institute (PMI) website.  In-depth mathematical derivations can be found there.

My best effort to explain in intuitive, non-mathematical terms why three-point estimates work appears on pp. 214-215 of Growth through Governance, during the discussion of the concept of variance.

For those who wish a little more depth on these derivations than what appears in my book, but who don’t want to read mathematical equations, here goes.  A project consists of many subtasks, which all share a couple important core statistical characteristics, even though they may differ broadly in their particulars.  Every subtask has a shape describing the likelihood of results in time or cost, with a hump or point in the middle, where the likelihood is highest, and a lower likelihood out at the tail ends of the best and worst cases.  In between those outlying cases, the likelihood generally gets higher as you proceed toward the middle, and lower as you proceed toward the outlying edges.  These shapes are called probability distributions.

Of course, there will be tasks whose probability distributions don’t actually fit this basic shape: they might be bimodal (having two most-likely peaks).  An everyday example is how long it takes me to get to work on the bus.  If the bus comes every 20 minutes, and the bus ride is 10 minutes, and I’ve looked at the bus schedule so I can try to be at the bus stop 3 minutes before the bus, you’ll see at least two peaks in the probability distribution of how long it takes me to get to work: one peak at 13 minutes, and then another smaller peak at 33 minutes, in case I miss that first bus.  The probability distribution will reach a low at 23 minutes, between the peaks, corresponding to a time when the bus is not scheduled to come.  Regardless of this and other cases where the single-peak distribution doesn’t quite fit, the single-peak shape turns out to be a pretty good model, and deviations from it in practice tend to average out.

So, project managers have found through experience that it’s a best practice to simplify the probability distribution of subtasks down to a “most likely” estimate at that peak of likelihood, as well as a low-likelihood estimate for “best case” and “worst case.”  We then draw a model of the subtask’s probability distribution as a triangle between those three points, or as a bell curve with a hump in the middle and low-likelihood tail ends at the best- and worst-case estimates.  The exact shape of the distribution doesn’t matter a whole lot, certainly not for the purposes of the relatively small projects Growth through Governance is mainly intended for; again, little discrepancies tend to average out.  (By “small” projects, I mean that you’re more likely to be planning a community event than building 100 airplanes.)

Note that this method, while mathematical, is imprecise.  It’s imprecise in a mathematical way.  The underlying math gives us a pretty good sense of just how imprecise it is, and it’s a good sight less imprecise than traditional one-point estimates.  I’ve found three-point estimates to be a powerful tool, an admittedly heavy tool, but one that has consistently worked for me for over 15 years.