## Monday, May 24, 2010

### All things bell shaped

Chapter 14 from the GAO Cost Estimating Manual on "Cost Risk and Uncertainty" is a good read, easily understood, and very practical in its examples.  Here's one illustration that I particularly like.  When you look at it, it's understood in a moment that the repeated random throw of two dice generates a probability density function [PDF] that has a bell-shape curve that is tending towards a true Normal distribution.

Statisticians call this phenomenon the Central Limit Theorem: random occurrences over a large population tend to wash out the asymmetry and uniformness of individual events.  A more 'natural' distribution ensues.  The name for it is the Normal distribution, more commonly: the bell curve.

Here's what it looks like to a project manager.  Notice that regardless of the distribution of cost adopted by  work package managers for each individual work package, in the bigger picture at the summation of the WBS there will tend to be a bell-shaped variation in the WBS budget estimate.  In part, use of these ideas addresses the project manager's need to understand the parameters of variation in the project budget as evidenced by the esitmates of WBS.  This diagram is (again) from Chapter 14 of GAO's manual:

If the risk analyst generates these data from a simulation, like a Monte Carlo simulation, then the numeric statistics like variance and standard deviation are usually reported, along with the cumulative probability more commonly called the "S" curve.  In the diagram, on the right side, we see the cumulative curve plotted and labeled on the vertical axis as the confidence level.  With a little inspection, you will realize that the cumulative curve is just the summation of the probabilities of the bell curve that is adjacent on the left.

The GAO manual, and especially Chapter 14, has a lot more information that is well explained.  Give it a read.