I was recently asked by a client why cost uncertainty is lognormal. While the normal distribution is familiar to every statistics student, the lognormal is often less encountered in an academic setting. The normal distribution is widely used because many natural and social phenomena have been observed to fit this distribution. However, cost uncertainty is not one of these. A better choice is a related distribution, namely the lognormal distribution. The lognormal distribution derives its name from the fact that for this distribution, the logarithm of the data is normally distributed. There are two reasons why the lognormal is a better choice for modeling cost uncertainty – asymmetry and extreme cost risk.
The normal distribution is often referred to as the bell curve because of its symmetric shape. However, cost is not symmetric. There is a strict lower bound at zero – a project cannot be less than free. However, on the other side, there is no strict upper limit on cost. While we may know for example that the procurement cost of a single missile will not be $1 billion, there is an asymmetry, as there is more risk that cost will grow than opportunities to reduce cost. Once funded, a project will likely not decrease it cost, but it could potentially increase by hundreds and even by an order of magnitude. The lognormal is asymmetric and is positively skewed. As the lognormal captures the phenomenon that more can go wrong than right, it is a better choice for modeling cost risk. See the graph below for a notional lognormal distribution.
The normal distribution allows for a limited variation. Extreme events, such as cost growth in excess of 100%, are extremely unlikely. The lognormal has the capability of providing larger chances for extremes. One study of cost growth for 289 NASA and DoD projects found that one in six projects doubles or more in cost from the initial plan to the actual final product. Thus, the lognormal is a better choice for modeling significant cost growth that occurs on a regular basis.
There is plentiful empirical evidence that supports the use of the lognormal for modeling cost risk. The Joint Agency Cost Schedule Risk and Uncertainty Handbook, which was endorsed by the cost leadership of the Army, Navy, Air Force, Marines, Missile Defense Agency, and NASA, recommends the lognormal as a default. It suggests the use of the lognormal unless there is convincing evidence another choice is better.
I have found that the lognormal is the best choice for modeling cost risk for satellites and planetary spacecraft. In a study of 289 DoD and NASA projects, I found that cost growth was found to closely follow a lognormal distribution. Paul Garvey, who wrote a textbook on cost risk analysis, recommends the lognormal for modeling cost risk at the system level John Hollman, an expert on cost risk quantification in the process industries has stated “life is lognormal” and recommends the use of the lognormal in modeling cost risk
Keep in mind that cost risk analysis is not at all normal and use the lognormal instead.