The topic of skewness of an uncertain variable is perhaps one of the most fundamental in risk assessment modelling. When it is believed that a (continuous) process is symmetric, the choice of distributions to use to represent that process is generally of less consequence than when the uncertainty is asymmetric. For example, a symmetric Triangular, PERT, and Normal distribution (with appropriately selected parameters e.g. so that the means and standard deviations for each are the same) will be broadly similar; of course there are some differences, but they are generally at the margin and of little significance in many practical risk analysis modelling situations for general business purposes (though such differences can still be important in cases where extremely accurate models are required).

Here, I briefly mention some sources of skewness that arise in real-life processes, or in the associated modelling of risk:

**Multiplicative processes.**A process in which random variables are multiplied will create a skewed output, tending to a Lognormal distribution when many such independent variables are multiplies, and often approximated by such a distribution in any case. Such process arise in cost budgeting (e.g. the total cost estimate as the product of an uncertain volume, unit cost, and perhaps a duration), in asset price forecasting (% changes to asset values over several periods work in a multiplicative sense), and in oil reservoir modelling (uncertain reserve estimation volume estimate being the product of uncertain spatial dimensions and some additional other factors, i.e. for exploration and production).**Compound processes with event risks.**When taking a pragmatic modelling approach in cost budgeting (e.g. using a Triangular distribution), one often simply assumes that the cost distribution of an item is asymmetric; that is we assume that (for unspecified reasons) the costs are more likely to be over the base estimate than below it. Often part of the underlying reason is the presence of event risks in the situation, where the occurrence of a specific event creates an additional (perhaps uncertain) set of costs in addition to a (perhaps uncertain and symmetric) base cost.**Parameter estimation for small sample sizes.**When estimating a probability from a set of observations (for example 5 occurrences of an event during 100 periods, or trials), one sometimes takes the “maximum likelihood” approach (i.e. assume 5%) or otherwise assumes that there is a distribution of possible probabilities (such as a Beta distribution). Either way, for small sample sizes, the distribution of the uncertainty of the true probability is not symmetric. Examples of this were given in the earlier blog about the difficulties in estimating the probability of low probability events.

Dr. Michael Rees

Director of Training and Consulting

I am particularly interested in this topic in relation to project cost and schedule forecasts. Have you come across any references to the emergence of skew in these circumstance?