Statistical Distributions Overview

If you are performing a Probabilistic analysis with Swedge, the following statistical distributions are available for defining random variables:

• Normal

• Uniform

• Triangular

• Beta

• Exponential

• Lognormal

• Gamma

• Fisher

The type of statistical distribution, together with the distribution parameters (mean, standard deviation, minimum and maximum values), define a probability density function (PDF) for a random variable. A PDF describes the distribution of possible values that a random variable may assume, for a hypothetical, infinite set of observations of the variable.

In most cases, very limited data is available, on which to decide what statistical distribution and standard deviation to use. Therefore, the engineer must often rely on "best estimates", when defining the PDF for a random variable. Some suggestions are noted below.

Distribution Selection

A Normal Distribution is commonly used for statistical analysis in geotechnical engineering. When the true distribution of a variable is not known, a Normal Distribution is often assumed. By making a best estimate of the minimum and maximum values of the variable, a standard deviation can be estimated, as described in the Normal Distribution topic.

Other distributions in Swedge may have the following applications.

• Variables which can only have positive values (e.g. cohesion, shear strength), often have PDFs which are NOT well modeled by a Normal Distribution. Such variables may have non-symmetric distributions, with a peak in the distribution at low values, and a gradual tapering off at higher values. For such variables, it is often more appropriate to use a Lognormal or Gamma distribution, rather than a Normal distribution.

• Some variables may be best modeled with an Exponential distribution, for example: waviness angle, persistence, trace length, water level.

• A Uniform Distribution can be useful, if you wish to specify an equal probability of the variable, taking on any value between the minimum and maximum values. Note: if you wish to uniformly vary an individual parameter between two values, you can also use a Sensitivity Analysis in Swedge.

• A Fisher distribution is applicable for orientation data in Swedge. See the Probabilistic Joint Orientation topic for more information.

Standard Deviation

The Standard Deviation of a random variable is a measure of the variance or scatter of the variable about the mean value. The larger the Standard Deviation, then the wider the range of values which the random variable may assume (within the limits of the Minimum and Maximum values). NOTE:

• The Standard Deviation is applicable for Normal, Lognormal, Beta, Gamma and Fisher distributions.

• It is NOT APPLICABLE for Uniform, Triangular, or Exponential distributions. If you are using one of these distributions, then you will NOT be able to enter a Standard Deviation.

• For tips on estimating values of Standard Deviation, see the Normal Distribution topic.

Minimum / Maximum Values

For each random variable, you must define a Minimum and Maximum allowable value. NOTE: for the purposes of data input, the Minimum / Maximum values are specified as Relative quantities (i.e. as distances from the Mean), rather than as absolute values. This simplifies the data input and is much less prone to error.

During the analysis, the Relative Minimum and Maximum values are converted to the actual Minimum and Maximum values, when the statistical sampling is carried out for each random variable, as follows:

MINIMUM = MEAN – Relative MINIMUM

MAXIMUM = MEAN + Relative MAXIMUM

EXAMPLE: if the Mean Friction Angle = 35, and the Relative Minimum = Relative Maximum = 10, then the actual Minimum = 25 degrees, and the actual Maximum = 45 degrees.

For each random variable, you must always specify non-zero values for the Relative Minimum and the Relative Maximum. If BOTH the Relative Minimum and Relative Maximum are equal to zero, no statistical samples will be generated for that variable, and the value of the variable will always be equal to the Mean.

In most cases, if you are using a Normal distribution (or other distribution which is symmetric about the Mean), the Relative Minimum and Relative Maximum values will be equal. However, they do not necessarily have to be equal, if your distribution is not symmetric.

Suggested Reading

• An excellent introduction to probability theory in a geotechnical engineering context, can be found in Chapter 2 of Hoek et.al. (1995).

• Chapter 6 of Law and Kelton (1991) is a useful guide to the selection of input probability distributions.

• Evans et.al. (1993) provides a summary of over 30 different probability density functions, in a quick-reference format.