Statistical Distribution Overview
If you are performing a Probabilistic Analysis with RocSlope2, the following statistical distributions are available for defining random variables:
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.
Distribution Selection
Normal Distributions are 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 RocSlope2 may have the following applications.
- Variables which can only have positive values (e.g. cohesion, shear strength), often have PDFs which are NOT well modelled 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 are best modelled 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 to all values between a minimum and maximum.
- A Fisher distribution is applicable for orientation data in RocSlope2. See the Joint Orientation Statistics topic for more information.
Standard Deviation
The Standard Deviation of a random variable is a measure of the variance or scatter of the parameter from the mean value. The larger the standard deviation, the wider the range of values that the random variable can assume (within the limits of the Minimum and Maximum values).
- 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.
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
For 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 an 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.