## Statistical methods for ground characterization

'[Use of statistics] demands a high order of statistical technique, available from very few designers who have committed their time to training and experience in geotechnical engineering.'24

The characteristic value of a material property Xk is defined in Chapter 2 as follows:

Xk = mx + Ksx where mX is the mean of X, sX its standard deviation, and kn a statistical coefficient that depends on the number of samples n.

This definition may be stated more simply as:

Characteristic value = mean value + epistemic x aleatory uncertainties

Statistical methods for determining the epistemic and aleatory uncertainties of man-made materials are presented in Chapter 2. The following subsections discuss the way natural materials such as the ground depart from the assumptions made there (and embodied in EN 1990).

5.5.1 Normal or log-normal distribution?

A normal (a.k.a. Gaussian) distribution arises when a physical property depends on the combination of a large number of individual, random effects.25 The normal distribution is encountered frequently in nature and is one of the most important probability density functions in the field of statistics. Many man-made materials have properties that follow a normal distribution, e.g. the compressive strength of concrete and the yield strength of steel.

A log-normal distribution arises when a physical property depends on the product of a large number of individual, random effects. The log-normal distribution is especially useful in cases where the data cannot take on negative values. Since this applies to many geotechnical parameters (e.g. the angle of shearing resistance of soils 9, the undrained shear strength of fine soils cu , and other parameters with a coefficient of variation greater than about 30%), the log-normal distribution is of particular interest to geotechnical engineers.

Consider the test results shown in Figure 5.15, which were obtained from cone penetration tests in clay tills.26 If we assume that the results follow a normal distribution (shown by the dashed bell-shaped curve on Figure 5.15), then the mean cone resistance is 1.99 MPa and its standard deviation 0.73 MPa. Using the equations given in Chapter 2, the lower ('inferior') and upper ('superior') characteristic values of qc are then:

qk w ] f0.78MPa lck,inf I = mqc + knsqc = 1.99 +1.647 x 0.73 = \

qck ,sup I qc nqc 13.20 MPa

However, the test results more closely follow a log-normal distribution, shown by the skewed bell-shaped curve of Figure 5.15 and defined by the expression:

where X in this case is the cone resistance; P(X, A, Q is the probability density function for X; A is the mean value of ln(X); and Z the standard deviation of ln(X).

Cone resistance, qt (MPa) Figure 5.15. Results of cone penetration tests in clay tills

The mean AX and standard deviation ZX of the log-normal distribution are related to the corresponding values for the normal distribution, as follows:27

where, in this instance, ZX = 0.357 and AX = 0.624. The inferior and superior characteristic values are then given by:

The log-normal distribution gives an inferior characteristic value that is 32% higher than obtained with the normal distribution and a superior value some 5% higher.

Many geotechnical parameters more closely follow a log-normal distribution and statistics based on the (more complicated) equations given in this chapter are preferred to statistics based on the equations given in Chapter 2.

5.5.2 Calculating the 95% confident mean value

Chapter 2 presents the statistical basis for selecting the upper and lower characteristic values of a material property, based on the 5% fractiles of the normal distribution — i.e. values that have a 5% probability of being exceeded.

As discussed in Section 5.3.5, many geotechnical designs involve a large volume of ground for which the spatial average of the material property is of greater relevance than a 5% fractile. In statistical terms, what we must calculate are the 95% confidence limits to the 50% fractile.

For this situation, if the variance of the population is known from prior knowledge (and hence does not need to be determined from the sample), the statistical definitions of Xkinf and Xk are:

where ]iX is the mean value of X, oX the standard deviation of the population, and ÔX its coefficient of variation (COV) — terms defined in Chapter 2. Here, however, the statistical coefficient kn is given by:

qck ,inf qck ,sup

Aqc+kniqc = p0.624+1.647x0.357

1.04 MPa 3.36 MPa

0 0