A second comparison index is proposed to account for the influence that the quality of the experimental results has on the reliability to be assessed to the "class A" predictions. The index is based on the measure of the half length "b" of the confidence interval centered at the predicted value "p" which contains the estimated true mean value " tex2html_wrap_inline124 " of the experiment with the probability tex2html_wrap_inline126 . In other words, tex2html_wrap_inline126 is the likelihood that the true value  tex2html_wrap_inline124 lies within the confidence interval of length 2b, centered at the predicted value p:

  equation14

This "b" index can only be evaluated when more than one experimental result is available for comparisons with "class A" predictions. The methodology is briefly described in the followings.

Before a series of N experiments are performed, their results tex2html_wrap_inline140 are random variables with expected values tex2html_wrap_inline142 and standard deviations tex2html_wrap_inline144 , tex2html_wrap_inline146 . The following assumptions are made:

  1. the experiments are performed under identical conditions;
  2. the experiment outcomes are mutually independent (random sampling), i.e. bias can be neglected;
  3. the results are normally distributed.
The validity of these assumptions is discussed in ref. [8] for laboratory soil tests, and for geotechnical centrifuge experiments in particular. The "random sampling" assumption is imposed by the lack of knowledge in quantifying the amount of correlation between experimental results and the bias in results given by centrifugal testing as compared to full-scale situations.

Under the above assumptions, the results tex2html_wrap_inline148 are identically distributed random variables, i.e. tex2html_wrap_inline150 , for all  tex2html_wrap_inline152 (assumption 1), the expected outcome equals the true value, tex2html_wrap_inline154 (assumption 2), and the average outcome tex2html_wrap_inline156 is a normally distributed random variable, even for small N (assumption 3). The mean and standard deviation of tex2html_wrap_inline160 are [12]: tex2html_wrap_inline162 and tex2html_wrap_inline164 , respectively. It follows that the random variable: tex2html_wrap_inline166 is standard normal. Since the real standard deviation of the test results is not known, it is replaced by its estimate - the sample standard deviation  tex2html_wrap_inline168 , evaluated after the experiments have been performed. The new random variable:

  equation35

can be shown to follow a Student's t distribution with N - 1 degrees of freedom [12]. Denoting by tex2html_wrap_inline174 the respective cumulative distribution, and replacing tex2html_wrap_inline124 in (5) using (6), one gets:

  equation45

with:

  equation51

After the experiments have been performed and the results tex2html_wrap_inline178 - realizations of tex2html_wrap_inline148  - are known at every time instant t, the random variable  tex2html_wrap_inline160 in (8) is replaced by its realization tex2html_wrap_inline186 , and, given a value for tex2html_wrap_inline126 , the equation (7) can be numerically solved for tex2html_wrap_inline190 .

The meaning of the size b of the tex2html_wrap_inline126 confidence interval is illustrated in the above figure. Before the experiments are performed, the random variable tex2html_wrap_inline160 has the probability distribution function tex2html_wrap_inline198 , shown in the figure. If the standard deviation of the test results tex2html_wrap_inline200 is known, tex2html_wrap_inline198 is derived from the normal distribution, and it comes from the Student's t distribution if tex2html_wrap_inline200 is approximated by the sample standard deviation tex2html_wrap_inline168 . After the experiments, tex2html_wrap_inline160 is replaced by its realization tex2html_wrap_inline212 , but the true value tex2html_wrap_inline124 remains unknown. Under the "random sampling" assumption, the uncertainty in locating the true value tex2html_wrap_inline124 relative to tex2html_wrap_inline212 can be expressed by the same probability distribution function. Finally, given the predicted value p, the tex2html_wrap_inline126 confidence interval has a size 2b, corresponding to an area which equals the likelihood  tex2html_wrap_inline126 (hatched area).

An overall measure of how close the predicted time history is to the experimental measurements is obtained by averaging the value  tex2html_wrap_inline228 , computed at every time instant t, over the analysis time interval tex2html_wrap_inline232 . Finally, to get a nondimensional index, the average value is normalized with respect to the initial effective vertical stress  tex2html_wrap_inline234 :

  equation71

The quantity tex2html_wrap_inline236 , evaluated for a certain tex2html_wrap_inline126 value, will be refered to as "size of tex2html_wrap_inline126 confidence interval".