## Zi mz

The line labelled 'best fit' on Figure 5.18 gives values of cu which have a 50% probability of being exceeded by the local value of cu at the same depth. This line is readily calculated by spreadsheet programs using linear regression.

The curve labelled 'lower characteristic' gives values of cu which have a 95% probability of being exceeded by the average value of cu over the depths considered. This is a 95% confident mean value and is the value most often needed in geotechnical designs which involve a large zone of the ground. The curvature of the lower characteristic line is a consequence of the chosen statistical method and signifies greater uncertainty at the ends of the data set.

The curve labelled 'upper characteristic' gives values of cu which have a 95% probability of not being exceeded by the average value of cu over the depths considered. This is needed when an upper value is critical to the design.

The line labelled '5% fractile' gives values of undrained strength cu which have a 95% probability of being exceeded by the local value of cu at the same depth. We will call this the 'local' characteristic value. It is needed when the ground is unable to redistribute stresses from highly to lightly loaded zones.

Finally, the line labelled 'mean less half standard error' gives values which are close to the 95% confident mean (lower characteristic) and are much easier to calculate, since the line is offset from the best-fit line and hence has the added benefit of being linear.

### 5.7.3 Shear box tests on Leighton Buzzard Sand

Example 5.3 considers the determination of the angle of shearing resistance of Leighton Buzzard Sand, based on the results of a small number of laboratory shear box tests. It also illustrates the combination of test results with previous knowledge, using Bayesian updating techniques.

Notes on Example 5.3

O The values of shearing resistance were obtained from a series of six laboratory shear box tests.

© The 'mean-deviation-from-the-mean' is an alternative measure of variation, which is simpler to calculate than the standard deviation and gives similar values. It can be used to give an 'alternative approximation' of based on half the mean-deviation-of-the-mean below the mean.

© The characteristic value here is the 95% confident mean value with variance unknown.

Example 5.3 Shear box tests on Leighton Buzzard Sand Determination of characteristic shearing resistance

Statistical analysis of laboratory measurements

### Data from laboratory tests

Consider the results of a series of shear box tests on Leighton Buzzard Sand, which measured angles of shearing resistance of: 48.7, 50.8, 50.9, 50.5, 48.3, and 46.3°. O Statistical analysis

The sum of the results of n = 6 tests is 9 = 295.5°

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