N i N
where t495% is Student's tvalue and N is the size of the population. The key difference between this definition of kn and the one given in Chapter 2 is the term inside the square root: (1/N) here, but (1/N + 1) for the 5% fractile.
Alternatively, if the variance of the population is unknown ab initio (and hence must be determined from the sample), the statistical definitions of Xkinf and Xksup change to:
where mX is the mean value of X, sX the sample's standard deviation, VX its coefficient of variation — terms defined in Chapter 2. The statistical coefficient kn is given by:
where tn195% is Student's tvalue for (n  1) degrees of freedom at a confidence level of 95% and n is the sample size.
Numerical values of Kn are given in Figure 5.16 by the lower solid line, labelled 'variance known', and vary between 0.164 for n = 100 and 1.163 for n = 2. Values of kn are given by the upper solid line, labelled 'variance unknown', and vary between 0.166 for n = 100 and 4.464 for n = 2. For comparison, the dashed lines on Figure 5.16 show the equivalent values of kn (for estimating the 5% fractile), which were discussed in Chapter 2.
Returning to the cone penetration test results of Figure 5.15 (which have mean 1.99 MPa and standard deviation 0.73 MPa), the lower and upper characteristic values of qc — based on 95% confident mean values — are:
(assuming the data follows a normal distribution), whereas the 5% fractiles were 0.78 and 3.20 MPa.
An important feature of Figure 5.16 is the rapid divergence between the curves for n < 5. The 'variance unknown' curve rises rapidly as the number of samples decreases.
Number of samples, n Figure 5.16. Statistical coefficients for determining the 50% fractile with 95% confidence (coefficient for 5% fractile from Chapter 2)
It is common in many ground investigations to gather just a handful of measurements in each stratum and so the uncertainty in any deductions made from those measurements (on the basis of statistics) is inherently large. This uncertainty may be reduced if the variance of the parameter being measured is known from previous observation — witness the gap between the curves labelled 'variance known' and 'unknown' in Figure 5.16. However, it is rare that we have sufficient prior knowledge of the strata encountered in ground investigations for this to be useful to us.
5.5.3 Statistics for parameters that vary with depth
Many geotechnical parameters vary with confining pressure and hence show a distinct correlation with depth below the ground surface. In such cases, the statistical methods described above  which apply to a single variable  must be replaced by multivariate statistics.
The characteristic value of a material property X that varies with depth z is given by:
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