Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 15

Table 1.4 Sorted Y data and the ECDF
Sorted Y data

Rank k ECDF F n ( y ) from Equation 1.12 Φ − 1 [F n ( y )]

71.43 1 0.067 − 1.50

74.40 2 0.163 − 0.98

81.15 3 0.260

− 0.64

89.81 4 0.356

− 0.37

90.52 5 0.452 − 0.12

105.41 6 0.548 0.12

113.35 7 0.644 0.37

124.06 8 0.740 0.64

133.54 9 0.837 0.98

134.75 10 0.933 1.50

If the actual standard deviation σ is known, the standardized ( m − μ)/(σ/ n 0.5 ) is distributed
as the standard normal distribution. One can then establish the 95% confidence interval
of μ by

mx + ⋅ ≤≤+ σ n

05 .

µ m x ⋅ σ n

05 .

(1.20)

0 025 . 0 975 .

where x 0.025 = −1.96 and x 0.975 = 1.96 are, respectively, the 0.025- and 0.975-percentiles of
the standard normal distribution. The confidence interval is more informative than simply
reporting a point estimate, because statistical uncertainty is quantified.
However, it is usually the case that the actual standard deviation σ is unknown. In this
case, if Y is indeed normally distributed (which may not be true), the standardized ( m − μ)/
( s / n 0.5 ) is distributed as the Student's t-distribution with n -1 degrees of freedom (DOF). An
empirical example of this t-distribution with 9 DOFs is shown in Figure 1.8a . One can then
establish the 95% confidence interval of μ by

140

130

120

Φ -1 [F n ( y )]
= ( y -101.81)/25.34

110

100

90

80

7 -1.5

-1 -0.5 0

Φ -1 [F n ( y )]

0.5 1 1.5

Figure 1.9 Normality plot: y versus Φ − 1 [F n ( y )].