Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 6

(a) 0.05 (b) 0.05

µ = 100, σ = 20
µ = 200, σ = 20

µ = 200, σ = 20
µ = 200, σ = 40

0.04 0.04

0.03 0.03

0.02 0.02

0.01 0.01

0 0 0 50

100 200 300 100 150 200 250 300 350

y y

Effect of µ Effect of σ

Figure 1.1 PDFs of normal random variables.

where f ( y ) can be evaluated using EXCEL function NORMDIST( y , μ, σ, 0) or MATLAB
function normpdf( y , μ, σ). Figure 1.1 shows some examples of the normal PDFs. It is clear
that the center of the PDF is equal to the mean value, μ. The spread of the PDF is controlled
by the standard deviation, σ. The larger the spread, the larger the value of σ. The normal dis-
tribution is a two-parameter probability model. It is widely used in practice, in part, because
the parameters (μ, σ) can be estimated easily and quite accurately from the sample sizes com-
monly encountered in geotechnical engineering practice, say between 30 and 100 data points.
The COV is defined as

COV = σ
µ

(1.2)

The COV is widely reported in the geotechnical engineering literature, because it is
dimensionless. However, one should not jump to the conclusion that the COV is a constant ,
because of the normalization in Equation 1.2 . It is possible that the standard deviation is a
constant, in which case, the COV must decrease with the mean value by definition as shown
in Figure 1.2 . It is important to develop a physical sense of COV based on the guidelines

100

- (sand)
- (clay)
- (?)

S.D. - = 5.0°
S.D. - = 1.5°

80

tan - (sand)
tan - (clay)

60

40

20

0 0

10 20

Mean - (degrees)

30 40 50

Figure 1.2 COV of the effective stress friction angle ( φ ) decreases with the mean value. (From Phoon,
K.K. and Kulhawy, F.H. 1999. Canadian Geotechnical Journal , 36(4), 612-624, reproduced with
permission of the Canadian Geotechnical Journal. )