Evaluating reliability in geotechnical engineering - Risk and Reliability in Geotechnical Engineering - page 133

3.3.4 Coefficient of variation
While the standard deviation is a useful indicator of the amount of scatter in a variable, the
degree of dispersion is easier to see in context if it is expressed i in terms of the COV, which
is the ratio of the standard deviation divided by the average (

σ/ x ).

COV = σ
x

(3.3)

where COV is the coefficient of variation.
The COV is a dimensionless measure of the amount of scatter, and is usually expressed
as a percentage. For the measured values of undrained shear strength listed in Table 3.1 , the
COV = (2.14/14.36) × 100% = 15%.

3.3.5 histograms and relative frequency diagrams
Histograms and relative frequency diagrams are graphical representations of a series of
measured values in the form of bar charts, as shown in Figure 3.2.
The range of measured values, from the lowest to the highest, is shown on the horizontal
axis. In order to display the data as vertical bars, the horizontal axis is divided into “bins.”
The height of each bar in the histogram indicates the number of values in the bin. The height
of each bar in the relative frequency diagram indicates the number of values in each bin
divided by the total number, 25 in this case.
Although the width of the bins is arbitrary, the bin width affects the appearance of the
histogram. The width of the bins used to plot Figure 3.2 i is 1 kPa. The numbers of values in
the bins, and the values of relative frequency, are shown in Table 3.1(c).

3.3.6 Probability and probability theory
The Random House Dictionary of the English Language defines probability as “the relative
possibility that an event will occur” and probability theory as “the theory of analyzing and
making statements concerning the probability of occurrence of uncertain events.”

3.3.7 Probability density function
Probability density functions (PDFs) are continuous distributions that indicate the probabil-
ity of occurrence of any value of the variable, within the range covered by the distribution.
Some PDFs extend from minus infinity to plus infinity; others cover finite ranges of values.
PDFs provide a means of making generalizations about distributions of variables, and can
be thought of as hypotheses regarding how the values would be found to be distributed if a
very large number of values were available.
The two most commonly assumed distributions are the normal distribution and the log-
normal distribution, which are shown in Figure 3.3. Both of these are theoretical distribu-
tions and can be calculated based solely on the values of the mean and standard deviation
of a variable. The distributions shown in Figure 3.3 were calculated for an average value of
s u = 14.36 kPa, and a standard deviation of 2.14 kPa, corresponding to the undrained shear
strength data in Table 3.1 and Figure 3.2.