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STRUCTURAL RELIABILITY AND RISK ANALYSIS – 4 Year FILS

the variation due to numerous factors (in the material, workmanship) whose influence cannot
be predicted, so that the variation must be regarded as a random variation.

1.1 Data samples

In most cases the inspection of each item of the production is prohibitively expensive and
time-consuming. Hence instead of inspecting all the items just a few of them (a sample) are
inspected and from this inspection conclusions can be drawn about the totality (the
population).

If X = random variable (concrete compressive strength)
x = value for random variables

If one performs a statistical experiment one usually obtains a sequence of observations. A
typical example is shown in Table 1.1. These data were obtained by making standard tests for
concrete compressive strength. We thus have a sample consisting of 30 sample values, so that
the size of the sample is n=30.

1.2 Indicators of the sample

One may compute measures for certain properties of the sample, such as the average size of
the sample values, the spread of the sample values, etc.
_
The mean value of a sample x 1, x 2, …, x n or, briefly, sample mean, is denoted by x (or m x) and
is defined by the formula:
_ 1 n
x   x (1.1)
n  j 1 j
It is the sum of all the sample values divided by the size n of the sample. Obviously, it
_
measures the average size of the sample values, and sometimes the term average is used for x .

The variance (dispersion) of a sample x 1, x 2, …, x n or, briefly, sample variance, is denoted by
2
s x and is defined by the formula:
1 n _
s x 2  n 1  x ( j  x) 2 (1.2)
 j 1
The sample variance is the sum of the squares of the deviations of the sample values from the
_
mean x , divide by n-1. It measures the spread or dispersion of the sample values and is
always positive.

2
The square root of the sample variance s is called the standard deviation of the sample and is
2
denoted by s x. s  s . The mean, m x and the standard deviation, s x has the same units.
x
x
The coefficient of variation of a sample x 1, x 2, …,x n is denoted by COV and is defined as the
ratio of the standard deviation of the sample to the sample mean
s
COV  (dimensionless) (1.3)
x

UTCB, Technical University of Civil Engineering, Bucharest 7