Structural reliability and risk analysis

1.1. Data samples

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.

Table 1.1. Sample of 30 values of the compressive strength of concrete, daN/cm2

320

380

340

350

340

350

370

390

370

320

350

360

380

360

350

420

400

350

360

330

360

360

370

350

370

400

360

340

360

390

The statistical relevance of the information contained in Table 1.1 can be revealed if one shall order the data in ascending order in Table 1.2 (320, 330 and so on). The number of occurring figures from Table 1.1 is listed in the second column of Table 1.2. It indicates how often the corresponding value x occurs in the sample and is called absolute frequency of that value x in the sample. Dividing it by the size n of the sample one obtains the relative frequency listed in the third column of Table 1.2.

If for a certain value x one sums all the absolute frequencies corresponding to the sample values which are smaller than or equal to that x, one obtains the cumulative frequency corresponding to that x. This yields the values listed in column 4 of Table 1.2. Division by the size n of the sample yields the cumulative relative frequency in column 5 of Table 1.2.

The graphical representation of the sample values is given by histograms of relative frequencies and/or of cumulative relative frequencies (Figure 1.1 and Figure 1.2).

If a certain numerical value does not occur in the sample, its frequency is 0. If all the n values of the sample are numerically equal, then this number has the frequency n and the relative frequency is 1. Since these are the two extreme possible cases, one has:

- the relative frequency is at least equal to 0 and at most equal to 1;

- the sum of all relative frequencies in a sample equals 1.

Structural Reliability and Risk Analysis - Lecture Notes 6

Table 1.2. Frequencies of values of random variable listed in Table 1.1

Compressive strength

Absolute frequency

Relative frequency

Cumulative frequency

Cumulative relative frequency

320

2

0.067

2

0.067

330

1

0.033

3

0.100

340

3

0.100

6

0.200

350

6

0.200

12

0.400

360

7

0.233

19

0.633

370

4

0.133

23

0.767

380

2

0.067

25

0.833

390

2

0.067

27

0.900

400

2

0.067

29

0.967

410

0

0.000

29

0.967

420

1

0.033

30

1.000

0.000.050.100.150.200.25320330340350360370380390400410420x, daN/cm2Relative frequency

Figure 1.1. Histogram of relative frequencies

Structural Reliability and Risk Analysis - Lecture Notes 7

0.00.10.20.30.40.50.60.70.80.91.0320330340350360370380390400410420x, daN/cm2Cumulative relative frequency

Figure 1.2. Histogram of cumulative relative frequencies

If a sample consists of too many numerically different sample values, the process of grouping may simplify the tabular and graphical representations, as follows (Kreyszig, 1979).

A sample being given, one chooses an interval I that contains all the sample values. One subdivides I into subintervals, which are called class intervals. The midpoints of these subintervals are called class midpoints. The sample values in each such subinterval are said to form a class. The number of sample values in each such subinterval is called the corresponding class frequency. Division by the sample size n gives the relative class frequency. This frequency is called the frequency function of the grouped sample, and the corresponding cumulative relative class frequency is called the distribution function of the grouped sample.

If one chooses few classes, the distribution of the grouped sample values becomes simpler but a lot of information is lost, because the original sample values no longer appear explicitly. When grouping the sample values the following rules should be obeyed (Kreyszig, 1979):

o all the class intervals should have the same length;

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