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3.3.1 Probability Theory

To enable the best use of probability theory, some important basic principles of mathematical statistics are inserted in to the discus­sion at this point. We will clarify these statistical concepts using an analysis of test results derived from crushing samples of cylinder concrete to measure its strength.

The main statistical parameters will be the following:

• Arithmetic average

• Standard deviation

• Coefficient of variation

Arithmetic average is the average value of a set of results and is represented in the following equation:

Подпись: (3.9)– X,+X2 + … + X„

A ———————————

П

where n is the number of results, and X is read of each test result.

As a practical example of the statistical parameter, assume that we have two groups of concrete mixture from different ready mix suppli­ers. The first group has a concrete compressive strength after 28 days for three samples, which are 310 kg/cm2,300 kg/cm2, and 290 kg/cm2. When we calculate the arithmetic average using equation (3.9), the arithmetic mean of these readings are 300 kg/cm2.

The second group has a test result for cube compressive strength after 28 days under the same conditions for the first group. The test results are 400 kg/cm2,300 kg/cm2, and 200 kg/cm2. When calculating the arithmetic mean, we find that it is equal to 300 kg/cm2.

Because the two groups have the same value of the arithmetic mean, does that mean that the same mixing has the same quality? Will you accept the two mixing? We find that this is unacceptable by engineering standards, but when we consider the mean the two groups are the same, so one should choose another criterion by which to compare the results as we cannot accept the second group based on our judgment, which will not support us in court.

Подпись: s = Подпись: (X1-X)2 + (X1-X)2 + + (X„-X)2 n Подпись: (3.10)

Standard deviation is a statistical factor that reflects near or far the reading results. From the arithmetic mean end, it is represented in the following equation:

The standard deviation for the first group sample is

Подпись:(310 – 300)2 + (300 – 300)2 + (290 – 300)2

Mixing one, S = 8.16 kg/cm2.

The standard deviation for the second group sample is

Подпись:(400 – 300)2 + (300 – 300)2 + (200 – 300)2

Mixing two, S = 81.6 kg/cm2.

One can find that the standard deviation in the second group has a higher value than the first group. So the distribution of test data results is far away from the arithmetic mean rather than group one. From equation (3.10) one can find that the ideal case is when S = 0.

We note that the standard deviation has units, as seen in the previous example. Therefore, standard deviation can be used to compare between the two groups of data as in the previous example where the two groups give the value of 300 kg/cm2 after 28 days. On the other hand, in the case of the comparison between the two different mixes of concrete, for instance, there is a resistance of 300 kg/cm2 in one concrete and 500 kg/cm2 in the second. In that case, the standard deviation is of no value. Therefore, we resort to the coefficient of variation.

The coefficient of variation is the true measure of quality control, as it determines the proportion after the readings for the average arithmetic profile. This factor has no units and is, therefore, used to determine the degree of product quality.

Подпись: (3.11)C. O.V = =

X

As another example, assume there is a third concrete mix at another site to provide concrete strength after 28 days of 500 kg/cm2. When you take three samples, it gives the results of strength after 28 days as 510 kg/cm2 and 500 kg/cm2 and 490 kg/cm2. When cal­culating the arithmetic mean and standard deviation, we find the following results:

• Arithmetic mean = 500 kg/cm2

• Standard deviation = 8.16 kg/cm2

Comparing concrete from one site with a mean concrete strength of 300 kg/cm2 and concrete from a second site with a mean concrete strength of 500 kg/cm2, in which both sites have the same standard
deviation as the above example, the coefficients of variation are as follows:

• Coefficient of variation of the first site = 0.03

• Coefficient of variation of the second site = 0.02

We note that the second site has a coefficient of variation less than the first location. That is, the standard deviation to the arithmetic average is less at the second site than at the first site. This means that the second site mix concrete has a higher quality. Therefore, the coefficient of variation is the standard quality control of concrete, and the closer to zero, the better the quality control is likely to be.

To illustrate probability distribution with a practical example, let us we have test results for 46 cube crushing strengths as shown in Table (3.7), and we need to define the statistical parameters for these numbers.

The raw data is collected in groups; the number of samples with a value between the range for every group is called the frequency. The frequency table from the raw data in Table (3.8) is tabulated in Table (3.9).

The data from Table (3.9) is presented graphically in Figure (3.5). To analyze the data, make a cumulative descending table, as shown

Table 3.8 Row data

340

298

422

340

305

356

320

382

297

267

355

312

340

366

349

311

306

368

382

404

326

350

322

448

350

358

384

346

365

303

398

306

298

339

344

378

282

320

360

360

367

341

326

325

352

384

Table 3.9 Frequency table

ID

Group

Average Value

Frequency

1

260-280

270

1

2

280-300

290

3

3

300-320

310

6

4

320-340

330

6

5

340-360

350

12

6

360-380

370

7

7

380^00

390

5

8

400-420

410

1

9

420^40

430

1

10

440-460

450

1

Total

43

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Figure 3.5 Frequency curve for concrete compressive strength data.

in Table (3.8). This table is presented graphically in Figure (3.6). From Table (3.10) one can find that from this set of data, sample results of concrete strength, one can see the cumulative descend­ing data. For example, the probability of having a concrete strength that is less than 300kg/cm2 is 9%.

From the cumulative descending curve, one may conclude that at some given point in time, 100% of the results of the samples have

a strength less than 459 kg/cm2. In the results of previous tests, we find that the samples have results less than or equal to 280 kg/cm2, which is about 2% of the number of tested samples.

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Concrete strength, kg/cm2

Figure 3.6 Cumulative distribution curve for concrete strength.

Table 3.10 Descending cumulative table

Group No.

Test Value

Reading Value Less than the Upper Limit

The Percentage Less than the Upper Limit

10

460

43

100

10

440

42

98

9

420

41

95

8

400

40

93

7

380

35

81

6

360

28

65

5

340

16

37

4

320

10

23

3

300

4

9

2

280

1

2

1

260

0

0