Project Economic Analysis
2.3 Economic Risk Assessment
2.3.1 Probability Theory
To enter into the theory of probability, there is some statistical informa- tion that is important and necessary to understand the probability theory and the probability distribution. To illustrate the statistical concepts, we will clarify them through a numerical analysis for test results for 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 repre- sented in the following equation:
X X X X
n
n
1 2
, (2.9)
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 cal- culate the arithmetic average using Equation 2.9, the arithmetic mean of these readings is 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 that the mean of the two groups are the same, one should choose another criteria by which to compare the results as we cannot accept the second group based on our judgment, which will not support us in court.
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:
S X X X X X X
n
( ) ( ) ( n )
1 .
2 1
2 2
(2.10) The standard deviation for the first group sample is
S ( ) ( ) ( )
310 300 300 300 290 300 . 3
2 2 2
Mixing one, S = 8.16 kg/cm2.
The standard deviation for the seconding group sample is S (400 300) (300 300) (200 300)
3
2 2 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 2.10, one can find that the ideal case is when S equals 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.
C O V S
. . X (2.11)
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, 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
So, when comparing between concrete from one site with a mean con- crete strength of 300 kg/cm2 and concrete from a second site with a mean concrete strength of 500 kg/cm2 and the two sites have the same standard deviation as the above example, the coefficients of variations 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.
To define a practical method of probability distribution, assume we have test results for 46 cube crushing strengths, as shown in Table 2.7, and we need to define the statistical parameters for these numbers.
The raw data is collected in groups and the number of samples with a value between the range for every group is called frequency. The frequency table from the raw data in Table 2.8 is tabulated in Table 2.9.
The data from Table 2.9 is presented graphically in Figure 2.5. To analyze the data, make a cumulative descending table, as shown in Table 2.8. This table is presented graphically in Figure 2.6. From Table 2.10, one can find that for this set of data, sample results of concrete strength can obtain the cumulative descending data. To understand this table, the probability of having a concrete strength that is less than 300kg/cm2 is nine percent.
Table 2.7 Net present value including inflation.
Year
(1) (2) (3) = (1) × (2) (4) (5) = (4) × (3) Net cash
flow
Inflation rate
Net cash flow after inflation
Discount rate
Net present value
0 51785 1.0 51785 1.0 51785
1 20000 0.96 19231 0.95 18182
2 20000 0.92 18491 0.89 16529
3 20000 0.89 17780 0.85 15026
4 20000 0.85 17096 0.80 13660
Sum (NPV) 11612
Table 2.8 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 2.9 Frequency table.
Frequency Average value Group ID
1 270 260–280 1
3 290 280–300 2
6 310 300–320 3
6 330 320–340 4
12 350 340–360 5
7 370 360–380 6
5 390 380–400 7
1 410 400–420 8
1 430 420–440 9
1 450 440–460 10
43 Total
0 5 10 15
270
Concrete strength, Kg/cm2
Frequency
290 310 330 350 370 390 410 430 450
Figure 2.5 Frequency curve for concrete compressive strength data.
0 20 40 60 80 100 120
260 280 300 320 340 360 380 400 420 440 460 Concrete strength, kg/cm2
Percentage less than concrete strength
Figure 2.6 Cumulative distribution curve for concrete strength.
Table 2.10 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
From the cumulative descending curve, one can find that 100 percent of the results of the samples have a strength less than 459 kg/cm2 at the same time. In the results of previous tests, we find that the samples have results less than or equal to 280 kg/cm2, which is about two percent of the number of tested samples.