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Figure 2. Flowchart for phase I-Initiation and planning. Illustration. This figure is a flowchart showing seven boxes representing steps in Phase I. Step 1 is "Identify need for the specification or specifications within the agency." This step includes three points:
1.1 Problems that need to be solved?
1.2 Innovative or progressive ideas?
1.3 Industry suggested need to improve?

Step 2 is "Define goal and expectations" and includes four subtasks:
2.1 Identify benefits to agency and industry.
2.2 What is expected of the final product?
2.3 Define criteria for success.

Step 3 is "Reach agency consensus." The subtasks in this step are:
3.1 Obtain/verify top management commitment and support.
3.2 Choose task force leaders and agency members.
3.3 Build consensus within task force.
3.4 Set target date for initial draft specification.

Step 4 is "Understand 'Best Practices" and includes two subtasks:
4.1 Review the literature.
4.2 Learn other States' experiences.

Step 5 is "Confirm interest and commitment." This step is
5.1 Reconfirming top management commitment and support.

Step 6 is "Establish industry contact" and includes two subtasks:
6.1 Present concepts to selected industry leaders, present potential benefits to industry and agency.
6.2 Select industry representatives for task force.

Step 7, "Hold first joint agency-industry task force meeting," involves eight subtasks:
7.1 Present concepts.
7.2 Build consensus among members.
7.3 Establish short-term and long-term goals.
7.4 Establish deadline for trial specification.
7.5 Establish schedule for the initiative.
7.6 Determine if outside expertise is needed.
7.7 Set frequency of task force meetings.
7.8 Make specific work assignments.

Finally, an arrow leaves step 7, indicating that the process can continue to Phase II: Specification Development.

Figure 3. Flowchart for phase II-Specifications development. Illustration. This figure is a five-page flowchart with boxes representing steps in the process. The first step is to "Select material or materials and/or construction specifications to develop." Step 2, "Procure outside assistance, if required (see Phase I, Item 7.6)," follows. After step 2 is completed, step 3, "Recognize need to identify current practices," can begin. After step 3 is finished, both step 4, "Search the literature (for specific material or materials selected)," and step 5, "Contact/interview other agencies and associations (for specific material or materials selected)," can begin simultaneously. Both of these steps must be completed before step 6, "Develop outline for the QA specification," can begin. The following examples of step 6 actions are presented: General Information, Definitions, Quality Assurance, Quality Control, Acceptance, Payment, and Conflict Resolution. Step 7 is to "Develop introductory information for the QA specification." The following examples of step 7 are presented: Responsibilities of agency and contractor, and Requirements for technician and laboratory qualification. Next, is step 8, "Begin to develop procedures." The flowchart concludes with arrows leaving step 8, indicating that both step 9, "Develop QC procedures and requirements," and step 19, "Develop acceptance procedures and requirements," can begin. The flowchart continues, beginning with step 9, "Develop QC procedures and requirements." After step 9 is completed, step 10, "Establish QC requirements," and step 11, "Determine quality characteristics to measure," can begin. Step 10 includes the following examples of QC requirements: QC plan, Qualified Technicians, Qualified Laboratories, Control Charts and Action Criteria. After step 11 is completed, step 12 indicates that steps 13 through 17 are to be completed for each QC quality characteristic. The steps, in order, are as follows. Step 13 is "Evaluate available data." Step 14 is to decide if there are sufficient valid data available. If the answer is no, then step 15, "Obtain data," is performed, and the process returns to step 13. If the answer is yes, then the process proceeds to step 16, "Determine sampling and testing procedures and test frequency." Step 17 asks whether or not the quality characteristic should be used for testing. If the answer is no, the quality characteristic is either eliminated or considered for acceptance testing. If the answer is yes, then the process proceeds to step 18, in which QC procedures and requirements are completed. This is not achieved until both steps 9 and 17 are completed. The flowchart concludes with an arrow leaving step 18, indicating that step 42, "Finalized initial draft specification," is the next step in the process.The flowchart continues, from step 8, "Begin to develop procedures," this time to step 19, "Develop acceptance procedures and requirements." From step 19, step 20 asks, "Will contractor tests be used in the acceptance decision?" If the answer is no, the process proceeds to step 23, which is described below. If the answer is yes, the process proceeds to step 21, "Use contractor to do acceptance testing with agency doing verification testing." After step 21 is completed, step 22, "Develop verification procedures," begins with the following examples of procedures given: F-test and T-test, allowable differences between tests and frequency of comparisons. After step 22 is completed, both step 42, "Finalized initial draft specification," and step 23, "Determine quality characteristics to measure," can begin. From step 23, step 24 indicates that steps 25 through 36 are to be completed for each acceptance quality characteristic. Step 25, "Evaluate available data," follows step 24. After step 25 is completed, step 26 is to decide if sufficient data are available. If the answer is no, then step 27, "Obtain data," is performed, and the process returns to step 25. If the answer is yes, then the process proceeds directly to step 28, "Analyze data for statistical parameters and distribution."After step 28 is completed, step 29 asks whether or not the quality characteristic is valid for acceptance. If the answer is no, the quality characteristic either is considered for QC testing or eliminated. If the answer is yes, the flowchart moves to step 30, which asks whether to use the characteristic for payment determination. If the answer to step 30 is yes, then the process proceeds to step 32, "Determine the quality measure to use." From step 32, the process proceeds to step 34, "Determine specification limits, decide on AQL and RQL." After step 34 is completed, the process proceeds to step 37, "Decide pay relationships" such as Performance-related pay, Incentive/Disincentive, Minimum pay provisions, Remove/replace provisions, and Retest provisions. The process then proceeds to step 38, "Determine sample size, lot size, sublot size." If the answer to step 30 is no, then the process proceeds to step 31, "Use as a Screening (pass/fail) test." The process then proceeds to step 33, "Determine the quality measure to use." The process then proceeds to step 35, "Determine specification limits, decide on AQL and RQL." After step 35 is completed, the process continues to step 36, "Determine acceptance/rejection procedures, including rework provisions."The process then proceeds to step 38, "Determine sample size, lot size, sublot size." Step 39, "Develop OC curve and evaluate risks," follows from step 38. Step 40 then asks "Are risks acceptable?" If the answer is no, then the process proceeds to step 41, "Modify specification limits, acceptance limits, pay schedule, sample size, and/or lot size." After this is done, the process returns to step 39, "Develop OC curve and evaluate risks," and step 40, "Are risks acceptable?" until the answer is yes. The process then proceeds to step 42, "Finalized initial draft specification." Step 42 also has arrows pointing to it, indicating that step 18, "QC procedures and requirements completed" and step 22, "Develop validation/verification procedures," also must be completed before step 42 can be completed.

Figure 4. Flowchart for phase III-Implementation. Illustration. This figure is a flowchart with boxes representing steps in the process. Step 1, "Simulate specification," is divided into five subtasks:
1.1. Use several projects under construction or recently completed.
1.2. Be sure random sampling was used.
1.3. Use same sampling and testing procedures as decided upon in Phase II.
1.4. Analyze simulated payment factor data.
1.5. Revise or fine-tune as necessary for a Draft Special Provision.

Step 2, "Begin/continue technician qualification training," is divided into two subtasks:
2.1. Include training in sampling and testing procedures.
2.2. Include training in basic statistical procedures.

Step 3, "Try specification on a limited number of pilot projects," is divided into three subtasks:
3.1. Let projects using Draft Special Provision (Item 1.5).
3.2. Apply only a percentage of the disincentives.
3.3. Bid prices may not reflect future project prices.

Step 4 is "Analyze pilot project results." After step 4 is completed, step 5 asks whether major revisions to the draft specification are needed. If the answer is yes, then the process proceeds to step 6, "Prepare new draft special provision." The process then returns to steps 3, 4, 5, and 6 until no major revisions are needed. The process then proceeds to step 7, "Phase in projects agency-wide," which has two subtasks:
7.1. Phase in until all major projects are included.
7.2 Phase in payment factors.
The flowchart concludes with step 8, "Ongoing monitoring of specification performance," which has six subtasks:
8.1. Look at quality levels achieved each year.
8.2. Look for administrative problems.
8.3. Consider contractor concerns.
8.4. Identify technology changes.
8.5. Tie results into the pavement management system.
8.6. Compare with established criteria for success.

Figure 6. Graphical presentation of survey results for the first ranking method. Chart. This figure is a bar chart presenting the results of the topic-ranking survey using the step-1 first-ranking method. The X-axis includes the 12 topic categories and the Y-axis is number of panel member votes. The topic category considered most important by those responding to the survey is number 10, which received 80 votes. The other topic categories, in order of selected importance, are number 7 (74 votes); number 1 (73 votes); number 8 (67 votes); number 6 (55 votes); number 2 (51 votes); number 4 (45votes); number 5 (41votes); number 3 (39 votes); number 11 (38 votes); number 9 (35votes); and number 12 (13 votes).

Figure 7. Graphical presentation of survey results for the second ranking method. Chart. This figure is a bar chart presenting the results of the topic-ranking survey using the step-2 second-ranking method. The X-axis includes the 12 topic categories and the Y-axis is number of panel member votes. The topic category considered most important by the panel members after the second ranking method is topic number 1, which received 138.5 votes. The other topic categories, in order of selected importance, are number 10 (133.5 votes); number 7 (127 votes); number 8 (105 votes); number 6 (84.5 votes); number 2 (83 votes); number 4 (66 votes); number 11 (54 votes); number 3 (53 votes); number 5 (52.5 votes); number 9 (41 votes); and number 12 (23 votes).

Figure 8. Plot of how the standard deviation of PWL estimates varies with the population PWL. Chart. This figure compares the standard deviation of the estimated percent within limits (PWL) with the population PWL for three sample sizes. The X-axis is actual PWL (0 to 100) and the Y-axis is standard deviation (0 to 30). The three sample sizes plotted on the chart are 3, 5, and 10. The graph indicates that variability in the PWL estimates decreases as the sample size increases. The chart also shows that the maximum standard deviation of the estimated PWL values occurs at an actual PWL value of 50, and decreases to a minimum as the actual PWL approaches either 100 or zero. The resulting semicircular curve is most pronounced for the smallest sample of 3 and becomes flatter as the sample size increases to 10.

Figure 13. Plot of average difference of simulated PWL values versus actual PWL values for samples sizes equal to 3, 5, and 10. Chart. This figure compares the average difference between the estimated and actual PWL values for three sample sizes of 3, 5, and 10. The X-axis is actual PWL (0 to 100) and the Y-axis is average difference (negative 0.6 to positive 0.4). The chart shows no obvious trends among the sample sizes, and wide variations above and below zero for the three sample sizes.

Figure 14a. Plots of the 95th percentile for the average estimated PWL minus the actual PWL at 50 versus the number of lots per project. Chart. The chart shows the results for 3 tests per lot, 5 tests per lot, and 10 tests per lot. The chart indicates that the smallest average difference is detected for the largest sample size, or the variability decreases as the number of tests on each lot increase. It also shows that the average difference is greatest for the lower numbers of lots per project, and that this difference decreases substantially as the number of lots increases.

Figure 14b. Plots of the 95th percentile for the average estimated PWL minus the actual PWL at 70 versus the number of lots per project. Chart. The chart is similar to the first chart for PWL value of 50 indicating that the smallest average difference is detected for the largest sample size, or the variability decreases as the number of test on each lot increase.

Figure 15. Plot of average difference of simulated AAD values versus actual AAD values for sample sizes equal to 3, 5, and 10. Chart. This figure compares the average difference between the estimated average absolute deviation (AAD) values and actual AAD values for three sample sizes. The X-axis is actual AAD (0.6 to 2.6) and the Y-axis is the average difference (negative 0.006 to positive 0.010). The chart shows no obvious trends among the sample sizes, and wide variations above and below zero for the three sample sizes.

Figure 16. Plot of how the standard deviation of AAD estimates varies with the population AAD value. Chart. This figure compares the actual AAD values versus the standard deviations for each of the three sample sizes. The X-axis is actual AAD values (0.60 to 2.6) and the Y-axis is the standard deviation (0 to 0.60). The chart indicates that the standard deviation decreases as the sample size increases. The chart also suggests that, for all sample sizes, there is a gradual increase in the standard deviation values as the actual AAD values increase.

Figure 17. Plot of average difference of simulated CI values versus actual CI values for sample sizes equal to 3, 5, and 10. Chart. This figure compares the average difference between the estimated conformal index (CI) values and actual CI values for three sample sizes. The X-axis is actual CI (1.0 to 3.0) and the Y-axis is the average difference (negative 0.10 to zero). The chart shows that the average difference is negative for all sample sizes and across all CI values, indicating that the average CI estimates are lower than the actual CI values. The chart also suggests that the average difference decreases with sample size. For the sample of 3, the average difference between estimated and actual CI values ranges between negative 0.10 and negative 0.05. The average difference for the sample size of 10 varies minimally between 0 and negative 0.03.

Figure 18. Plot of how the standard deviation of CI estimates varies with the population CI value. Chart. This figure compares the actual CI values versus the standard deviations for each of the three sample sizes. The X-axis is actual CI values (1.0 to 3.0) and the Y-axis is the standard deviation (0 to 0.60). The chart indicates that the standard deviation decreases as the sample size increases. The chart also suggests that, for all sample sizes, there is a gradual increase in the standard deviation values as the actual CI values increase.

Figure 19a. Plots of bias versus actual PWL for 10,000 simulated lots with 3 tests per lot and one-sided limits showing positive skewness. Chart. The X-axis is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). The chart reveals that the bias in the PWL estimates increases as the skewness coefficient increases. Maximum bias at a skewness coefficient of positive 3 is positive 4. The plots in the top chart change from a negative bias to a positive bias at an actual PWL value of about 75.

Figure 19b. Plots of bias versus actual PWL for 10,000 simulated lots with 3 tests per lot and one-sided limits showing negative skewness. Chart. The X-axis is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). The chart reveals that the bias in the PWL estimates increases as the skewness coefficient increases. Maximum bias at a skewness coefficient of negative 3 is negative 4. The plots in the bottom chart change from a negative bias to a positive bias at an actual PWL value of about 25.

Figure 20a. Plots of bias versus actual PWL for 10,000 simulated lots with 5 tests per lot and one-sided limits showing positive skewness. The X-axis on the chart is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). The chart reveals that the bias in the PWL estimates increases as the skewness coefficient increases. Maximum bias at a skewness coefficient of positive 3 and negative 3 is positive or negative 7, respectively. The plots in this chart change from a negative bias to a positive bias at an actual PWL value of about 80.

Figure 20b. Plots of bias versus actual PWL for 10,000 simulated lots with 5 tests per lot and one-sided limits showing negative skewness. The X-axis on the chart is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). The chart reveals that the bias in the PWL estimates increases as the skewness coefficient increases. The plots in this chart show a negative bias to a positive bias at an actual PWL value of about 20.

Figure 21a. Plots of bias versus actual PWL for 10,000 simulated lots with 10 tests per lot and one-sided limits with positive skewness. Chart. The X-axis is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). The chart reveals that the bias in the PWL estimates increases as the skewness coefficient increases. Maximum bias at a skewness coefficient of positive 3 and negative 3 is positive or negative 11, respectively. The plots in this chart change from a negative bias to a positive bias at an actual PWL value of about 75.

Figure 21b. Plots of bias versus actual PWL for 10,000 simulated lots with 10 tests per lot and one-sided limits with negative skewness. Chart. The X-axis is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). The chart reveals that the bias in the PWL estimates increases as the skewness coefficient increases. The plots in this chart change from a negative bias to a positive bias at an actual PWL value of about 25.

Figure 22a. Plot of bias versus actual PWL for 10,000 simulated lots with various tests per lot and one-sided limits with +1 skewness. Chart. This chart presents the plots using a skewness coefficient of positive 1. The X-axis is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). With a skewness coefficient of positive 1, the maximum bias is approximately positive or negative 5. The plots in this chart change from a negative bias to a positive bias at an actual PWL value of about 78, and from a positive bias to a negative bias at a PWL value of about 8.

Figure 22b. Plot of bias versus actual PWL for 10,000 simulated lots with various tests per lot and one-sided limits with +2 skewness. Chart. This chart presents the plots using a skewness coefficient of positive 2. The X-axis is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). For a skewness coefficient of positive 2, the maximum bias is positive or negative 8. The plots in this chart change from a negative bias to a positive bias at an actual PWL values of about 75 and 8.

Figure 22c. Plot of bias versus actual PWL for 10,000 simulated lots with various tests per lot and one-sided limits with +3 skewness. Chart. This chart presents the plots using a skewness coefficient of positive 3. The X-axis is actual PWL values (99 to 1) and the Y-axis is bias (negative 14 to positive 14). The chart shows that a skewness coefficient of positive 3 results in a maximum bias of about positive or negative 11. The change from negative bias to a positive bias occurs at a PWL value of about 72, and from positive to negative at a PWL of about 5.

Figure 28a. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD equals 30, skewness equals positive 1, and two-sided limits. Chart. The chart presents the plots using a skewness coefficient of positive 1and reveals that the bias in the PD estimates increases as the sample size increases. With a skewness coefficient of positive 1, the maximum bias shown on the top chart is positive 6, occurring at the 1-slash-7 split.

Figure 28b. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD equals 30, skewness equals positive 2, and two-sided limits. Chart. The chart reveals that the bias in the PD estimates increases as the sample size increases. With a skewness coefficient of positive 2, the maximum bias is positive 14, occurring at the 1-slash-7 split.

Figure 29a. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD equals 50, skewness equals positive 1, and two-sided limits. Chart. The chart presents the plots using a skewness coefficient of positive 1 and reveals that the bias in the PD estimates increases as the sample size increases. With a skewness coefficient of positive 1, the maximum bias shown on the top chart is about positive 8, occurring at the 1-slash-7 split.

Figure 29b. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD equals 50, skewness equals positive 2, and two-sided limits. Chart. The chart reveals that the bias in the PD estimates increases as the sample size increases. On this chart, for which the skewness coefficient is positive 2, the maximum bias is just above positive 15, also occurring at the 1-slash-7 split.

Figure 30. Comparison of a normal population with a population with a skewness coefficient of positive 1.0. Chart. This figure compares a normally distributed population with a population having a skewness coefficient of positive 1. The X-axis is the range of values (1.5 to 10.5) characterizing the population. This figure shows that the mean value of 5 is the same for the normal population and the skewed population, and that the median value is the same as the mean for the normal population. For the skewed population, the median value is 4.85, somewhat smaller than the mean.

Figure 31. Sample program output screen for a population with PD equals 10, skewness coefficient equals 0.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen for a normal population of 1,000, characterized by 10 PD, a sample size of 5, skewness coefficient of 0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each of the PD splits. All of the histograms are asymmetrical, bars to the right, and most of the bias values are distributed near zero. On each histogram, the average bias value and the standard error are also given. The standard error for a sample size of 5 is 0.11 for all PD splits. The average bias for the 8-slash-0 split is positive 0.07. For the 7-slash-1 split, the average bias is negative 0.02, and for the 6-slash-2 split, the average bias is negative 0.29. For the 5-slash-3 split, the average bias is negative 0.23. For the 4-slash-4 split, the average bias is negative 0.02, and for the 3-slash-5 split, the average bias is negative 0.36. The average bias for the 2-slash-6 split is positive 0.13. For the 1-slash-7 split, the average bias is negative 0.01, and for the 0-slash-8 split, the average bias is positive 0.11.

Figure 32a. Portions of program output screens for PD equals 10, skewness coefficient equals 0.00, and sample sizes equal to 3. Chart. For a sample size of 3, the standard error is 0.15. For the 8-slash-0 split, the average bias is negative 0.22; for the 7-slash-1 split, the average bias is positive 0.27; and for the 6-slash-2 split, the average bias is negative 0.14.

Figure 32b. Portions of program output screens for PD equals 10, skewness coefficient equals 0.00, and sample sizes equal to 5. Chart. The standard error for a sample size of 5 on this group of program output screens is 0.11. The average bias for the 8-slash-0 split is positive 0.07; for the 7-slash-1 split, the average bias is negative 0.02; and for the 6-slash-2 split, the average bias is negative 0.29.

Figure 32c. Portions of program output screens for PD equals 10, skewness coefficient equals 0.00, and sample sizes equal to 10. Chart. For a sample size of 10, the standard error is 0.08. The average bias for the 8-slash-0 split is positive 0.05; for the 7-slash-1 split, the average bias is positive 0.06; and for the 6-slash-2 split, the average bias is positive 0.11.

Figure 33. Sample program output screen for a population with pd equals 30, skewness coefficient equals 1.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen for a population of 1,000, characterized by a skewness coefficient of 1.0, a population PD of 30, a sample size of 5, and 10,000 replications. The program output screen shows nine histograms of individual bias values, one for each of the PD splits. On each histogram, the average bias value and the standard error are also given. All of the histograms are asymmetrical with most of the values distributed around zero. The average bias for the 8-slash-0 split is negative 1.96 and the standard error is 0.15. For the 7-slash-1 split, the average bias is negative 2.02 and the standard error is 0.15. For the 6-slash-2 split, the average bias is negative 1.31 and the standard error is 0.16. For the 5-slash-3 split, the average bias is negative 0.66 and the standard error is 0.17. The average bias for the 4-slash-4 split is positive 0.88 and the standard error is 0.18. For the 3-slash-5 split, the average bias is positive 1.97 and the standard error is 0.18. The average bias for the 2-slash-6 split is positive 3.19 and the standard error is 0.19. For the 1-slash-7 split, the average bias is positive 3.99 and the standard error is 0.20. The average bias for the 0-slash-8 split is positive 1.81 and the standard error is 0.19.

Figure 34a. Portions of program output screens for PD equals 30, skewness coefficient equals 1.00, and sample sizes equal to 3. Chart. For a sample size of 3 and an 8-slash-0 split, the average bias is negative 0.86 and the standard error is 0.23; for the 7-slash-1 split, the average bias is negative 0.61 and the standard error is 0.22; and for the 6-slash-2 split, the average bias is negative 0.68 and the standard error is 0.23.

Figure 34b. Portions of program output screens for PD equals 30, skewness coefficient equals 1.00, and sample sizes equal to 5. Chart. For a sample size of 5, the average bias for the 8-slash-0 split is negative 1.96 and the standard error is 0.15; for the 7-slash-1 split, the average bias is negative 2.02 and the standard error is 0.15; and for the 6-slash-2 split, the average bias is negative 1.31 and the standard error is 0.16.

Figure 34c. Portions of program output screens for PD equals 30, skewness coefficient equals 1.00, and sample sizes equal to 10. Chart. For a sample size of 10, the standard error is 0.10 for all splits. The average bias for the 8-slash-0 split is negative 2.40; for the 7-slash-1 split, the average bias is negative 2.97; and for the 6-slash-2 split, the average bias is negative 2.21.

Figure 36a. Charts. Comparison of the shapes and spread of estimated AAD values for populations centered on the target and with various skewness coefficients. Charts. Population number 1, which is not skewed, is normally distributed with a standard deviation of 1 and an average AAD value of 0.7977. Population number 2 is characterized by a skewness coefficient of 0.5, a standard deviation of 1.0, and an average AAD value of 0.7934. Population number 3 is characterized by a skewness coefficient of 1.0, a standard deviation of 1.0, and an average AAD value of 0.7776.

Figure 36b. Comparison of the shapes and spread of estimated AAD values for populations centered on the target and with various skewness coefficients. Charts. With a sample size of three, the mean is 0.8019, the standard deviation is 0.3502, the standard error is 0.0035, the minimum value is 0.0455, and the maximum value is 2.4312. In the next sample, the mean becomes 0.7982, the standard deviation is 0.3524, the standard error is 0.0035, the minimum value is 0.0419, and the maximum value is 3.0949. In the last sample in this trio, the mean becomes 0.7857, the standard deviation is 0.3659, the standard error is 0.0037, the minimum value is 0.0240, and the maximum value is 3.3132.

Figure 36c. Comparison of the shapes and spread of estimated AAD values for populations centered on the target and with various skewness coefficients. With a sample size of five, the mean decreases to 0.7981, the standard deviation is 0.2693, the standard error is 0.0027, the minimum value is 0.0478, and the maximum value is 2.1776. The next chart shows the mean decreases to 0.7941, the standard deviation is 0.2709, the standard error is 0.0027, the minimum value is 0.1175, and the maximum value is 2.2893. The last chart shows, the mean decreases to 0.7768, the standard deviation is 0.2811, the standard error is 0.0028, the minimum value is 0.1043, and the maximum value is 2.3642.

Figure 36d. Comparison of shapes and spread of estimated AAD values for populations centered on the target and with various skewness coefficients, sample size = 10. Charts. In the first chart, the mean is reduced to 0.7968, the standard deviation is 0.1899, the standard error is 0.0019, the minimum value is 0.1714, and the maximum value is 1.7266. In the second chart, the mean is reduced to 0.7912, the standard deviation is 0.1919, the standard error is 0.0019, the minimum value is 0.2243, and the maximum value is 1.7471. In the final chart, the mean is reduced to 0.7794, the standard deviation is 0.1975, the standard error is 0.0020, the minimum value is 0.2239, and the maximum value is 1.7835.

Figure 37. Comparison of the shapes and spread of estimated AAD values for normal populations centered on and offset from the target. Charts. This figure compares three normal populations, one population having a mean centered on the target value and two populations having the means offset by 1.0 and 2.0 from the target value. The three normal populations, with means of 0, 1.0, and 2.0, have normal bell-shaped distributions with standard deviations of 1.0. The AAD sampling distributions for the three populations are slightly skewed to the right. The histograms show that the actual population AAD value increases as the mean moves away from the target value. The standard deviation increases as the mean offset increases.

Figure 38. Example of populations that are very dissimilar in shape, but have approximately the same AAD. Charts. This figure presents samples of a simulation output screen comparing three populations with similar AAD values, but with very different distributions histograms. The three populations to be sampled are population number 1, which is a normal population with typical bell-shaped distribution, a standard deviation of 1.0, and an average AAD value of 0.7977; population number 2, which is characterized by a skewness coefficient of 1.5, a standard deviation of 1.06, and an average AAD value of 0.7996; and population number 3, which is characterized by a skewness coefficient of 3.0, a standard deviation of 1.19, and an average AAD value of 0.7994. Although the means of the sampling distributions are similar, the skewed distributions have a greater spread of AAD values, than the normal population, which ranges between 0.0825 and 1.9390. The sampling distribution range of population number 2, with a skewness coefficient of 1.5, is 0.0812 to 3.6974 and the range for population number 3, with a skewness coefficient of 3.0, is 0.1170 to 3.8438.

Figure 42. Sample output screen from the simulation program for bimodal distributions. Charts. This figure is a sample screen of the results from a computer simulation program developed for bimodal distributions. Three populations, two identical and one of the combined identical populations, are represented in this sample. On the left side of the screen, the characteristics of the populations are listed. Alongside these characteristics are the associated bell-shaped normal population curves, and on the right side of the screen is a summary of the results of the simulation and a histogram revealing the differences between the individual sample PD values and the true population PD value. The two identical populations, referred to as population number 1 and population number 2, are characterized by a size of 500, a mean of 10, a standard deviation of 1, and a percent defective value of 9.99. The combined population is characterized by a size of 1,000, a mean of 10, a standard deviation of 1, a percent defective value of 9.99, a lower specification limit of 8.72, and an upper specification limit of 999. In the center of the screen are the three bell-shaped curves associated with the populations described on the left. The curves for populations number 1 and 2 are identical and identify the lower specification limit of 8.72 by a vertical division on the left side of the curve. The third curve representing the combined populations shows the same distribution as the individual populations but is twice the height of the individual population curves, reflecting the doubled sample size. On the right of the screen, the simulation results indicate that the sampling distribution of PD estimates is characterized by a sample size of 5, 1,000replications, a mean of 10.16, a standard deviation of 11.46, a standard error of 0.36, a no-bias T value of 0.45, and a T value, where A divided by 2 equals 0.025, of 1.96. The summary of results lists the shift as 0, the population PD as 9.99, the average bias as 0.16, and the 0.95 confidence interval as plus or minus 0.71.

Figure 45. OC surface for the maximum allowable difference test method verification method (assuming the smaller sigma equals sigma subscript test). Chart. This figure is a three-dimensional plot of the operating characteristics (OC) surface to evaluate the performance of the maximum allowable difference method for test method verification. This figure illustrates a case where the minimum standard deviation between the contractor's population and the agency's population equals the test standard deviation. The X-axis is the mean difference in units of test standard deviation (0 to 3), the Y-axis is the probability of detecting a difference in percent (0 to 80), and the Z-axis is the standard deviation ratio (0 to 5). The combination of these three axes forms a surface that is nearly rectangular in shape and is bounded by the following four corners. The values defining the lower right corner of the surface are zero-percent probability of detecting a difference for a mean difference of zero and a standard deviation ratio of zero. The upper right corner of the surface is defined by a 60-percent probability of detecting a difference for a mean difference of 0 and a standard deviation of 5. The values defining the lower left corner of the surface are a 58-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 0. The upper left corner of the surface is defined by a 65-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 5.

Figure 46. OC surface for the maximum allowable difference test method verification method (assuming the smaller sigma equals 0.5 sigma subscript test). Chart. This figure is a three-dimensional plot of the OC surface to evaluate the performance of the maximum allowable difference method for test method verification. This figure illustrates a case where the minimum standard deviation between the contractor's population and the agency's population equals one-half of the test standard deviation. The X-axis is the mean difference in units of test standard deviation (0 to 3), the Y-axis is the probability of detecting a difference in percent (0 to 80), and the Z-axis is the standard deviation ratio (0 to 5). The combination of these three axes forms a surface that is nearly rectangular in shape and is bounded by the following four corners. The values defining the lower right corner of the surface are zero-percent probability of detecting a difference for a mean difference of zero and a standard deviation ratio of zero. The upper right corner of the surface is defined by a 25-percent probability of detecting a difference for a mean difference of zero and a standard deviation of 5. The values defining the lower left corner of the surface are a 62-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 0. The upper left corner of the surface is defined by a 55-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 5.

Figure 47. OC surface for the maximum allowable difference test method verification method (assuming the smaller sigma equals 2 sigma subscript test). Chart. This figure is a three-dimensional plot of the OC surface to evaluate the performance of the maximum allowable difference method for test method verification. This figure illustrates a case where the minimum standard deviation between the contractor's population and the agency's population equals twice the test standard deviation. The X-axis is the mean difference in units of test standard deviation (0 to 3), the Y-axis is the probability of detecting a difference in percent (0 to 80), and the Z-axis is the standard deviation ratio (0 to 5). The combination of these three axes forms a surface that is nearly rectangular in shape and is bounded by the following four corners. The values defining the lower right corner of the surface are an 18-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 0. The upper right corner of the surface is defined by a 79-percent probability of detecting a difference for a mean difference of zero and a standard deviation of 5. The values defining the lower left corner of the surface are a 55-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of zero. The upper left corner of the surface is defined by a 79-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 5.

Figure 48a. Example 1 of some of the cases considered in the average run length analysis for the maximum allowable difference method. Charts. The first chart shows a single bell-shaped curve characterized by contractor population means and standard deviations equal to agency population means and standard deviations. The difference between the agency's population mean and the contractor's population mean is represented by small italicized I and is equal to zero. The ratio of the contractor's population standard deviation to the agency's population standard deviation is represented as small italicized J and is equal to 1. The second chart presents two overlapping curves of the same shape, showing the contractor's distribution offset one standard deviation to the right. For this example, the contractor population mean is larger than the agency population mean, the standard deviations of the contractor and agency populations are equal, small italicized I is equal to 1, and small italicized J is equal to 1. The third chart illustrates two overlapping curves of the same shape, showing the contractor's curve offset to the left. This example is characterized by the contractor population mean smaller than the agency mean, the standard deviation of the contractor population equal to the agency population, small italicized I equals 2, and small italicized J equals 1.

Figure 48b. Example 2 of some of the cases considered in the average run length analysis for the maximum allowable difference method. The first chart in this example shows two curves of different shapes. The contractor population distribution is taller and narrower than the agency population distribution curve. The agency population mean and the contractor population mean are equal, the standard deviation of the contractor population is lower than the standard deviation of the agency population, small italicized I equals zero, and small italicized J equals 0.5. The second chart also illustrates two curves of different shapes. The agency population curve is shorter and wider than the agency population curve. This chart is characterized by equal agency and contractor population means, contractor standard deviation larger than the agency population standard deviation, small italicized I equal to zero, and small italicized J equal to 2.

Figure 48c. Example 3 of some of the cases considered in the average run length analysis for the maximum allowable difference method. The first chart illustrates two bell-shaped curves, different in shape and overlapping. The contractor population curve is taller and narrower than the agency population curve, and is offset to the right. This chart is characterized by a contractor population mean larger than the agency population mean, the contractor population standard deviation less than the agency population standard deviation, small italicized I equal to 2, and small italicized J equal to 0.5. The second chart shows two curves different in shape and overlapping. The contractor population curve is lower and wider than the agency population curve. For this chart, the means of the agency and contractor populations are not equal, the standard deviation of the contractor population is larger than the standard deviation for the agency population, small italicized I is equal to 1, and small italicized J is equal to 2.

Figure 49. OC curves for a two-sided T-test, alpha equals 0.05. Chart. This figure is a plot of the OC curves for a two-sided T-test at an alpha value of 0.05. The X-axis is the standardized difference from 0 to 3 and the Y-axis is the probability of not detecting a difference in beta values from 0 to 1.0. The chart includes 13 curves, one for each of the following N values, where N is the sample size: 2, 3, 4, 5, 7, 10, 15, 20, 30, 40, 50, 75, and 100. All of the curves begin at 0.95 probability of not detecting a difference for a standardized difference of zero. For each sample-size curve, the probability of not detecting a difference decreases as the standardized difference increases. The slope of the curve for a sample size of 2 is the least steep and is at the top of the chart. The slope of the curve for a sample size of 100 is the steepest and is at the bottom of the chart. For a sample size of 2, the probability of not detecting a difference is about 0.73 for a standardized difference of 3. For a sample size of 100, the probability of not detecting a difference is 0 at a standardized difference of 0.5.

Figure 50. OC curves for a two-sided T-test, alpha equals 0.01. Chart. This figure is a plot of the OC curves for a two-sided T-test at an alpha value of 0.01. The X-axis is the standardized difference from 0 to 3.2 and the Y-axis is the probability of not detecting a difference in beta values from 0 to 1.0. The chart includes 12 curves, one for each of the following N values, where N is the sample size: 3, 4, 5, 7, 10, 15, 20, 30, 40, 50, 75, and 100. All of the curves begin at 0.99 probability of not detecting a difference for a standardized difference of zero. For each sample-size curve, the probability of not detecting a difference decreases as the standardized difference increases. The slope of the curve for a sample size of 3 is the least steep and is at the top of the chart. The slope of the curve for a sample size of 100 is the steepest and is at the bottom of the chart. For a sample size of 3, the probability of not detecting a difference is about 0.72 for a standardized difference of 3.2. For a sample size of 100, the probability of not detecting a difference is 0 at a standardized difference of 0.5.

Figure 51a. OC surfaces (also called i) for the appendix G method for 5 contractor tests compared to a single agency test. Chart. For this first example of five contractor tests and one agency test, the nearly rectangular surface formed by the three axes is bounded by four corners. The values defining the lower right corner of the surface are zero-percent probability of detecting a difference for a mean difference of zero and a standard deviation ratio of 0.5. The upper right corner of the surface is defined by a 42-percent probability of detecting a difference for a mean difference of 3 and a standard deviation of 0.5. The values defining the lower left corner of the surface are a 25-percent probability of detecting a difference for a mean difference of zero and a standard deviation ratio of 3. The upper left corner of the surface is defined by a 42-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 3.

Figure 51b. OC surfaces (also called power surfaces) for the appendix G method for 6 contractor tests compared to a single agency test. Chart. For this example of six contractor tests and one agency test, the values defining the lower right corner of the surface are zero-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 0.5. The upper right corner of the surface is defined by a 40-percent probability of detecting a difference for a mean difference of 3 and a standard deviation of 0.5. The values defining the lower left corner of the surface are a 30-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 3. The upper left corner of the surface is defined by a 50-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 3.

Figure 51c. OC surfaces (also called power surfaces) for the appendix G method for 7 contractor tests compared to a single agency test. Chart. For this example of seven contractor tests and one agency test, the values defining the lower right corner of the surface are zero-percent probability of detecting a difference for a mean difference of zero and a standard deviation ratio of 0.5. The upper right corner of the surface is defined by a 45-percent probability of detecting a difference for a mean difference of 3 and a standard deviation of 0.5. The values defining the lower left corner of the surface are a 35-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 3. The upper left corner of the surface is defined by a 52-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 3.

Figure 51d. OC surfaces (also called power surfaces) for the appendix G method for 8 contractor tests compared to a single agency test. Chart. For this example of eight contractor tests and one agency test, the values defining the lower right corner of the surface are zero-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 0.5. The upper right corner of the surface is defined by a 50-percent probability of detecting a difference for a mean difference of 3 and a standard deviation of 0.5. The values defining the lower left corner of the surface are a 35-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 3. The upper left corner of the surface is defined by a 55-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 3.

Figure 51e. OC surfaces (also called power surfaces) for the appendix G method for 9 contractor tests compared to a single agency test. Chart. For the example of nine contractor tests and one agency test, the values defining the lower right corner of the surface are zero-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 0.5. The upper right corner of the surface is defined by a 58-percent probability of detecting a difference for a mean difference of 3 and a standard deviation of 0.5. The values defining the lower left corner of the surface are a 38-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 3. The upper left corner of the surface is defined by a 58-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 3.

Figure 51f. OC surfaces (also called power surfaces) for the appendix G method for 10 contractor tests compared to a single agency test. Chart. For the last example of 10 contractor tests and 1 agency test, the values defining the lower right corner of the surface are zero-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 0.5. The upper right corner of the surface is defined by a 59-percent probability of detecting a difference for a mean difference of 3 and a standard deviation of 0.5. The values defining the lower left corner of the surface are a 38-percent probability of detecting a difference for a mean difference of 0 and a standard deviation ratio of 3. The upper left corner of the surface is defined by a 57-percent probability of detecting a difference for a mean difference of 3 and a standard deviation ratio of 3. Overall, a comparison of the charts shows that the power of the Appendix-G method is low. Even if the number of contractor tests is as high as 10, the probability of detecting a difference is still less than 60 percent.

Figure 52. OC curves for the two-sided F-test for level of significance alpha equals 0.05. Chart. This figure is a plot of the OC curves for a two-sided F-test at an alpha value of 0.05. The X-axis is lambda from 0 to 4.0 and the Y-axis is the probability of accepting the null hypothesis, H subscript 0, from 0 to 1.0. The chart includes 13 curves, one for each of the following N values, where N subscript X equals N subscript Y values: 3, 4, 5, 6, 7, 8, 9, 10, 16, 21, 31, 51, and 101. The curve for N equals 3 is at the top of the chart and the curve for N equals 101 is at the bottom. At a lambda value of 1.0, all of the curves pass through a value of 0.95 probability of accepting the null hypothesis. To the left of this point, each of the curves decreases steeply to reach a zero probability of accepting the null hypothesis. The value of lambda ranges from 0 for the N-equals-3 curve to approximately 0.62 for the N-equals-101 curve. To the right of this point, each of the curves decreases more gradually to reach a probability of accepting the null hypothesis values between 0 and 0.69, and lambda values between approximately 1.55 and 4.00. The probability of accepting the null hypothesis for the N-equals-3 curve is about 0.69 at a lambda value of 4.0. The probability of accepting the null hypothesis for the N-equals-4 curve is about 0.50 at a lambda value of 4.0. The probability of accepting the null hypothesis for the N-equals-101 curve is 0 at a lambda value of about 1.55.

Figure 53. OC curves for the two-sided F-test for level of significance alpha equals 0.01. Chart. This figure is a plot of the OC curves for a two-sided F-test at an alpha value of 0.01. The X-axis is lambda from 0 to 4.0 and the Y-axis is the probability of accepting the null hypothesis, H subscript 0, from 0 to 1.0. The chart includes 13 curves, one for each of the following N values, where N subscript X equals N subscript Y values: 3, 4, 5, 6, 7, 8, 9, 10, 16, 21, 31, 51, and 101. The curve for N equals 3 is at the top of the chart and the curve for N equals 101 is at the bottom. At a lambda value of 1.0, all of the curves pass through a value of 0.99 probability of accepting the null hypothesis. To the left of this point, each of the curves decreases steeply to reach a zero probability of accepting the null hypothesis. The value of lambda ranges from zero for the N-equals-3 curve to 0.60 for the N-equals-101 curve. To the right of this point, each of the curves decreases more gradually to reach a probability of accepting the null hypothesis values between 0 and 0.92, and lambda values between approximately 1.6 and 4.0. The probability of accepting the null hypothesis for the N-equals-3 curve is about 0.92 at a lambda value of 4.0. The probability of accepting the null hypothesis for the N-equals-4 curve is about 0.81 at a lambda value of 4.0. The probability of accepting the null hypothesis for the N-equals-101 curve is zero at a lambda value of about 1.6.

Figure 57. Illustration of measuring AAD in standard deviation units, Z subscript Targ, from the mean. Chart. This figure illustrates the relationship between standard deviation units, when used to measure AAD, and the distance that the population mean is offset from the target value. Shown is a normal bell-shaped curve. The population mean, mu, is identified by a vertical line through the center of the bell curve. The target value, capital T, is shown a distance of capital Z subscript TARG, from the mean. The units of standard deviation are shown as sigma. The relationship is described in the equation: capital Z subscript capital TARG equals mu minus capital T divided by sigma.

Figure 58. Example illustrating the PWL specification limits and the offset in sigma units between the population mean and the target. Chart. This figure shows a normal bell-shaped curve identifying the PWL upper and lower specification limits, the target value, the mean, and the distance between the population mean and the target value. The population mean, mu, is identified by a vertical line through the center of the bell curve and the standard deviation, sigma, is identified as 1.0. The target value, capital T, is shown to the left 0.7403 units from the mean. The lower specification limit is identified at the far left of the curve, at a value equal to the target value minus 1.645 standard deviation units. The upper specification limit is identified to the right of the mean at a value equal to the target value plus 1.645 standard deviation units.

Figure 59. EP curves for matched PWL and AAD payment equations for sample size equal to 5. Chart. This figure presents the results of a computer simulation comparing the performance of a PWL payment plan with an AAD payment plan. The X-axis is the mean offset from the target value (0 to 3.0) and the Y-axis is the expected payment in percent (50 to 100). Expected payment curves for PWL and AAD are similar, except for the mean offset range between 1.5 and 2.5, where the expected payment is slightly higher for the AAD plan. For both plans, the expected payment is 100 percent at a mean value equal to the target value. With the mean offset by 1.5 from the target value, the expected payment is approximately 84 percent. With the mean offset by 3.0, the expected payment for both PWL and AAD drops to about 60 percent.

Figure 60. Standard deviations for individual payment factors for matched PWL and AAD payment equations for samples size equal to 5. Chart. This figure presents the results of a computer simulation comparing the performance of a PWL payment plan with an AAD payment plan. The X-axis is the mean offset from the target value (0 to 3.0) and the Y-axis is the payment standard deviation in percent (0 to 10). Standard deviations are shown for both the PWL and AAD plans. Overall, the payment standard deviations are slightly higher for the PWL plan than for the AAD plan. The PWL standard deviations increase to a peak of about 9.5 percent at a mean offset of about 1.6 and then decline to about 5.5 percent at an offset of 3.0. The AAD standard deviations increase to a peak of about 8.4 percent at a mean offset of about 1.4 and then decline to about 6.8 percent.

Figure 61a. Illustration 1 of two normal variables with various values for correlation coefficient. Charts. The first example illustrates no correlation between the two variables. The two histograms show normally distributed populations, where the minimum and maximum values of the variable X are negative 3.48 and positive 4.17, respectively, and the minimum and maximum values for the variable Y are negative 3.82 and 3.93, respectively. Because there is no correlation, the scatter plot reveals no pattern for the simulated data. The actual correlation of the simulation is negative 0.012.

Figure 61b. Illustration 2 of two normal variables with various values for correlation coefficient. Charts. The second example is for a desired correlation coefficient of positive 1.0. The two histograms show normally distributed populations, where the minimum and maximum values of both variable X and variable Y are negative 3.53 and positive 3.64, respectively. The scatter plot reveals a straight line increasing at a 45-degree angle through the intersection point of the X and Y axes. The actual correlation is positive 1.0.

Figure 61c. Illustration 3 of two normal variables with various values for correlation coefficient. Charts. The third example illustrates a desired correlation of positive 0.25. The two histograms show normally distributed populations, where the minimum and maximum values of the variable X are negative 3.52 and positive 3.75, respectively, and the minimum and maximum values for the variable Y are negative 3.32 and positive 3.74, respectively. The scatter plot reveals a very slight positive correlation. The actual correlation after the simulation is positive 0.246.

Figure 61d. Illustration 4 of two normal variables with various values for correlation coefficient. Charts. The fourth example illustrates a desired correlation of negative 0.25. The two histograms show normally distributed populations, where the minimum and maximum values of the variable X are negative 3.3 and positive 4.32, respectively, and the minimum and maximum values for the variable Y are negative 3.51 and positive 3.77, respectively. The scatter plot reveals a very slight negative correlation. The actual correlation after the simulation is negative 0.244.

Figure 61e. Illustration 5 of two normal variables with various values for correlation coefficient. Charts. The fifth example illustrates a desired correlation of positive 0.5. The two histograms show normally distributed populations, where the minimum and maximum values of the variable X are negative 4.47 and positive 3.44, respectively, and the minimum and maximum values for the variable Y are negative 4.0 and positive 3.40, respectively. The scatter plot reveals a positive correlation. The actual correlation after the simulation is positive 0.503.

Figure 61f. Illustration 6 of two normal variables with various values for correlation coefficient. Charts. This example is for a desired correlation of negative 0.5. The two histograms show normally distributed populations, where the minimum and maximum values of the variable X are negative 3.32 and positive 3.74, respectively, and the minimum and maximum values for the variable Y are negative 3.08 and positive 3.32, respectively. The scatter plot reveals a negative correlation. The actual correlation after the simulation is negative 0.493.

Figure 61g. Illustration 7 of two normal variables with various values for correlation coefficient. Charts. The example illustrates a desired correlation of positive 0.75. The two histograms show normally distributed populations, where the minimum and maximum values of the variable X are negative 3.39 and positive 3.72, respectively, and the minimum and maximum values for the variable Y are negative 3.56 and positive 3.70, respectively. The scatter plot reveals a fairly well defined positive correlation. The actual correlation after the simulation is positive 0.757.

Figure 61h. Illustration 8 of two normal variables with various values for correlation coefficient. Charts. The last example illustrates a desired correlation of negative 0.75. The two histograms show normally distributed populations, where the minimum and maximum values of the variable X are X.X and X.X, respectively, and the minimum and maximum values for the variable Y are Y.Y and Y.Y, respectively. The scatter plot reveals a fairly well defined negative correlation. The actual correlation after the simulation is negative X.X.

Figure 62. Expected combined weighted average payment factors for various weights and correlation coefficients based on PWL. Chart. This figure compares the expected payment value with various weighting values for each of the two variables for three different correlation coefficients, based on the PWL. The X-axis is the weighting value of the variables number one and number 2 in determining the pay factor. The scale is 1.0, 0 to 0, 1.0. The Y-axis is the expected payment in percent from 89.0 to 91.0. Three correlation coefficients are plotted: 0, positive 0.5, and negative 0.5. Text on the chart presents the following values: P subscript small I equals 55 plus 0.5 times PWL; PF equals the weighted average of P subscript small I values; small N equals 5; PWL for both variables equals 70. Data on the chart do not reveal any obvious trends, and suggests that neither the correlation coefficients nor the variable weightings affect the expected payment factors. The expected payment values are mostly around 90 percent.

Figure 63. Standard deviations of weighted average payment factors for various weights and correlation coefficients based on PWL. Chart. This figure compares the standard deviations with various weighting values for each of the two variables for three different correlation coefficients, based on the PWL. The X-axis is the weighting value of the variables number one and number 2 in determining the pay factor. The scale is 1.0, 0 to 0, 1.0. The Y-axis is the standard deviation in percent from 4 to 9. Three correlation coefficients are plotted: 0, positive 0.5, and negative 0.5. Text on the chart presents the following values: small N equals 5; PWL for both variables equals 70; P subscript small I equals 55 plus 0.5 times PWL; PF equals the weighted average of P subscript small I values. Data on the chart indicate that both the correlation coefficients and the variable weightings influence the variability. The chart shows that, at both ends of the weighting scale (the 1.0, 0 weighting and at the 0, 1.0 weighting), the percent standard deviation for each of the three different correlation coefficients is essentially similar at about 8.75 percent. The chart also indicates that a set of equal weights of 0.5 results in the lowest variability for each of the three correlation coefficients. The lowest value of 5 percent is associated with the negative 0.5 correlation coefficient. The lowest value for the positive 0.5 correlation coefficient is 7.5 percent. The low value for the 0 correlation coefficient is about 6.1 percent.

Figure 64. Expected combined weighted average payment factors for various weights and correlation coefficients based on AAD. Chart. This figure compares the expected payment value with various weighting values for each of the two variables for three different correlation coefficients, based on the AAD. The X-axis is the weighting value of the variables number one and number 2 in determining the pay factor. The scale is 1.0, 0 to 0, 1.0. The Y-axis is the expected payment in percent from 92.8 to 94.8. Three correlation coefficients are plotted: 0, positive 0.5, and negative 0.5. Text on the chart presents the following values: P subscript small I equals the remainder of 105 minus 24.75 times the remainder of AAD minus 0.798; PF equals the weighted average of P subscript small I values; small N equals 5; AAD for both variables equals 1.238. Data on the chart do not reveal any obvious trends, and suggests that neither the correlation coefficients nor the variable weightings affect the expected payment factors. The expected payment values are mostly near 93.7 percent.

Figure 65. Standard deviations of weighted average payment factors for various weights and correlation coefficients based on AAD. Chart. This figure compares the standard deviations with various weighting values for each of the two variables for three different correlation coefficients, based on the AAD. The X-axis is the weighting value of the variables number one and number 2 in determining the pay factor. The scale is 1.0, 0 to 0, 1.0. The Y-axis is the standard deviation in percent from 4 to 9. Three correlation coefficients are plotted: 0, positive 0.5, and negative 0.5. Text on the chart presents the following values: small N equals 5; AAD for both variables equals 1.238; P subscript small I equals the remainder of 105 minus 24.75 times the remainder of AAD minus 0.798; PF equals the weighted average of P subscript small I values. Data on the chart indicate that both the correlation coefficients and the variable weightings influence the variability. The chart shows that, at both ends of the weighting scale (the 1.0, 0 weighting and at the 0, 1.0 weighting), the percent standard deviation for each of the three different correlation coefficients is essentially similar at about 8.5 percent. The chart also indicates that a set of equal weights of 0.5 results in the lowest variability for each of the three correlation coefficients. The lowest value of 5 percent is associated with the negative 0.5 correlation coefficient. The lowest value for the positive 0.5 correlation coefficient is about 7.6 percent. The low value for the 0 correlation coefficient is about 6.1 percent.

Figure 66. Expected average payment factors for two populations with various correlation coefficients based on PWL. Chart. This figure compares the expected average payment factor with the correlation coefficients for two populations, based on PWL. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 75 to 105. Three PWL values, 90, 70, and 50, are plotted. Text on the chart presents the following values: P subscript small I equals 55 plus 0.5 times PWL; PF equals the average of P subscript small I values; small N equals 5; PWL values for the populations are equal. The chart shows that the expected payment value is not affected by the correlation between the individual variables. The plots for the three values of PWL are straight horizontal lines across the correlation coefficient scale. For a PWL value of 90, the expected payment factor is 100 percent; for PWL of 70, the expected payment factor is 90 percent; and for PWL equal to 50, the expected payment factor is 80 percent.

Figure 67. Expected average payment factors for two populations with various correlation coefficients based on AAD. Chart. This figure compares the expected average payment factor with various correlation coefficients for two populations, based on AAD. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 75 to 105. Three AAD values, 0.798, 1.238, and 1.688, are plotted. Text on the chart presents the following values: P subscript small I equals the remainder of 105 minus 24.75 times the remainder of AAD minus 0.798; PF equals the average of P subscript small I values; small N equals 5; PWL values for both populations are equal. The chart shows that the expected payment value is not affected by the correlation between the individual variables. The plots for the three values of AAD are straight horizontal lines across the correlation coefficient scale. For an AAD value of 0.798, the expected payment factor is about 102 percent; for an AAD value of 1.238, the expected payment factor is about 94 percent; and for an AAD value of 1.688, the expected payment factor is about 83 percent.

Figure 68. Standard deviations of individual payment factors for two populations with various correlation coefficients based on PWL. Chart. This figure compares the effect of the correlation between two variables on the variability of individual payment factors for two populations, based on PWL. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation of the expected payment in percent from 0 to 10. Three values of PWL, 90, 70, and 50, are plotted. Text on the chart presents the following values: P subscript small I equals 55 plus 0.5 times PWL; PF equals the average of P subscript small I values; small N equals 5; PWL for both populations are equal. Data on the chart suggest that the variability of the payment estimates is related to the correlation between the two variables. For a PWL of 50 and 70, the variability about the average payment increases as the correlation between the two variables increases from negative 1.0 to positive 1.0. For the PWL value of 90, the variability about the average payment decreases slightly as the correlation coefficient approaches 0.

Figure 69. Standard deviations of individual payment factors for two populations with various correlation coefficients based on AAD. Chart. This figure compares the effect of the correlation between two variables on the variability of individual payment factors for two populations, based on AAD. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation of the expected payment in percent from 0 to 10. Three values of AAD, 0.798, 1.238, and 1.688 are plotted. Text on the chart presents the following values: P subscript small I equals the remainder of 105 minus 24.75 times the remainder of AAD minus 0.798; PF equals the average of P subscript small I values; small N equals 5; PWL values for both populations are equal. Data on the chart suggest that the variability of the payment estimates is related to the correlation between the two variables. For an AAD of 1.238 and 1.688, the variability about the average payment increases as the correlation between the two variables increases from negative 1.0 to positive 1.0. For the AAD value of 0.798, the variability about the average payment decreases slightly as the correlation coefficient approaches zero.

Figure 70. Bias for the average payment for two populations with various actual PWL values. Chart. This figure illustrates the bias in the expected average payment for two populations with different PWL quality values, and for three correlation coefficients (negative 0.5, 0, and positive 0.5). The X-axis is the PWL of the two populations, ranging from an increment of 90 to 80 to an increment of 60 to 50. The Y-axis is the payment bias in percent, from negative 1.0 to positive 1.0. Text on the chart presents the following values: P subscript small I equals 55 plus 0.5 times PWL; PF equals the average of P subscript small I values; and small N equals 5. Data on the chart indicate that the payment biases using PWL are small and are generally centered at the correct value, regardless of the quality level of the two populations. For all correlation coefficients plotted, the payment bias is generally between negative 0.2 and positive 0.2.

Figure 71. Bias for the average payment for two populations with various actual AAD values. Chart. This figure illustrates the bias in the expected average payment for two populations with different AAD quality values, and for three correlation coefficients (negative 0.5, 0, and positive 0.5). The X-axis is the AAD of the two populations, ranging from a quality increment of 0.798 to 1.019 to an increment of 1.461 to 1.688. The Y-axis is the payment bias in percent, from negative 2.0 to positive 2.0. Text on the chart presents the following values: P subscript small I equals the remainder of 105 minus 24.75 times the remainder of AAD minus 0.798; PF equals the average of P subscript small I values; small N equals 5. Data on the chart indicate that the payment biases using AAD are larger than for PWL, are negative for higher quality levels, and are positive for lower quality levels. For all correlation coefficients plotted, the payment bias increases from approximately negative 1.2 at the highest quality increment to approximately positive 1.2 at the lowest quality increment.

Figure 72. Standard deviation for the individual average payment values for two populations with various actual PWL values. Chart. This figure compares the effect of the standard deviation for the individual average payment values on two populations with various PWL values. The X-axis is the PWL of the two populations, ranging from a quality increment of 90 to 80 to an increment of 60 to 50. The Y-axis is the standard deviation of the individual expected payment value in percent from 3 to 9. Three correlation coefficients, negative 0.5, 0, and positive 0.5, are plotted. Text on the chart presents the following values: P subscript small I equals 55 plus 0.5 times PWL; PF equals the average of P subscript small I values; and small N equals 5. Data on the chart suggest that the standard deviation values are lower for higher quality PWL increments. From a PWL increment of about 90 to 50, the quality level decreases, the standard deviation values become larger, and differences in the correlation coefficients can be detected. The highest standard deviation value of about 8.5 percent is associated with the correlation coefficient of positive 0.5 at the lowest PWL quality increment of 60 to 50. The lowest standard deviation value at the same quality increment is associated with a correlation coefficient of negative 0.5.

Figure 73. Standard deviation for the individual average payment values for two populations with various actual AAD values. Chart. This figure compares the effect of the standard deviation for the individual average payment values on two populations with various AAD quality values. The X-axis is the AAD of the two populations, ranging from a quality increment of 0.798 to 1.019 to an increment of 1.461 to 1.688. The Y-axis is the standard deviation of the individual expected payment value in percent from 3 to 9. Three correlation coefficients, negative 0.5, 0, and positive 0.5, are plotted. Text on the chart presents the following values: P subscript small I equals the remainder of 105 minus 24.75 times the remainder of AAD minus 0.798; PF equals the average of P subscript small I values; and small N equals 5. Data on the chart suggest that the standard deviation values are lower for higher quality AAD increments. From an AAD increment of about 1.019 to 1.238, the quality level decreases, the standard deviation values become larger, and differences in the correlation coefficients can be detected. The highest standard deviation value of about 8.5 percent is associated with the correlation coefficient of positive 0.5 at the lowest AAD quality increment of 1.461 to 1.688. The lowest standard deviation value at the same AAD quality increment is associated with a correlation coefficient of negative 0.5.

Figure 74. OC curve for an acceptance plan that calls for rejection if the estimated PWL is less than 60, for sample size equal to 4. Chart. This figure illustrates the results of a computer simulation of an acceptance plan showing the relationship between the actual PWL quality value and the probability of its acceptance. The X-axis is the actual PWL values from 0 to 100. The Y-axis is the probability of acceptance from 0 to 100 percent. A horizontal line at 97.5 percent probability of acceptance is the 2.5 percent alpha risk. The vertical line at a PWL of 90 is included for reference because it is used to determine the alpha risk. The OC curve begins at zero probability of acceptance for a PWL value near 15 and increases nearly proportionately to a probability of acceptance near 1.0 at a PWL value near 95.

Figure 75. OC curves for the probabilities of receiving at least some payment and at least 100-percent payment, for sample size equal to 4. Chart. This figure illustrates the results of a computer simulation of an acceptance plan showing the probability of receiving at least 100 percent payment and the probability of acceptance at any price. The X-axis is the actual PWL values from 0 to 100. The Y-axis is the probability of acceptance from 0 to 1.0. The horizontal line at 97.5 percent probability of acceptance is the 2.5 percent alpha risk. The horizontal line at 61 percent probability is the 39 percent risk of contractor receiving less than 100 percent pay for AQL material. The OC curve for receiving at least 100 percent payment begins at 0 probability of acceptance for a PWL value near 35 and increases nearly proportionately to a 100-percent probability of acceptance at a PWL value near 100. The OC curve for receiving a payment greater than zero is the same as shown in the previous figure 74, and begins at a zero probability of acceptance for a PWL value near 15 and increases to a 100-percent probability of acceptance at a PWL value near 95. The data indicate that the probability of receiving a payment of 100 percent or more is approximately 60 percent.

Figure 76. OC curves for the probability of receiving various payments, sample size equal to 4. Chart. This figure illustrates the results of a computer simulation of an acceptance plan showing the probability of receiving payments greater than zero, greater than 100 percent, and greater than 104.5 percent. The X-axis is the actual PWL values from 0 to 100. The Y-axis is the probability of acceptance from 0 to 1.0. A horizontal line near 99 percent probability of acceptance is the alpha risk. Horizontal lines at about 50 and 60 percent probability are included for reference back to the OC curves. The OC curve for receiving payment greater than 0 begins at zero probability of acceptance for an actual PWL value near 15 and increases nearly proportionately to a 100-percent probability of acceptance at an actual PWL value near 95. The OC curve for receiving a payment greater than 100 percent begins at a 0 probability of acceptance for an actual PWL value near 35 and increases nearly proportionately to a 100-percent probability of acceptance at an actual PWL value near 100. The OC curve for receiving a payment equal to or greater than 104.5 percent begins at a 0 probability of acceptance for an actual PWL value near 40 and increases to a 100-percent probability of acceptance at an actual PWL value near 100. The data indicate that the probability of receiving a payment of more than 0 is about 98 percent; the probability of receiving a payment of 100 percent of more is about 60 percent; and the probability of receiving a payment of 104.5 percent or more is less than 50 percent.

Figure 77. EP curve for the payment relationship pay equals 55 plus 0.5PWL, with an RQL provision, sample size equal to 4. Chart. This figure illustrates the results of a computer simulation of the effects of the actual PWL values on the expected payment. The X-axis is the actual PWL from 0 to 100 and the Y-axis is the percent expected payment from 0 to 110. A vertical line extends upwards from an actual PWL value of 90 to intersect the curve. At this point of intersection, a horizontal line extends across to the Y-axis at a value just under 100 percent expected payment. The curve begins at an expected payment of zero percent for an actual PWL value of around 20 and increases to just under 100 percent at an actual PWL value of 90.

Figure 78a. Distribution of estimated PWL values for an AQL population. Chart. The X-axis is divided into increments of 10 from 0 to 100. The height of the bars on the histogram represents the relative frequency of PWL estimates for each sample represented on the horizontal scale. The frequency is highest at a value of about 99, indicated by a single tall bar. The remaining values are distributed fairly evenly between 50 and 100. The low end of the range is about 39.

Figure 78b. Distribution of payment factors for an AQL population. Chart. This histogram is for the payment factors corresponding to the PWL values in the first histogram. The X-axis is divided into increments of 10 from 0 to 110. The height of the bars on the histogram represents the relative frequency of the payment factors. The frequency is highest at a value of about 105, indicated by a single tall bar. Although a short bar occurs near 0, most of the values are distributed between 85 and 105.

Figure 79. EP curve for the payment relationship pay equals 55 plus 0.5PWL, sample size equal to 4. Chart. This figure illustrates the results of a computer simulation of the effects of the actual PWL values on the expected payment. The X-axis is the actual PWL from 0 to 100 and the Y-axis is the percent expected payment from 0 to 110. A vertical line, representing the AQL, extends upwards from an actual PWL value of 90 to intersect the curve. At this point of intersection, a horizontal line extends across to the Y-axis at a value of 100 percent expected payment. The curve is a straight line that begins at an actual PWL value of 0 and an expected payment of 55 percent, and ends at an actual PWL value of 100 at an expected payment of about 105. The chart shows that the expected payment at the AQL is 100 percent.

Figure 80. Distributions of sample PWL estimates for a population with 90 PWL. Charts. This figure consists of two histograms showing the distribution of estimated PWL values with one-sided and two-sided specification limits for a population with an actual PWL of 90 and a sample size of 4. The first histogram is for one-sided specifications. The X-axis on this histogram is divided into increments of five, from 5 to 100. A vertical line, marked 90 PWL, occurs between the 90 and 95 values on the X-axis. The height of the bars on the histogram represents the relative frequency of sample PWL values for each sample represented on the horizontal scale. The frequency is highest at a value of about 100, indicated by a single tall bar marked 526. Occurrences at other values of PWL are: 75 at a PWL of 95, 85 at a value of 90, 84 at a value of 85, 73 at a value of 80, 67 at a value of 75, 47 at a value of 70, 25 at a value of 65, 9 at a value of 60, 6 at a value of 55, and one each at values of 50, 45, and 35. The second histogram is for two-sided specifications. The X-axis on this histogram is divided into increments of five, from 5 to 100. A vertical line, marked 90 PWL, occurs between the 90 and 95 values on the X-axis. The height of the bars on the histogram represents the relative frequency of estimated PWL values for each sample represented on the horizontal scale. The frequency is highest at a value of about 100, indicated by a single tall bar marked 537. Occurrences at other values of estimated PWL are: 74 at a PWL of 95, 80 at a value of 90, 62 at a value of 85, 74 at a value of 80, 67 at a value of 75, 48 at a value of 70, 30 at a value of 65, 15 at a value of 60, 8 at a value of 55, 3 at a value of 50, and 2 at a value of 45. These two histograms suggest that the distribution is skewed to the right for both the one-sided and two-sided specifications.

Figure 81a. Distributions of sample PWL estimates for a population with 50 PWL and one-sided speculations. Chart. The X-axis on this histogram is divided into increments of five, from 5 to 100. The height of the bars on the histogram represents the relative frequency of sample PWL values for each sample represented on the horizontal scale. The distribution of the bars is somewhat symmetrical and approximates a bell shape. The frequency is highest near the middle at PWL values of 50 and 60, indicated by a two tall bars marked 114 and 115, respectively. Occurrences at other values of PWL are: 103 at a PWL of 55, 92 at a value of 45, 79 at a value of 65, 73 at a value of 35, 66 at a value of 40, 57 at a value of 70, 43 at a value of 75, 40 at a value of 30, 32 at a value of 80, 29 at a value of 25, 23 at a value of 20, 22 at a value of 85, 17 at a value of 90, 15 at a value of 15, 13 at a value of 95, 7 at a value of 10, 30 at a value of 100, and 30 at a value of 5.

Figure 81b. Distributions of sample PWL estimates for a population with 50 PWL and two-sided speculations. Chart. The X-axis on this histogram is divided into increments of five, from 5 to 100. The height of the bars on the histogram represents the relative frequency of estimated PWL values for each sample represented on the horizontal scale. Unlike the histogram above with one-sided specifications, the distribution of the bars on this histogram is skewed to the right. The frequency is highest near the middle of the estimated PWL scale, at a value of 45, indicated by a tall bar marked 121. Occurrences at other values of estimated PWL are: 117 at a PWL of 35, 113 at a value of 50, 101 at a value of 40, 97 at a value of 55, 89 at a value of 60, 85 at a value of 30, 66 at a value of 65, 37 at a value of 70, 29 at values of 75 and 80, 34 at a value of 85, 24 at a value of 25, 11 at a value of 90, 7 at a value of 95, 32 at a value of 100, 4 at a value of 20, 3 at a value of 15, and 1 at a value of 10.

Figure 83. EP contours for the values in table 53. Chart. This figure presents expected payment contours for the EP values presented in table 53. The EP values are represented as the actual PWL values for two variables. The X-axis of the chart is the PWL for variable number 1 and is scaled from 0 to 100 in increments of 10. The Y-axis is the PWL for variable number 2 and is scaled from 0 to 100 in increments of 10. Seven contours are plotted, each in a direction from upper left to lower right on the chart. The contour for an EP value of 40 is the line that connects between the point defined by Y equals 23 and X equals 10, and the point defined by Y equals 10 and X equals 23. The contour for 50 is the line that connects between the point defined by Y equals 58 and X equals 10, and the point defined by Y equals 10 and X equals 55. The contour for 60 is the line that connects between the point defined by Y equals 90 and X equals 10, and the point defined by Y equals 10 and X equals 90. The contour for 70 is the line that connects between the point defined by Y equals 95 and X equals 28, and the point defined by Y equals 28 and X equals 95. The contour for 80 is the line that connects between the point defined by Y equals 95 and X equals 48, and the point defined by Y equals 48 and X equals 95. The contour for 90 is the line that connects between the point defined by Y equals 95 and X equals 65, and the point defined by Y equals 65 and X equals 95. The contour for 100 is the line that connects between the point defined by Y equals 95 and X equals 85, and the point defined by Y equals 85 and X equals 95.

Figure 84. EP surface for the values in table 53. Chart. This figure is a three-dimensional graph that presents an expected payment surface for the values presented in table 53. The X-axis is the PWL for variable number 1 and is scaled from 0 to 100 in increments of 20. The Y-axis is the PWL for variable number 2 and is scaled from 0 to 100 in increments of 20. The Z-axis is the EP in values from 15 to 105. The EP surface is depicted as a two-dimensional plane within a three-dimensional cube. The EP surface ranges from an EP of about 25, for low X- and Y-axis PWL values, to 105, for high X- and Y-axis PWL values of the two of is ranges from is and illustrates the variability of the EP with the PWL values.

Figure 85. EP curves for the values in table 53. Chart. Using the data from table 53, this figure presents expected payment curves for two variables. The X-axis of the chart is the PWL for variable number 1 and is scaled from 0 to 100 in increments of 10. The Y-axis is the expected payment in percent from 15 to 105. The eight EP curves are shown as slightly divergent straight lines. The eight lines begin at a PWL value of 10, between 35 and 63 percent EP and end at a PWL value of 95, between EP values of 62 and 105 percent.

Figure 86. OC curve for an acceptance plan that calls for rejection if the estimated PWL is less than 60, sample sizes equal to 5. Chart. This figure associates the probability of acceptance with actual PWL values. The X-axis shows the actual PWL values from 0 to 100 and the Y-axis presents the probability of acceptance from 0 to 1.0. The association is an elongated S-shaped curve beginning at a probability of 0 for a PWL value of about 20 and ending at a probability of 1.0 for a PWL value of about 90.

Figure 87a. Simulation results of expected payment for the averaging method, combining two populations with equal PWL values. Chart. This chart shows that the expected payment value is not affected by the correlation between two acceptance variables. The X-axis shows the correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 75 to 105. The plots for the three values of PWL are straight horizontal lines across the correlation coefficient scale. For a population PWL value of 90, the expected payment factor is 100 percent; for PWL of 70, the expected payment factor is 90 percent; and for a population PWL equal to 50, the expected payment factor is 80 percent.

Figure 87b. Simulation results of standard deviation values for the averaging method, combining two populations with equal PWL values. Chart. This chart compares the effect of the correlation between two variables on the variability of individual payment factors for two populations with equal PWL values. The X-axis shows the correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 10. Data on the chart suggest that the variability of the payment estimates is related to the correlation between the two variables. For a PWL of 50 and 70, the variability about the average payment increases from 0 and 4 percent, respectively, to about 9 percent as the correlation between the two variables increases from negative 1.0 to positive 1.0. For the PWL value of 90, the variability about the average payment decreases slightly from 5.5 percent to about 4 percent as the correlation coefficient approaches 0, and then increases to about 5 percent as the correlation coefficient approaches positive 1.0.

Figure 88a. Simulation results of expected payment for the weighted average method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 75 to 105. The top chart shows that the expected payment value is not affected by the correlation between two acceptance variables. The plots for the three values of PWL are straight horizontal lines across the correlation coefficient scale. For a population PWL value of 90, the expected payment factor is 100 percent; for PWL of 70, the expected payment factor is 90 percent; and for a population PWL equal to 50, the expected payment factor is 80 percent.

Figure 88b. Simulation results of standard deviation values for the weighted average method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 10. This chart compares the effect of the correlation between two variables on the variability of individual payment factors for two populations with equal PWL values. Data on the chart suggest that the variability of the payment estimates is related to the correlation between the two variables. For a PWL of 50 and 70, the variability about the average payment increases from about 5.5 percent to between 8.5 and 9.5 percent as the correlation between the two variables increases from negative 1.0 to positive 1.0. For the PWL value of 90, the variability about the average payment decreases slightly from 5.5 percent to about 4.5 percent as the correlation coefficient approaches 0, and then increases to 5.5 percent as the correlation approaches positive 1.0.

Figure 89a. Simulation results of expected payment for the multiplication method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 60 to 105. The top chart shows that the expected payment value is not affected by the correlation between two acceptance variables. The plots for the three values of PWL are straight horizontal lines across the correlation coefficient scale. For a population PWL value of 90, the expected payment factor is 100 percent; for PWL of 70, the expected payment factor is 80 percent; and for a population PWL equal to 50, the expected payment factor is between 63 and 65 percent.

Figure 89b. Simulation results of standard deviation values for the multiplication method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 16. This chart compares the effect of the correlation between two variables on the variability of individual payment factors for two populations with equal PWL values. Data on the chart suggest that the variability of the payment estimates is related to the correlation between the two variables. For a PWL of 50 and 70, the variability about the average payment increases from about 1.5 and 7 percent, respectively, to 16 percent as the correlation between the two variables increases from negative 1.0 to positive 1.0. For the PWL value of 90, the variability about the average payment decreases from about 10.5 percent to about 8 percent as the correlation coefficient approaches 0, and then increases to 10.5 percent as the correlation approaches positive 1.0.

Figure 90a. Simulation results of expected payment for the summation method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 55 to 105. This chart shows that the expected payment value is not affected by the correlation between two acceptance variables. The plots for the three values of PWL are mostly straight horizontal lines across the correlation coefficient scale. For a population PWL value of 90, the expected payment factor is 100 percent; for PWL of 70, the expected payment factor is 80 percent; and for a population PWL equal to 50, the expected payment factor is about 60 percent.

Figure 90b. Simulation results of standard deviation values for the summation method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 20. This chart compares the effect of the correlation between two variables on the variability of individual payment factors for two populations with equal PWL values. Data on the chart suggest that the variability of the payment estimates is related to the correlation between the two variables. For a PWL of 50 and 70, the variability about the average payment increases from about 1.5 and 7.5 percent, respectively, to 17 and 19 percent, respectively, as the correlation between the two variables increases from negative 1.0 to positive 1.0. For the PWL value of 90, the variability about the average payment decreases from about 11 percent to about 8 percent as the correlation coefficient approaches 0, and then increases to 11 percent as the correlation approaches positive 1.0.

Figure 91a. Simulation results of expected payment for the maximum method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 75 to 105. This chart shows that the expected payment value is minimally affected by the correlation between two acceptance variables. The plots for the three values of PWL decline very slightly across the correlation coefficient scale. For a population PWL value of 90, the expected payment factor increases slightly from 100 percent at a correlation coefficient of negative 1.0 to about 103 percent at a correlation coefficient of 0, and declines slightly to about 100 percent at a correlation coefficient of positive 1.0. For a PWL value of 70, the expected payment factor declines gradually from about 97 percent at a correlation coefficient of negative 1.0 to about 90 percent at a correlation coefficient of positive 1.0. For a population PWL value equal to 50, the expected payment factor declines gradually from about 87 percent at a correlation coefficient of negative 1.0 to about 80 percent at a correlation coefficient of positive 1.0.

Figure 91b. Simulation results of expected payment for the maximum method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 10. This chart compares the effect of the correlation between two variables on the variability of individual payment factors for two populations with equal PWL values. Data on the chart suggest that the variability of the payment estimates is related to the correlation between the two variables. For a PWL of 50 and 70, the variability about the average payment increases gradually from about 6 percent to about 8.5 and 9.5 percent, respectively, as the correlation between the two variables increases from negative 1.0 to positive 1.0. For the PWL value of 90, the variability about the average payment decreases from about 5.8 percent to about 3.2 percent as the correlation coefficient approaches 0, and then increases to 5.8 percent as the correlation approaches positive 1.0.

Figure 92a. Simulation results of expected payment for the minimum method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 70 to 105. This chart shows that the expected payment value is minimally affected by the correlation between two acceptance variables. For a population PWL value of 90, the expected payment factor remains fairly constant around 95 to 100 percent across the correlation coefficient scale. For a PWL value of 70, the expected payment factor increases very slightly from about 84 percent at a correlation coefficient of negative 1.0 to about 90 percent at a correlation coefficient of positive 1.0. For a population PWL value equal to 50, the expected payment factor increases very gradually from about 73 percent at a correlation coefficient of negative 1.0 to about 80 percent at a correlation coefficient of positive 1.0.

Figure 92b. Simulation results of standard deviation values for the minimum method, combining two populations with equal PWL values. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 10. This chart compares the effect of the correlation between two variables on the variability of individual payment factors for two populations with equal PWL values. Data on the chart suggest that the variability of the payment estimates is minimally related to the correlation between the two variables for the populations with PWL values of 50 and 70. For a PWL of 90, the payment estimate does not vary with correlation coefficients and is shown as a fairly straight line between 5.8 and 6 percent across the correlation coefficient scale. For populations with PWL values of 50 and 70, the variabilities about the average payment increase gradually from about 6 percent to about 8.5 and 10 percent, respectively, as the correlation between the two variables increases from negative 1.0 to positive 1.0.

Figure 93a. Comparison of simulation results for various methods for combining individual expected payment factors for two populations with PWL equal to 90. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 94 to 104. This chart reveals that the expected payment value does not vary with the correlation between two acceptance variables when the two populations are combined by the averaging, weighted average, multiplication, or summing methods. On the chart, this is illustrated by somewhat straight lines at 100 percent expected payment across the correlation coefficient scale. The expected payment value does vary with the correlation coefficient when the populations are combined using either the minimum or the maximum individual payment factors. When the populations are combined using the maximum individual payment factors, the expected payment factor increases from 100 percent at a correlation coefficient of negative 1.0 to about 104 percent at a correlation coefficient of 0, and then decreases to 100 percent at a correlation coefficient of positive 1.0. When the populations are combined using the minimum individual payment factors, the expected payment factor decreases from 100 percent at a correlation coefficient of negative 1.0 to about 97 percent at a correlation coefficient of 0, and then decreases to 100 percent at a correlation coefficient of positive 1.0.

Figure 93b. Comparison of simulation results for various methods for combining individual standard deviation payment factors for two populations with PWL equal to 90. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 12. This chart compares the effect of the correlation between two variables on the variability of individual payment factors. Two populations with equal PWL values of 90, combined by the six methods described in the text, are plotted. Data on the chart suggest that the variability of the payment estimates is not related to the correlation coefficients for populations combined by the minimum individual payment factor method. This is illustrated on the chart by a straight line at 6 percent across the correlation coefficient scale. A minimal relationship between the variability of the payment estimates and the correlation between the two variables is shown for the populations combined by the average, weighted average, multiplication, summation, and maximum individual payment factor methods. Slight decreases from about 5.5 percent at a correlation coefficient of negative 1.0 to about 4 percent are shown as the correlation coefficient approaches zero. Between 0 and a correlation coefficient of positive 1.0, the variabilities about the average payment gradually increase to 5.5 percent.

Figure 94a. Comparison of simulation results for various methods for combining individual expected payment factors for two populations with PWL equal to 70. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 78 to 98. This chart reveals that the expected payment value does not vary with the correlation between two acceptance variables when the two populations are combined by the averaging, weighted average, multiplication, or summing methods. On the chart, this is illustrated by somewhat straight lines across the correlation coefficient scale. The expected payment value does vary with the correlation coefficient when the populations are combined using either the minimum or the maximum individual payment factors. When the populations are combined using the maximum individual payment factors, the expected payment factor decreases from 96 percent at a correlation coefficient of negative 1.0 to about 90 percent at a correlation coefficient of positive 1.0. When the populations are combined using the minimum individual payment factors, the expected payment factor increases from 84 percent at a correlation coefficient of negative 1.0 to about 90 percent at a correlation coefficient of positive 1.0.

Figure 94b. Comparison of simulation results for various methods for combining individual standard deviation payment factors for two populations with PWL equal to 70. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 20. This chart compares the effect of the correlation between two variables on the variability of individual payment factors. Two populations with equal PWL values of 70, combined by the six methods described in the text, are plotted. Data on the chart suggest that the variability of the payment estimates is minimally related to the correlation coefficients for populations combined by the average, weighted average, maximum individual payment factor, or minimum individual payment factor methods. This is illustrated on the chart by slightly increasing variabilities from 4 to 6.5 percent at a correlation coefficient of negative 1.0 to about 8.5 percent at a correlation coefficient of positive 1.0. When populations are combined by the multiplication or summation methods, the variability of the individual payment factors is affected by the correlation coefficients. On the chart, this is illustrated by increases from about 7 percent at a correlation coefficient of negative 1.0 to 15.5 and 17.5 percent for the multiplication and summation methods, respectively, at a correlation coefficient of possitive 1.0.

Figure 95a. Comparison of simulation results for various methods for combining individual expected payment factors for two populations with PWL equal to 50. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the expected payment factor in percent from 55 to 90. This chart reveals that the expected payment value does not vary with the correlation between two acceptance variables when the two populations are combined by the averaging, weighted average, multiplication, or summing methods. On the chart, this is illustrated by generally straight lines across the correlation coefficient scale. The expected payment value does vary with the correlation coefficient when the populations are combined using either the minimum or the maximum individual payment factors. When the populations are combined using the maximum individual payment factors, the expected payment factor decreases from about 87 percent at a correlation coefficient of negative 1.0 to about 80 percent at a correlation coefficient of positive 1.0. When the populations are combined using the minimum individual payment factors, the expected payment factor increases from about 73 percent at a correlation coefficient of negative 1.0 to about 80 percent at a correlation coefficient of positive 1.0.

Figure 95b. Comparison of simulation results for various methods for combining individual standard deviation payment factors for two populations with PWL equal to 50. Chart. The X-axis shows correlation coefficients from negative 1.0 to positive 1.0. The Y-axis is the standard deviation in percent of the payment factor, from 0 to 20. This chart compares the effect of the correlation between two variables on the variability of individual payment factors. Two populations with equal PWL values of 50, combined by the six methods described in the text, are plotted. Data on the chart suggest that the variability of the payment estimates is related to the correlation coefficients for populations combined by all of the six methods. This is illustrated on the chart by increasing variabilities across the correlation coefficient scale. For populations combined by the multiplication and summation methods, however, the relationship between the correlation of the variables and the variability of the payment factors is more pronounced. On the chart, this is shown as an increase from about 1 percent to 16 and 19 percent, respectively for the multiplication and summation methods, across the correlation coefficient scale.

Figure 96. Flowchart of the PRS process. Diagram. This figure diagrams the performance-related specifications (PRS) process. At the upper left corner of the figure is an arrow pointing diagonally to a text box. Below the box, also at a diagonal, is another arrow pointing to a second text box. Below this box is a third arrow pointing away from the figure. Text associated with the first arrow includes design variables and construction variables, both representing inputs to the performance prediction models, which is the text contained in the first box. Text associated with the second arrow below the first box includes predicted distress, and occurrence and extent. Text associated with the second box is maintenance cost models. The third arrow below this box is labeled life-cycle cost and represents the output from the model.

Figure 97. Illustration of the net impact of rescheduling an overlay 2 years earlier than originally planned. Diagram. This figure illustrates the net effect of scheduling a pavement overlay at 18 years rather than the 20 years originally planned. It shows a time line, in years, on the horizontal axis, starting at zero, or the beginning of the pavement life, and including 18 years, representing the time of a premature pavement overlay and 20 years, or the expected planned overlay of the pavement. Just above the number zero on the horizontal axis are two short vertically aligned arrows, one solid line and one broken line, enclosed within a circle and pointing down to the number zero. A solid vertical arrow, labeled a premature overlay, pointing down occurs just above the number 18 on the X-axis and a broken line vertical arrow, labeled as planned overlay, also pointing down, occurs just above the number 20 on the X-axis. From the tops of each of these two taller arrows is another set of arrows pointing diagonally to the two shorter arrows enclosed within the circle. Another arrow, pointing away from the circle above the number zero, points to text that reads: net loss.

Figure 98. Spread of possible sample means for a normal distribution with 10 PD below the lower specification limit and sample size equal to 3. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 3. The histogram shows a normal distribution centered at the population mean. Text alongside the histogram indicates that the skewness is 0, 10 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 10, and none of the distribution is above the upper specification limit, or PD subscript uppercase U equals zero. Below the histogram is a bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 99. Spread of possible sample means for a normal distribution with 10 PD below the lower specification limit and sample size equal to 10. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 10. The histogram shows a normal distribution centered at the population mean. Text alongside the histogram indicates that the skewness is 0, 10 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 10, and none of the distribution is above the upper specification limit, or PD subscript uppercase U equals 0. Below the histogram is a tall narrow bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. The third component of this figure is again the bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 100. Spread of possible sample means for a normal distribution with 5 PD outside each specification limit and sample size equal to 3. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 3. The histogram shows a normal distribution centered at the population mean. Text alongside the histogram indicates that the skewness is 0, 5 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 5, and 5 percent of the distribution is shaded and above the upper specification limit, or PD subscript uppercase U equals 5. Below the histogram is a bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the largest mean from the original population can occur above the upper specification limit of the normal distribution of sample means.

Figure 101. Spread of possible sample means for a normal distribution with 5 PD outside each specification limit and sample size equal to 10. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 10. The histogram shows a normal distribution centered at the population mean. Text alongside the histogram indicates that the skewness is 0, 5 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 5, and 5 percent of the distribution is shaded and above the upper specification limit, or PD subscript uppercase U equals 5. Below the histogram is a tall narrow bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. The third component of this figure is again the tall narrow bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 102. Spread of possible sample means for a normal distribution with 50 PD below the lower specification limit and sample size equal to 3. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 3. The histogram shows a normal distribution centered at the population mean. Text alongside the histogram indicates that the skewness is 0, 50 percent of the distribution is shaded or below the lower specification limit, or PD subscript upper case L equals 50, and none of the distribution is above the upper specification limit, or PD subscript uppercase U equals 0. For this example, the mean is equal to the specification limit. Below the histogram is a bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 103. Spread of possible sample means for a normal distribution with 50 PD below the lower specification limit and sample size equal to 10. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 10. The histogram shows a normal distribution centered at the population mean. Text alongside the histogram indicates that the skewness is 0, 50 percent of the distribution is shaded or below the lower specification limit, or PD subscript upper case L equals 50, and none of the distribution is above the upper specification limit, or PD subscript uppercase U equals 0. For this example, the mean is equal to the specification limit. Below the histogram is a tall narrow bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 10, shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the tall narrow bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 10, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 104. Spread of possible sample means for a normal distribution with 25 PD outside each specification limit and sample size equal to 3. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 3. The histogram shows a normal distribution centered at the population mean. Text alongside the histogram indicates that the skewness is 0, 25 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 25, and 25 percent of the distribution is shaded and above the upper specification limit, or PD subscript uppercase U equals 25. Below the histogram is a bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the largest mean from the original population can occur above the upper specification limit of the normal distribution of sample means.

Figure 105. Spread of possible sample means for a normal distribution with 25 PD outside each specification limit and sample size equal to 10. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 10. The histogram shows a normal distribution centered at the population mean. Text alongside the histogram indicates that the skewness is 0, 25 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 25, and 25 percent of the distribution is shaded and above the upper specification limit, or PD subscript uppercase U equals 25. Below the histogram is a tall narrow bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the largest mean from the original population can occur above the upper specification limit of the normal distribution of sample means.

Figure 106. Spread of possible sample means for a distribution with skewness equals 1.0, 10 PD below the lower specification limit, and sample size equals 3. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the mean distribution with the spread of the sample mean distribution for a sample size of 3. The histogram shows a distribution skewed to the left. Text alongside the histogram indicates that the skewness is positive 1.0, 10 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 10, and none of the distribution is above the upper specification limit, or PD subscript uppercase U equals 0. Below the histogram is a bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped normal distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 107. Spread of possible sample means for a distribution with skewness equals 1.0, 10 PD below the lower specification limit, and sample size equals 10. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the mean distribution with the spread of the sample mean distribution for a sample size of 10. The histogram shows a distribution skewed to the left of the mean. Text alongside the histogram indicates that the skewness is positive 1.0, 10 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 10, and none of the distribution is above the upper specification limit, or PD subscript uppercase U equals 0. Below the histogram is a tall narrow bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped normal distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 108. Spread of possible sample means for a distribution with skewness equals 1.0, 5 PD outside each specification limit, and sample size equals 3. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the mean distribution with the spread of the sample mean distribution for a sample size of 3. The histogram shows a distribution skewed to the left of the mean. Text alongside the histogram indicates that the skewness is positive 1.0, 5 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 5, and 5 percent of the distribution is above the upper specification limit, or PD subscript uppercase U equals 5. Below the histogram is a bell-shaped distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the distribution with the smallest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the largest mean from the original population can occur above the upper specification limit of the normal distribution of sample means.

Figure 109. Spread of possible sample means for a distribution with skewness equals 1.0, 5 PD outside each specification limit, and sample size equals 10. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the mean distribution with the spread of the sample mean distribution for a sample size of 10. The histogram shows a distribution skewed to the left of the mean. Text alongside the histogram indicates that the skewness is positive 1.0, 5 percent of the distribution is shaded and below the lower specification limit, or PD subscript upper case L equals 5, and 5 percent of the distribution is above the upper specification limit, or PD subscript uppercase U equals 5. Below the histogram is a tall narrow bell-shaped distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the distribution with the smallest mean that can be obtained from the original population. The third component of this figure is again the bell-shaped distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 110. Spread of possible sample means for a distribution with skewness equals 1.0, 50 PD below the lower specification limit, and sample size equals 3. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 3. The histogram shows a distribution skewed to the left of the mean. Text alongside the histogram indicates that the skewness is positive 1.0, 50 percent of the distribution is shaded or below the lower specification limit, or PD subscript upper case L equals 50, and none of the distribution is above the upper specification limit, or PD subscript uppercase U equals 0. Below the histogram is a bell-shaped distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 111. Spread of possible sample means for a distribution with skewness equals 1.0, 50 PD below the lower specification limit, and sample size equals 10. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 10. The histogram shows a distribution skewed to the left of the mean. Text alongside the histogram indicates that the skewness is positive 1.0, 50 percent of the distribution is shaded or below the lower specification limit, or PD subscript upper case L equals 50, and none of the distribution is above the upper specification limit, or PD subscript uppercase U equals 0. Below the histogram is a tall narrow bell-shaped distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the tall narrow bell-shaped distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population.

Figure 112. Spread of possible sample means for a distribution with skewness equals 1.0, 25 PD outside each specification limit, and sample size equals 3. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 3. The histogram shows a distribution skewed to the left of the mean. Text alongside the histogram indicates that the skewness is positive 1.0, 25 percent of the distribution is shaded or below the lower specification limit, or PD subscript upper case L equals 25, and 25 percent of the distribution is shaded and above the upper specification limit, or PD subscript uppercase U equals 25. Below the histogram is a bell-shaped distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the bell-shaped distribution of the sample means for a sample size of 3. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 3, shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the largest mean from the original population can occur above the upper specification limit of the normal distribution of sample means.

Figure 113. Spread of possible sample means for a distribution with skewness equals 1.0, 25 PD outside each specification limit, and sample size equals 10. Charts. This figure consists of a histogram and two bell-shaped curves, each comparing the normal mean distribution with the spread of the sample mean distribution for a sample size of 10. The histogram shows a distribution skewed to the left of the mean. Text alongside the histogram indicates that the skewness is positive 1.0, 25 percent of the distribution is shaded or below the lower specification limit, or PD subscript upper case L equals 25, and 25 percent of the distribution is shaded and above the upper specification limit, or PD subscript uppercase U equals 25. Below the histogram is a tall narrow bell-shaped distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the left is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the smallest mean that can be obtained from the original population. This curve illustrates that the mean of the distribution using the smallest mean from the original population can occur below the lower specification limit of the normal distribution of sample means. The third component of this figure is again the tall narrow bell-shaped distribution of the sample means for a sample size of 10. The distribution is centered at the population mean. Overlapping this distribution on the right is a second bell-shaped curve for a sample size of 10, much shorter and wider than the distribution of the sample means, representing the normal distribution with the largest mean that can be obtained from the original population. This curve shows that the mean of the distribution using the largest mean from the original population can occur above the upper specification limit of the normal distribution of sample means.

Figure 114. Sample output screen for a population with PD equals 10, skewness coefficient equals 0.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a normal population of 1,000, characterized by 10 PD, a sample size of 5, skewness coefficient of 0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. All of the histograms are asymmetrical, bars to the right, and most of the bias values are distributed near zero. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The standard error for a sample size of 5 is 0.11 for all PD splits. The average bias for the 8-slash-0 split is positive 0.07. For the 7-slash-1 split, the average bias is negative 0.02, and for the 6-slash-2 split, the average bias is negative 0.29. For the 5-slash-3 split, the average bias is negative 0.23. For the 4-slash-4 split, the average bias is negative 0.02, and for the 3-slash-5 split, the average bias is negative 0.36. The average bias for the 2-slash-6 split is positive 0.13. For the 1-slash-7 split, the average bias is negative 0.01, and for the 0-slash-8 split, the average bias is positive 0.11. Overall, the nine histograms indicate that, for a fixed sample size and no skewness, there is no visible change in the distribution of the bias values as the PD splits change from 8-slash-0 to 0-slash-8.

Figure 115. Portions of output screens for PD equals 10, skewness coefficient equals 0.00, and sample sizes equal to 3, 5, and 10. Charts. This figure presents an example of a simulation output screen that illustrates the effect of various sample sizes on the variability of PD estimates. The simulation is for a normal population of 1,000; characterized by 10 PD; sample sizes of 3, 5, and 10; skewness coefficient of 0; and 10,000 replications. The program output screen shows nine histograms of actual bias values arranged three across in three rows. The scale of the horizontal axis is negative 90 to positive 90. Each row across includes histograms for three PD splits (8-slash-0; 7-slash-1; 6-slash-2) for one sample size. The three rows down are for the three sample sizes. In the first row for a sample size of 3, all of the histograms are similar and asymmetrical, bars to the right, and most of the bias values are distributed at 0, with the remaining in the bars to the right. On each histogram, the average bias value and the standard error are also given. For a sample size of 3, the standard error is 0.15. The average bias is negative 0.22 for the 8-slash-0 split; positive 0.27 for the 7-slash-1 split, and negative 0.14 for the 6-slash-2 split. In the second row for a sample size of 5, all of the histograms are similar and asymmetrical, bars to the right. Most of the bias values are distributed at zero and then decreasing in the three bars to the right. For a sample size of 5, the standard error is 0.11. The average bias is positive 0.07 for the 8-slash-0 split, negative 0.02 for the 7-slash-1 split, and negative 0.29 for the 6-slash-2 split. In the third row for a sample size of 10, all of the histograms are similar and asymmetrical, bars to the right. Most of the bias values are distributed at zero and the bar adjacent on the right. The remaining values are distributed among three smaller bars to the right. For a sample size of 10, the standard error is 0.08. The average bias is positive 0.05 for the 8-slash-0 split, positive 0.06 for the 7-slash-1 split, and positive 0.11 for the 6-slash-2 split. Overall, the nine histograms show that the distribution of the bias values becomes more spread out as the sample size increases.

Figure 116. Sample output screen for a population with PD equals 10, skewness coefficient equals 1.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen that illustrates the effect of sample size on the variability of PD estimates. The simulation is for a population of 1,000, characterized by 10 PD, a sample size of 5, skewness coefficient of 1.0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. All of the histograms are asymmetrical, bars to the right, and most of the bias values are distributed near zero. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The average bias for the 8-slash-0 split is positive1.56 and the standard error is 0.10. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 7-slash-1 split, the average bias is positive 1.44 and the standard error is 0.10. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 6-slash-2 split, the average bias is positive 1.41 and the standard error is 0.11. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 5-slash-3 split, the average bias is positive 1.48 and the standard error is 0.12. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 4-slash-4 split, the average bias is positive 1.76 and the standard error is 0.12. Most of the bias values are distributed at zero and the adjacent bar to the right. For the 3-slash-5 split, the average bias is positive 1.80 and the standard error is 0.13. Most of the bias values are distributed at zero and the adjacent bar to the right. The average bias for the 2-slash-6 split is positive 1.79 and the standard error is 0.13. Most of the bias values are distributed at zero. The remaining values are distributed among the bars to the right. For the 1-slash-7 split, the average bias is positive 1.98 and the standard error is 0.14. Most of the bias values are distributed at zero. The remaining values are distributed among the bars to the right. For the 0-slash-8 split, the average bias is negative 0.36 and the standard error is 0.13. Most of the bias values are distributed at zero. The remaining values are distributed among the bars to the right. Overall, the nine histograms show that the distribution of the bias values become less spread out as the PD splits change from 8-slash-0 to 0-slash-8.

Figure 117. Portions of output screens for PD equals 10, skewness coefficient equals 1.00, and sample sizes equal to 3, 5, and 10. Charts. This figure presents an example of a simulation output screen that illustrates the effect of various sample sizes on the variability of PD estimates. The simulation is for a population of 1,000, characterized by 10 PD, sample sizes of 3, 5, and 10, skewness coefficient of positive 1, and 10,000 replications. The program output screen shows nine histograms of actual bias values arranged three across in three rows. Each row across includes histograms for three PD splits (8-slash-0; 7-slash-1; 6-slash-2) for one sample size. The three rows down are for the three sample sizes. In the first row for a sample size of 3, all of the histograms are similar and asymmetrical, bars to the right, and most of the bias values are distributed at 0. On each histogram, the average bias value and the standard error are also given. For a sample size of 3, the average bias is positive 0.73 and the standard error is 0.14 for the 8-slash-0 split; the average bias is positive 0.63 and the standard error is 0.15 for the 7-slash-1 split, and the average bias is positive 1.03 and the standard error is 0.15 for the 6-slash-2 split. In the second row for a sample size of 5, all of the histograms are similar and asymmetrical, bars to the right. Most of the bias values are distributed at zero, with decreasing amounts in the bars to the right. For a sample size of 5, the average bias is positive 1.56 and the standard error is 0.10 for the 8-slash-0 split; the average bias is positive 1.44 and the standard error is 0.10 for the 7-slash-1 split; and the average bias is positive 1.41 and the standard error is 0.11 for the 6-slash-2 split. In the third row for a sample size of 10, all of the histograms are similar and asymmetrical, bars to the right. Most of the bias values are somewhat evenly distributed between zero and the adjacent bar on the right. For a sample size of 10, the standard error is 0.07 for the three splits. The average bias is positive 2.22 for the 8-slash-0 split, positive 1.76 for the 7-slash-1 split, and positive 1.66 for the 6-slash-2 split. Overall, the nine histograms show that the distribution of the bias values at zero decreases as the sample size increases.

Figure 118. Sample output screen for a population with PD equals 10, skewness coefficient equals 2.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a population of 1,000, characterized by 10 PD, a sample size of 5, skewness coefficient of 2.0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. All of the histograms are asymmetrical, bars to the right, and most of the bias values are distributed near zero. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The average bias for the 8-slash-0 split is positive 3.63 and the standard error is 0.10. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 7-slash-1 split, the average bias is positive 3.43 and the standard error is 0.10. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 6-slash-2 split, the average bias is positive 3.81 and the standard error is 0.12. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 5-slash-3 split, the average bias is positive 4.47 and the standard error is 0.13. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 4-slash-4 split, the average bias is positive 4.66 and the standard error is 0.14. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 3-slash-5 split, the average bias is positive 5.47 and the standard error is 0.15. Most of the bias values are distributed at zero and the adjacent bar to the right. The average bias for the 2-slash-6 split is positive 5.61 and the standard error is 0.16. Most of the bias values are distributed at zero. The remaining values are distributed in decreasing amounts among the bars to the right. For the 1-slash-7 split, the average bias is positive 6.19 and the standard error is 0.18. Most of the bias values are distributed at zero. The remaining values are distributed among the bars to the right. For the 0-slash-8 split, the average bias is positive 0.45 and the standard error is 0.14. Most of the bias values are distributed at zero. The remaining values are distributed among the bars to the right. Overall, the nine histograms show that the distribution of the bias values become less spread out as the PD splits change from 8-slash-0 to 0-slash-8.

Figure 119. Portions of output screens for PD equals 10, skewness coefficient equals 2.00, and sample sizes equal to 3, 5, and 10. Charts. This figure presents an example of a simulation output screen that illustrates the effect of various sample sizes on the variability of PD estimates. The simulation is for a population of 1,000, characterized by 10 PD, sample sizes of 3, 5, and 10, skewness coefficient of positive 2, and 10,000 replications. The program output screen shows nine histograms of actual bias values arranged three across in three rows. Each row across includes histograms for three PD splits (8-slash-0; 7-slash-1; 6-slash-2) for one sample size. The three rows down are for the three sample sizes. In the first row for a sample size of 3, all of the histograms are similar and asymmetrical, bars to the right, and most of the bias values are distributed at zero. On each histogram, the average bias value and the standard error are also given. For a sample size of 3, the average bias is positive 2.14 and the standard error is 0.14 for the 8-slash-0 split; the average bias is positive 2.13 and the standard error is 0.15 for the 7-slash-1 split, and the average bias is positive 2.61 and the standard error is 0.16 for the 6-slash-2 split. In the second row for a sample size of 5, all of the histograms are similar and asymmetrical, bars to the right. Most of the bias values are distributed at zero and the two adjacent bars to the right. For a sample size of 5, the average bias is positive 3.63 and the standard error is 0.10 for the 8-slash-0 split; the average bias is positive 3.43 and the standard error is 0.10 for the 7-slash-1 split; and the average bias is positive 3.81 and the standard error is 0.12 for the 6-slash-2 split. In the third row for a sample size of 10, the histograms are generally similar and asymmetrical, bars to the right. Most of the bias values are distributed at the bar adjacent to zero to the right, and the remaining values somewhat evenly divided among zero and the second bar to the right. For a sample size of 10, the average bias is positive 5.07 and the standard error is 0.07 for the 8-slash-0 split; the average bias is positive 4.94 and the standard error is 0.07 for the 7-slash-1 split; the average bias is positive 5.03 and the standard error is 0.08 for the 6-slash-2 split. Overall, the nine histograms show that the distribution of the bias values at zero decreases as the sample size increases.

Figure 120. Sample output screen for a population with PD equals 30, skewness coefficient equals 0.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a normal population of 1,000, characterized by 30 PD, a sample size of 5, skewness coefficient of 0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. All histograms are asymmetrical and have similar distributions. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The standard error is 0.17 for all histograms except the 0-slash-8 split, for which the standard error is 0.18. The average bias for the 8-slash-0 split is negative 0.07. For the 7-slash-1 split, the average bias is positive 0.35, and for the 6-slash-2 split, the average bias is positive 0.13. For the 5-slash-3 split, the average bias is positive 0.20. For the 4-slash-4 split, the average bias is positive 0.04, and for the 3-slash-5 split, the average bias is negative 0.02. The average bias for the 2-slash-6 split is negative 0.26. For the 1-slash-7 split, the average bias is positive 0.02, and for the 0-slash-8 split, the average bias is negative 0.03. Overall, the nine histograms indicate that, for a fixed sample size and no skewness, there is little visible change in the distribution of the bias values as the PD splits change from 8-slash-0 to 0-slash-8.

Figure 121. Sample output screen for a population with PD equals 50, skewness coefficient equals 0.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a normal population of 1,000, characterized by 50 PD, a sample size of 5, skewness coefficient of 0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. The 8-slash-0 and the 0-slash-8 splits are similar, and the 7-slash-1 and 1-slash-7 splits are similar, both sets with nearly symmetrical distributions. The remaining histograms are asymmetrical and similar to each other, with most of the bias values distributed at zero and the bars on either side of zero. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The average bias for the 8-slash-0 split is negative 0.15 and the standard error is 0.19. For the 7-slash-1 split, the average bias is negative 0.01 and the standard error is 0.17. For the 6-slash-2 split, the average bias is positive 0.06 and the standard error is 0.17. For the 5-slash-3 split, the average bias is positive 0.25 and the standard error is 0.16. For the 4-slash-4 split, the average bias is negative 0.19 and the standard error is 0.16. For the 3-slash-5 split, the average bias is positive 0.03 and the standard error is 0.16. The average bias for the 2-slash-6 split is negative 0.02 and the standard error is 0.17. For the 1-slash-7 split, the average bias is 0 and the standard error is 0.17. For the 0-slash-8 split, the average bias is positive 0.07 and the standard error is 0.19. Overall, the nine histograms indicate that, for a fixed sample size and no skewness, there is a slight change in the distribution of the bias values as the PD splits change from 8-slash-0 to 4-slash-4, and then the change reverses.

Figure 122. Sample output screen for a population with PD equals 30, skewness coefficient equals 0.00, and sample size equals 10. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a normal population of 1,000, characterized by 30 PD, a sample size of 10, skewness coefficient of 0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. The 8-slash-0 and the 0-slash-8 splits are similar, with nearly symmetrical distributions. The remaining histograms are all asymmetrical and similar to each other, with most of the bias values distributed at zero and the bars on either side of zero. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The average bias for the 8-slash-0 split is positive 0.03 and the standard error is 0.12. For the 7-slash-1 split, the average bias and the standard error are both positive 0.12. For the 6-slash-2 split, the average bias is positive 0.07 and the standard error is 0.11. For the 5-slash-3 split, the average bias is negative 0.10 and the standard error is 0.11. For the 4-slash-4 split, the average bias is negative 0.01 and the standard error is 0.11. For the 3-slash-5 split, the average bias is positive 0.06 and the standard error is 0.11. The average bias for the 2-slash-6 split is negative 0.23 and the standard error is 0.11. For the 1-slash-7 split, the average bias is negative 0.01 and the standard error is 0.12. For the 0-slash-8 split, the average bias is positive 0.16 and the standard error is 0.12. Overall, the nine histograms indicate that, for a fixed sample size and no skewness, only slight changes can be detected in the distribution of the bias values as the PD splits change from 8-slash-0 to 0-slash-8.

Figure 123. Sample output screen for a population with PD equals 50, skewness coefficient equals 0.00, and sample size equals 10. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a normal population of 1,000, characterized by 50 PD, a sample size of 10, skewness coefficient of 0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. The 8-slash-0 and the 0-slash-8 splits are similar, with nearly symmetrical distributions. The remaining histograms are all asymmetrical and similar to each other, with most of the bias values distributed at zero and the bars on either side of zero. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The average bias for the 8-slash-0 split is negative 0.07 and the standard error is 0.13. For the 7-slash-1 split, the average bias is negative 0.17 and the standard error is 0.11. For the 6-slash-2 split, the average bias is positive 0.15 and the standard error is 0.11. For the 5-slash-3 split, the average bias is positive 0.26 and the standard error is 0.10. For the 4-slash-4 split, the average bias is positive 0.17 and the standard error is 0.10. For the 3-slash-5 split, the average bias is negative 0.01 and the standard error is 0.10. The average bias for the 2-slash-6 split is 0 and the standard error is 0.11. For the 1-slash-7 split, the average bias is negative 0.05 and the standard error is 0.11. For the 0-slash-8 split, the average bias is negative 0.0.08 and the standard error is 0.13. As a group, the nine histograms show that the distribution for 8-slash-0, approaching a bell-shaped curve, is similar to the distribution for 0-slash-8. The distributions for 7-slash-1 and 1-slash-7 are similar; the distributions for 6-slash-2 and 2-slash-6 are similar; and the distributions for 5-slash-3 and 3-slash-5 are similar.

Figure 124. Portions of output screens for PD equals 50, sample size equals 5, and skewness coefficients equal 0.00, 1.00, 2.00, and 3.00. Charts. This figure presents an example of a simulation output screen that illustrates the effect of the skewness coefficient on the distribution of the bias values with changes in the PD split. The simulation is for a population of 1,000, characterized by 50 PD; a sample size of 5; skewness coefficient of 0, 1, 2, and 3; and 10,000 replications. The program output screen shows 12 histograms of actual bias values arranged three across in four rows. Each row across includes histograms for three PD splits (8-slash-0; 7-slash-1; 6-slash-2) and one skewness coefficient. The four rows down are for the four skewness coefficients. In the first row for a skewness coefficient of zero, the histograms are similar and nearly symmetrical about zero. On each histogram, the average bias value and the standard error are also given. In the first row for a skewness coefficient of 0, the average bias for the 8-slash-0 split is negative 0.15 and the standard error is 0.19. For the 7-slash-one split, the average bias is negative 0.01 and the standard error is 0.17. The average bias for the 6-slash-2 split is positive 0.06 and the standard error is 0.17. In the second row for a skewness coefficient of 1.0, the histograms are similar and nearly symmetrical about 0. Most of the bias values are distributed at or adjacent to zero. For a skewness coefficient of positive 1.0 and an 8-slash-0 split, the average bias is negative 3.06 and the standard error is 0.19. For the 7-slash-1 split, the average bias is negative 3.56 and the standard error is 0.17. For the 6-slash-2 split, the average bias is negative 2.70 and the standard error is 0.16. In the third row for a skewness coefficient of 2.0, the histograms are similar, with most of the bias values distributed near zero and decreasing amounts on either side. For a skewness coefficient of 2.0 and 8-slash-0 split, the average bias is negative 5.64 and the standard error is 0.19. For the 7-slash-1 split, the average bias is negative 5.80 and the standard error is 0.17. For the 6-slash-2 split, the average bias is negative 3.79 and the standard error is 0.17. In the fourth row for a skewness coefficient of 3.0, the three histograms are similar and asymmetrical. Most of the bias values are distributed at zero and the adjacent bars. For the 8-slash-0 split, the average bias is negative 7.55 and the standard error is 0.19. For the 7-slash-2 split, the average bias is negative 6.57 and the standard error is 0.17. The average bias for the 6-slash-2 split is negative 3.97 and the standard error is 0.18.

Figure 125. Sample output screen for a population with PD equals 10, skewness coefficient equals 3.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a population of 1,000, characterized by 10 PD, a sample size of 5, skewness coefficient of 3.0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. All of the histograms are asymmetrical, bars to the right, and most of the bias values are distributed near zero. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The average bias for the 8-slash-0 split is positive 5.64 and the standard error is 0.09. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 7-slash-1 split, the average bias is positive 5.78 and the standard error is 0.11. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 6-slash-2 split, the average bias is positive 6.39 and the standard error is 0.12. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 5-slash-3 split, the average bias is positive 6.82 and the standard error is 0.14. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 4-slash-4 split, the average bias is positive 7.77 and the standard error is 0.16. Most of the bias values are distributed at zero and the adjacent two bars to the right. For the 3-slash-5 split, the average bias is positive 8.28 and the standard error is 0.17. Most of the bias values are distributed at zero and the adjacent bar to the right. The remaining values are distributed in decreasing amount in the bars to the right. The average bias for the 2-slash-6 split is positive 9.76 and the standard error is 0.19. Most of the bias values are distributed at zero. The remaining values are distributed in decreasing amounts among the bars to the right. For the 1-slash-7 split, the average bias is positive 10.03 and the standard error is 0.19. Most of the bias values are distributed at zero. The remaining values are distributed in decreasing amounts in the bars to the right. For the 0-slash-8 split, the average bias is positive 0.64 and the standard error is 0.16. Most of the bias values are distributed at zero. The remaining values are distributed among the bars to the right. Overall, the nine histograms show that the distribution of the bias values become less spread out as the PD splits change from 8-slash-0 to 0-slash-8.

Figure 126. Sample output screen for a population with PD equals 30, skewness coefficient equals 3.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a population of 1,000, characterized by 30 PD, a sample size of 5, skewness coefficient of 3.0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. All of the histograms are asymmetrical. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The average bias for the 8-slash-0 split is negative 3.08 and the standard error is 0.13. Most of the bias values are distributed at zero and the adjacent two bars. For the 7-slash-1 split, the average bias is negative 2.32 and the standard error is 0.15. Most of the bias values are distributed at zero and the adjacent two bars. For the 6-slash-2 split, the average bias is negative 0.27 and the standard error is 0.17. Most of the bias values are distributed at zero and the remaining values are spread out in decreasing amounts among the bars to the left and right. For the 5-slash-3 split, the average bias is positive 1.80 and the standard error is 0.20. Most of the bias values are distributed at zero, with decreasing amount in the bars to the left and right. For the 4-slash-4 split, the average bias is positive 5.25 and the standard error is 0.22. Most of the bias values are distributed at zero and the adjacent two bars to the left and right. For the 3-slash-5 split, the average bias is positive 7.79 and the standard error is 0.23. The bias values are distributed somewhat evenly around zero and the adjacent bars to the left and right. The average bias for the 2-slash-6 split is positive 11.22 and the standard error is 0.25. The bias values are fairly evenly distributed among all bars. For the 1-slash-7 split, the average bias is positive 14.07 and the standard error is 0.25. The bias values are fairly evenly distributed among all the bars, except those to the far right. For the 0-slash-8 split, the average bias is positive 5.99 and the standard error is 0.22. Most of the bias values are distributed in the bar to the far left of the chart. The remaining values are distributed at zero and bars to the right.

Figure 127. Sample output screen for a population with PD equals 50, skewness coefficient equals 3.00, and sample size equals 5. Charts. This figure presents an example of a simulation output screen that illustrates the changes in the distribution of the bias values with changes in the PD split. The simulation is for a population of 1,000, characterized by 50 PD, a sample size of 5, skewness coefficient of 3.0, and 10,000 replications. The program output screen shows nine histograms of actual bias values, one for each PD split. All of the histograms are asymmetrical. The scale of the horizontal axis is negative 90 to positive 90. On each histogram, the average bias value and the standard error are also given. The average bias for the 8-slash-0 split is negative 7.55 and the standard error is 0.19. Most of the bias values are distributed at zero and the adjacent two bars. For the 7-slash-1 split, the average bias is negative 6.57 and the standard error is 0.17. Most of the bias values are distributed at zero and the adjacent two bars. For the 6-slash-2 split, the average bias is negative 3.97 and the standard error is 0.18. Most of the bias values are distributed at zero and the adjacent bar. The remaining values are spread out in decreasing amounts among the bars to the left and right. For the 5-slash-3 split, the average bias is negative 0.34 and the standard error is 0.20. Most of the bias values are distributed in the bars to the left of zero, at zero, and in decreasing amounts in the bars to the right of zero. For the 4-slash-4 split, the average bias is positive 3.74 and the standard error is 0.21. Most of the bias values are distributed fairly evenly among zero, the adjacent bar to the left, and the three bars to the right. For the 3-slash-5 split, the average bias is positive 7.51 and the standard error is 0.22. The bias values are distributed in increasing amounts in the bars from the left of zero to the right of zero. The average bias for the 2-slash-6 split is positive 11.5 and the standard error is 0.22. The bias values are distributed in increasing amounts in the bars from the left of zero to the right of zero. For the 1-slash-7 split, the average bias is positive 15.06 and the standard error is 0.21. The bias values are distributed in increasing amounts in the bars from the left of zero to the right of zero. For the 0-slash-8 split, the average bias is positive 7.43 and the standard error is 0.19. Most of the bias values are distributed in increasing amounts in the bars from the left of zero to the right of zero.

Figure 128a. Distribution of sample means for 1000 samples from a normal population with mu equal to 0.00, sigma equal to 1.00. Chart. This chart is a histogram of the sample means. Text on the chart states that small N equals 4, and that mu is greater than or equal to 489 and less than 511. The horizontal axis presents the sample means from negative 1.8 to positive 1.8. The distribution of the bars is bell-shaped with 162, or most of the mean values, at a mean of positive 0.2, followed by 154 at 0, and 148 at a mean of 0.4. Other occurrences include 131 means at negative 0.2; 102 at negative 0.4; 81 at positive 0.6; 76 at positive 0.8; 60 at negative 0.6; 28 at negative 0.8; 25 at positive 1.0; 13 at positive 1.2; 9 at negative 1.0; 4 at positive 1.4; 3 at negative 1.2; and 1 occurrence at each of the following values: positive 1.6, positive 1.8, negative 1.4, and negative 1.8.

Figure 128b. Distribution of standard deviations for 1000 samples from a normal population with mu equal to 0.00, sigma equal to 1.00. Chart. This chart is a histogram of the sample standard deviations. Text on the chart states that small N equals 4, and that sigma is greater than or equal to 605 and less than 395. The horizontal axis is the standard deviation from 0.2 to 2.8. The distribution of the bars is asymmetrical, with 346 of the values having a standard deviation of 0.8. Other occurrence include 196 values at 1.0; 138 values at 1.2; 108 values at 1.4; 86 values at 1.6; 49 values at 0.4; 40 values at 1.8; 15 values at 0.2; 14 values at 2; 6 values at 2.2; 2 values at 2.4, and 1 value at 2.8.

Figure 129a. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 90, one-sided specifications. Chart. This chart is a histogram of the sample PWL values and one-sided specifications. Text on the chart states that the actual PWL equals 90; small N equals 4, one-sided specs; and that 90 PWL is greater than or equal to 399 and less than 601. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical, with 526 or most of the values at a PWL of 100. Other occurrences include 75 values at 95; 85 values at 90; 84 values at 85; 73 values at 80; 67 values at 75; 47 values at 70; 25 values at 65; 9 values at 60; 6 values at 55; and one occurrence each at PWL values of 50, 45, and 35.

Figure 129b. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 90, two-sided specifications. Chart. This chart is a histogram of the sample PWL values and two-sided specifications. Text on the chart states that the actual PWL equals 90; small N equals 4, two-sided specs; and that 90 PWL is greater than or equal to 389 and less than 611. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical, with 537 or most of the values at a PWL of 100. Other occurrences include 74 values at 95; 80 values at 90; 62 values at 85; 74 values at 80; 67 values at 75; 48 values at 70; 30 values at 65; 15 values at 60; 8 values at 55; 3 values at 50; and two values at 45. A comparison of the two charts suggests that there is little visible difference in the distributions of the sample PWL values with either one-sided or two-sided specifications.

Figure 130a. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 80, one-sided specifications. Chart. This chart is a histogram of the sample PWL values and one-sided specifications. Text on the chart states that the actual PWL equals 80; small N equals 4, one-sided specs; and that 80 PWL is greater than or equal to 484 and less than 516. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical, with 309 or most of the values at a PWL of 100. Other occurrences include 59 values at 95; 71 values at 90; 77 values at 85; 86 values at 80; 94 values at 75; 98 values at 70; 75 values at 65; 59 values at 60; 40 values at 55; 15 values at 50; 10 values at 45; two each at 40 and 25; and one each at 35, 30, and 20.

Figure 130b. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 80, two-sided specifications. Chart. This chart is a histogram of the sample PWL values and two-sided specifications. Text on the chart states that the actual PWL equals 80; small N equals 4, two-sided specs; and that 80 PWL is greater than or equal to 492 and less than 508. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical, with 307 or most of the values at a PWL of 100. Other occurrences include 49 values at 95; 64 values at 90; 88 values at 85; 89 values at 80; 92 values at 75; 76 values at 70; 87 values at 65; 62 values at 60; 48 values at 55; 18 values at 50; 12 values at 45, and 8 values at 40. A comparison of the two charts suggests that the difference in the distributions of the sample PWL values with either one-sided or two-sided specifications is small.

Figure 131a. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 70, one-sided specifications. Chart. This chart is a histogram of the sample PWL values and one-sided specifications. Text on the chart states that the actual PWL equals 70; small N equals 4, one-sided specs; and that 70 PWL is greater than or equal to 510 and less than 490. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical, with 168 or most of the values at a sample PWL of 100. Other occurrences include 32 values at 95; 60 values at 90; 61 values at 85; 80 values at 80; 89 values at 75; 97 values at 70; 84 values at 65; 117 values at 60; 78 values at 55; 48 values at 50; 37 values at 45; 22 values at 40; 13 values at 35; 5 values at 5; 4 values at 30; 2 each at 25 and 20; and 1 value at 15.

Figure 131b. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 70, two-sided specifications. Chart. This chart is a histogram of the sample PWL values and two-sided specifications. Text on the chart states that the actual PWL equals 70; small N equals 4, two-sided specs; and that 70 PWL is greater than or equal to 535 and less than 465. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical, with 161 or most of the values at a sample PWL of 100. Other occurrences include 49 values at 95; 50 values at 90; 57 values at 85; 55 values at 80; 93 values each at 75 and 70; 89 values at 65; 95 values at 60; 87 values at 55; 77 values at 50; 57 values at 45; 12 values at 40; 15 at 35; and 1 at 30. A comparison of the two charts suggests that the difference in the distributions of the sample PWL values with either one-sided or two-sided specifications is small.

Figure 132a. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 60, one-sided specifications. Chart. This chart is a histogram of the sample PWL values and one-sided specifications. Text on the chart states that the actual PWL equals 60; small N equals 4, one-sided specs; and that 60 PWL is greater than or equal to 502 and less than 498. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical but approaching a bell-shaped curve, with 106 or most of the values at a sample PWL of 65. Other occurrences include 104 values at 55; 95 values at 60; 94 values at 50; 93 values at 70; 78 values at 100; 70 values at 75; 68 values at 45; 59 values at 80; 50 values at 40; 38 values at 85; 34 values at 35; 31 values at 90; 23 values at 95; 19 values at 30; 13 values each at 25 and 5; 6 values at 20; 5 values at 15; and 1 value at 10.

Figure 132b. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 60, two-sided specifications. Chart. This chart is a histogram of the sample PWL values and two-sided specifications. Text on the chart states that the actual PWL equals 60; small N equals 4, two-sided specs; and that 60 PWL is greater than or equal to 578 and less than 422. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical but approaching a bell-shaped curve, with 121 or most of the values at a sample PWL of 50. Other occurrences include 114 values at 60; 105 values at 45; 95 values at 55; 85 values at 100; 76 values at 65; 75 values at 40; 61 values at 70; 60 values at 75; 51 values at 85; 40 values at 35; 38 values at 80; 29 values at 95; 23 values at 30; 22 values at 90; and 5 values at 25. A comparison of the two charts suggests that the one-sided specifications result in a smoother curve approaching a bell shape.

Figure 133a. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 50, one-sided specifications. Chart. This chart is a histogram of the sample PWL values and one-sided specifications. Text on the chart states that the actual PWL equals 50; small N equals 4, one-sided specs; and that 50 PWL is greater than or equal to 489 and less than 511. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is approaching a bell-shaped curve, with 115 or most of the values at a sample PWL of 60. Other occurrences include 114 values at 50; 103 values at 55; 92 values at 45; 79 values at 65; 73 values at 35; 66 values at 40; 57 values at 70; 43 values at 75; 40 values at 30; 32 values at 80; 30 values each at 5 and 100; 29 values at 25; 23 values at 20; 22 values at 85; 17 values at 90; 15 values each at 15; 13 values at 95; and 7 values at 10.

Figure 133b. Distribution of sample PWL values for 1000 samples from a normal population with PWL equal to 50, two-sided specifications. Chart. This chart is a histogram of the sample PWL values and two-sided specifications. Text on the chart states that the actual PWL equals 50; small N equals 4, two-sided specs; and that 50 PWL is greater than or equal to 569 and less than 431. The horizontal axis presents the sample PWL values from 5 to 100. The distribution of the bars is asymmetrical but approaching a bell-shaped curve, with 121 or most of the values at a sample PWL of 45. Other occurrences include 117 values at 35; 113 values at 50; 101 values at 40; 97 values at 55; 89 values at 60; 85 values at 30; 66 values at 65; 37 values at 70; 32 values at 100; 34 values at 85; 29 values each at 75 and 80; 24 values at 25; 11 values at 90; 7 values at 95; 4 values at 20; 3 values at 15; and 1 value at 10. A comparison of the two charts suggests that the one-sided specifications result in a smoother curve with a more normal distribution.

Figure 134. Illustration of the populations for which the distributions of sample PWL values are shown in figures 129 through 133. Charts. This figure consists of five sets of bell-shaped curves illustrating sample populations with PWL values ranging from 90 to 50, for both one-sided and two-sided specifications. The first set shows the 90 PWL populations and the last set shows the 50 PWL populations. Each set shows the one-sided specification on the left and the two-sided specification on the right. A vertical line through each bell-shaped curve on the left indicates the boundary of the one-sided specification. A set of vertical lines through each bell-shaped curve on the right indicates the upper and lower boundaries for the two-sided specifications. From top to bottom, or from 90 PWL to 50 PWL, the one-sided vertical boundary moves nearer to the center of the curve, and the section of the curve within the upper and lower boundaries becomes smaller as the PWL value decreases.