PWLPay  

User’s Manual

Developed by:

Technical:

James L. Burati, Jr.

Civil Engineering Dept.

Clemson University

Programming:

Thomas P. McAfee, Jr.

Clemson University

FHWA Technical

Representative:

Bruce E. Wasill

Western Federal

Lands Highway Division

March 2009 Western Federal Lands Highway Division 

US Department of Transportation – Federal Highway Administration 

Table of Contents Table of Contents

Table of Contents

1 Getting Started.................................... 1‐1

Overview....................................................................1‐1

Typographical Conventions in this Manual................1‐2

Installing PWLPay ......................................................1‐2

Starting PWLPay ........................................................1‐2

The Process of Using PWLPay....................................1‐3

Format of this Manual ...............................................1‐3

2 The Home Screen ................................ 2‐1

The Layout .................................................................2‐1

The File Menu ............................................................2‐1

The Tools Menu .........................................................2‐2

The Navigate Menu ...................................................2‐3

Help............................................................................2‐3

Recent Projects..........................................................2‐3

The Navigation Pane..................................................2‐5

Miscellaneous Items..................................................2‐5

3 Setting up Specs.................................. 3‐1

The Empty Setup Specs Window ...............................3‐1

Select a State / Agency ..............................................3‐1

Item Numbers............................................................3‐2

Characteristics ...........................................................3‐3

Pay Information .........................................................3‐4

The Completed Setup Specs Window......................3‐15

The Expected Life Approach ....................................3‐17

4 Setting Up Projects.............................. 4‐1

The Setup Project Window ........................................4‐1

Adding a Project.........................................................4‐1

Adding Project Details ...............................................4‐4

PWLPay User’s Manual    i
Table of Contents Table of Contents

5 Entering Test Results............................5‐1

The Add Samples Window .........................................5‐1

Entering Test Results .................................................5‐2

Working with Test Results .........................................5‐9

6 Analyzing the Data .............................. 6‐1

The Analysis Window.................................................6‐1

Types of Analysis........................................................6‐2

Descriptive Statistics..................................................6‐3

Tests for Normality ....................................................6‐7

Process Control........................................................6‐10

Verification Tests.....................................................6‐15

Pay Factors...............................................................6‐19

Risk Analysis.............................................................6‐23

Saving an Analysis....................................................6‐29

7 Random Sampling ................................7‐1

Background................................................................7‐1

The Random Sampling Window.................................7‐1

8 Navigating PWLPay  .............................8‐1

Navigation Options....................................................8‐1

The Navigation Tree...................................................8‐1

The Navigation Pane ..................................................8‐3

9 Reports................................................9‐1

Saving Reports ...........................................................9‐1

Deleting a Saved Report ............................................9‐3

10 Working with Charts ..........................10‐1

Printing, Saving, and Copying Charts .......................10‐1

Modifying Chart Properties .....................................10‐2

 

PWLPay User’s Manual    ii
Table of Contents Table of Contents

11 Computational Methods.................... 11‐1

Overview..................................................................11‐1

Distribution Calculator.............................................11‐1

Correlation...............................................................11‐5

Normality Testing ..................................................11‐15

 

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PWLPay User’s Manual    iv

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Forward Forward

Forward

PWLPay was developed at Clemson University under contract from the 

Federal Highway Administration Western Federal Lands Highway Division 

(FHWA-WFLHD). James L. Burati, Jr. was the principal investigator and 

provided all technical guidance. Thomas P. McAfee, Jr. performed the 

computer programming for PWLPay. Bruce E. Wasill was the technical 

representative for FHWA-WFLHD. 

PWLPay allows users to input any Percent Within Limits (PWL) acceptance 

plan, along with the corresponding quality characteristics and their test 

results. The software then allows the user to compute payment factors; to 

consider the potential impacts of various changes in the process mean, 

standard deviation, or both; and to develop and display control charts. 

PWLPay should provide a powerful tool for contractors to use to track their 

processes, develop control charts, determine payment factors, analyze risks, 

and make informed decisions necessary to maximize profit while providing 

agencies with the quality product they desire. The software will provide 

agencies that do not already have a materials database a simple way of 

collecting and tracking contractor and agency test data. 

PWLPay User’s Manual    v
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PWLPay User’s Manual    vi

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Chapter 1 Getting Started

1  Getting Started

Overview

PWLPay was developed at Clemson University under contract from the 

Federal Highway Administration Western Federal Lands Highway Division 

(FHWA-WFLHD). The software allows users to input any Percent Within 

Limits (PWL) acceptance plan, along with the corresponding quality 

characteristics and their test results. The software then allows the user to 

compute payment factors; to consider the potential impacts of various 

changes in the process mean, standard deviation, or both; and to develop and 

display control charts. 

PWLPay should provide a powerful tool for contractors to use to track their 

processes, develop control charts, determine payment factors, analyze risks, 

and make informed decisions necessary to maximize profit while providing 

agencies with the quality product they desire. The software will provide 

agencies that do not already have a materials database a simple way of 

collecting and tracking contractor and agency test data. 

PWLPay Capabilities PWLPay is not intended to allow development of a statistical acceptance plan, 

but rather to allow the determination of the risks associated with a particular 

acceptance plan. Other software, such as SpecRisk, is available from FHWA 

to assist in developing PWL acceptance plans. PWLPay is intended to be used 

as a tool to assist both owners and contractors in their quality assurance (QA) 

tasks, including both acceptance and process control. PWLPay allows the user 

to 

• Input PWL acceptance plans, including acceptable quality limits 

(AQL), rejectable quality limits (RQL), sample sizes (n), payment 

relationships, specification limits, etc. 

• Input test results for the contractor and agency. 

• Conduct verification tests, including F-tests and t-tests for both 

independent and paired data sets. 

• Determine descriptive statistics, including mean, standard deviation, 

skewness, and kurtosis, for the input test results. 

• Test the input data for normality using both graphical and statistical 

hypothesis tests. 

• Compute current payment factors. 

• Compute projected payment factors. 

• Develop and plot Run Charts, as well as Control Charts for 

individuals and moving ranges, and for sample means and ranges. 

• Play “what if games” to determine the potential impacts of changes 

in the process mean, standard deviation, or both. 

PWLPay User’s Manual    1‐1
Chapter 1 Getting Started

Typographical Conventions in this Manual

File ` Exit Denotes a menu command. In this case choose Exit from the 

File menu. Here is another example: File ` Open Project `

Show All Projects means open the File menu, then open the 

Open Project submenu, and finally choose Show All

Projects. 

Click OK. Bold text clarifies dialog box items and buttons and PWLPay

commands. 

Enter Test 1. Italic text specifies text the user needs to enter. 

Installing PWLPay

PWLPay is written to run on all versions of the Microsoft Windows operating 

system, including Windows 2000, XP, and Vista. Installing PWLPay is 

simple—just run PWLPay’s Setup program and follow its instructions. 

If Obtained Online If you downloaded PWLPay from a Website, you have probably already run 

PWLPay Setup. In case you have not, run the file you downloaded 

(SETUP.EXE) and follow the instructions. 

If Obtained on CD If you are installing PWLPay from a CD-ROM, PWLPay Setup will usually 

start automatically when you insert the CD-ROM into the drive. If it does 

not, run SETUP.EXE and follow the instructions. You can run it using the 

Run command on the Start menu, or from the My Documents window. 

Administrator You must have administrator rights to install PWLPay. Users whose 

Rights organizations have particularly strict security policies may need to have their 

information technology department revise the user’s permission rights for 

PWLPay.

Starting PWLPay

The Setup program installs a PWLPay shortcut on your Desktop. Run this 

shortcut to start PWLPay. The program opens to the Home screen. All of 

PWLPay’s capabilities can be accessed from this screen, including: 

Home

Setup Specs

Setup Project

Analysis 

Random Sampling 

PWLPay User’s Manual    1‐2
Chapter 1 Getting Started

Navigation Tree

Reports

These selections are discussed in the following chapters. 

The Process for Using PWLPay

The general steps for beginning to use PWLPay are shown below: 

1. Use Setup Specs to enter the acceptance characteristics and payment 

equations for the various items (e.g., PCC, hot mix asphalt) in the 

agency’s specifications. 

2. Use Setup Project to enter the items and their specification limits for 

a project. 

3. Use Random Sampling to determine random sample locations or 

times for a project. 

4. Add contractor acceptance and quality control test results, as well as 

any agency acceptance or verification tests for the project. 

5. Use Analysis to perform various analyses on the data for one project 

or multiple projects. These analyses could include, for example, 

calculation of basic statistics, checks for normality of the entered 

data, preparation of run charts or process control charts, statistical 

tests to verify contractor test results, and pay factor calculations. 

Detailed explanations of the features of PWLPay, as well as instructions on 

how to use each of these features, are presented in the following chapters of 

this manual. 

Format of this Manual

This manual is provided as a PDF file and it is likely that it will most often 

be used as an electronic file. For users who wish to print a copy, the manual 

is formatted for two-sided printing with a larger left margin to accommodate 

binding. If printed on only one side, even and odd pages will be slightly 

offset from one another. 

PWLPay User’s Manual    1‐3
Chapter 1 Getting Started

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Chapter 2 The Home Screen

2  The Home Screen

The Layout

PWLPay opens to a Home screen similar to that shown in Figure 2.1. The 

Recent Projects section that forms the majority of the screen shows the most 

recently created or edited projects. The first time PWLPay is opened, this area 

will be empty. 

Figure 2.1

Drop down menus for File, Tools, Navigate, and Help appear along the top of 

the Home screen. The Navigation Pane is the blue shaded region on the far 

right side of the Home screen in Figure 2.1. Note: PWLPay is designed for 

use with the Windows default blue Appearance under Display Properties. It 

is recommended that PWLPay be used with these default settings. 

The File Menu

The options accessible from the File menu are shown in Figure 2.2. 

Figure 2.2 

From the File submenu the user can display projects, e.g., File ` Open Project

` Show All Projects, that can then be selected for editing or data entry. The 

user can also Exit the program from this menu. 

PWLPay User’s Manual    2‐1
Chapter 2 The Home Screen

The Tools Menu

The options accessible from the Tools menu are shown in Figure 2.3. 

Figure 2.3 

On the Tools menu, Database Options opens a submenu with 4 options. 

Locate Database opens a window (see Figure 2.4) that allows the user to 

browse for and then load an existing database. Save Copy of Database

allows the user to save a copy of the current database. Clear Database

empties the contents of the current database, leaving a blank database. Load 

Sample Database loads the sample database that was used to create the 

figures in this user’s manual. 

Figure 2.4 

Normality Check on the Tools menu opens a submenu (see Figure 2.5). 

Figure 2.5 

The submenu under Normality Check provides the option of selecting Show

Normality Check (selected in Figure 2.5) or Hide Normality Check. This 

determines whether or not the program automatically checks data as they are 

entered and warns the user when the data appear to have not come from a 

normal distribution. This warning is explained in detail later in this manual. 

PWLPay User’s Manual    2‐2
Chapter 2 The Home Screen

Highlighting Alpha Value opens another submenu that allows the user to 

select the alpha value (α) that the program uses when performing the 

automatic test for the normality of the input data. In Figure 2.5 the default 

value, α = 0.05, is selected. 

Tools ` Computation Routines is discussed in detail later in this manual. 

The Navigate Menu

The options accessible from the Navigate menu are shown in Figure 2.6. 

Figure 2.6 

The options on the Navigate menu include Home, which takes the user to the 

Home screen, and Open Navigation Tree. The Navigation Tree is discussed 

in detail later in this manual. 

Help

Clicking on Help (see) opens the submenu shown in Figure 2.7. PWL Pay

Manual opens a copy of this user’s manual as a PDF file in another window. 

About opens the window shown in Figure 2.8. 

Figure 2.7 

Figure 2.8 

Recent Projects

The first time PWLPay is opened, the Recent Projects section will be empty. 

As projects are entered into the program, this area shows the most recently 

created or edited projects, as shown in the example screen in Figure 2.9. 

PWLPay User’s Manual    2‐3
Chapter 2 The Home Screen

Figure 2.9

The projects are shown in order of the most recent date they were modified. 

The projects are created by selecting the Setup Project option. The program 

will not allow the Setup Project option to be selected until at least one 

specification has been input to the program using the Setup Specs option. 

In Figure 2.9, eight projects have been created. Either left or right clicking 

the mouse on a project opens the menu shown in Figure 2.10. 

Figure 2.10 

Edit Project opens a window that allows changes to be made to the 

information, such as item numbers and specification limits, associated with 

the project. Add Test Results opens a window that allows test data to be 

input or edited. These windows are discussed in detail later in this manual. 

The user can go directly to the test result window by clicking the blue Add

Test Result button associated with each project. Open Project and Delete

Project are self-explanatory. 

PWLPay User’s Manual    2‐4
Chapter 2 The Home Screen

The Navigation Pane

The Navigation pane is the blue shaded region on the far right side of Figure

2.8. It shows the most recent locations that have been visited in the program. 

The Navigation pane can be used to go directly to any of these locations by 

clicking the mouse on that location. 

Miscellaneous Items

The user can create a new or edit an existing project directly by clicking on 

the green Create New / Edit Project link (see Figure 2.8) on the Home page. 

The storage location of the current database file is shown in the lower left 

hand corner of the Home screen. 

If the computer has Internet access, clicking on the FHWA logo in the upper 

right corner opens the FHWA website in a new browser window, while the 

Search box in the upper right corner allows the user to perform a key word 

search of the FHWA website. 

PWLPay User’s Manual    2‐5
Chapter 2 The Home Screen

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Chapter 3 Setting Up Specs

3  Setting Up Specs

The Empty Setup Specs Window

The first thing that must be done to use PWLPay is to setup one or more 

specifications. The first time that the Setup Specs icon is selected the 

window in Figure 3.1 is displayed. 

Figure 3.1

Select a State / Agency

The user must first select or input the state and/or agency to which the 

specification applies. The user selects Edit List to open a window with a list 

of states and agencies from which to choose. The user can also add to the list 

an agency that is not currently included. Any number of states or agencies 

can be added at this time. Figure 3.2 shows an example of the window with 

the selection list. In Figure 3.2, two states and one Federal agency have been 

selected. The Western Federal Lands agency was added by entering the 

name into the Add State / Agency box and selecting Add. 

When Done is selected the program returns to the Setup Specs window as 

shown in Figure 3.3. 

PWLPay User’s Manual    3‐1
Chapter 3 Setting Up Specs

Figure 3.2 

Figure 3.3

Item Numbers

Next, select Add New Item Number to open the Add Item Number window 

shown in Figure 3.4 to enter the item number to which the specification 

applies. 

PWLPay User’s Manual    3‐2
Chapter 3 Setting Up Specs

Figure 3.4 

Characteristics

    The user can select any characteristics from the Existing Characteristics box 

to associate with this item number. If the Existing Characteristics box does 

not contain the necessary characteristics for the item number, then the user 

can select Add Characteristic to open the window in Figure 3.5 and enter the 

required quality characteristic and information to be associated with this item 

number. Any number of characteristics can be associated with a given item 

number. Note that the user can also add characteristics from the Setup Specs

window by clicking Add New Characteristic (see Figure 3.3). 

Figure 3.5 

Spec Type Since different characteristics have different specification requirements, 

selecting Spec Type: opens a drop down menu that allows the user to select 

the type of specification that is appropriate for the selected quality 

characteristic. As shown in Figure 3.6, there are five different specification 

types from which to choose. 

PWLPay User’s Manual    3‐3
Chapter 3 Setting Up Specs

Figure 3.6 

Target Value ± Tolerance. This applies to a characteristic such as asphalt 

content where there is a target value with allowable tolerances both above 

and below the target value. 

Minimum Limit Only. This applies to a characteristic such as concrete 

compressive strength where there is only a lower specification limit to be 

met. 

Maximum Limit Only. This applies in the case of a characteristic such as 

percent passing the 0.075 mm (No. 200) sieve for which there is only an 

upper specification limit to be met. 

Bounded Mean ± Tolerance. This might apply in the case of a characteristic 

such as aggregate gradation where there is a target value and allowable 

tolerances above and below the target, but there might also be a master band 

that cannot be exceeded by the allowable tolerances. For example, there 

might be a characteristic for which the target was 75 with ± tolerances of 10. 

But, if there is a master band that requires specification limits to be between 

60 and 80, then the allowable lower specification limit would be 75 – 10 =

65, while the allowable upper specification limit would not be 75 + 10 = 85, 

but would be limited by the master band to be 80. 

Minimum and Maximum Limit. This might apply to a case where there is 

not a specific target value, but where there are stipulated lower and upper 

specification limits. For example, the specification limits for air content of 

portland cement concrete might be 3% to 5%. 

Pay Information

    In most cases a pay relationship will be assigned to the quality characteristic, 

and this relationship can be selected from the list in the Equations: box, or it 

can be entered by selecting Add New Pay Relationship. In some cases, such 

as if the lot payment factor is to be determined by an expected life approach, 

there is no pay relationship needed and the This characteristic requires an 

equation box can be unchecked (see Figure 3.5). 

PWLPay User’s Manual    3‐4
Chapter 3 Setting Up Specs

Selecting Add New Pay Relationship allows the user to select from two 

options for the pay relationship: Tabular Pay Factors or Pay Equations(s)

(see Figure 3.7). 

Figure 3.7 

Tabular Pay Factors If Tabular Pay Factors is selected, the Add Tabular Equation window shown 

in Figure 3.8 is opened so that the pay relationship can be entered. 

Figure 3.8

PWLPay User’s Manual    3‐5
Chapter 3 Setting Up Specs

The user can enter a name for the pay relationship in the Tabular Equation 

Name: box. The user can then either import a CSV file with the pay 

relationship information, or create the payment table with the options in this 

window. 

Entering a Pay The Number of Pay Columns box can be increased to account for the total 

Relationship number of sample sizes for which there are pay factors in the table, and the 

Edit item range for selected column can be selected to enter the range of 

sample sizes for which each pay factor applies. Each time the number of 

columns is increased, the window in Figure 3.9 pops up to allow the user to 

enter the sample size range for the new column. The example in Figure 3.9

shows that the column will include the pay factors for a sample size of n = 3. 

Figure 3.9 

Figure 3.10 shows the input values for an example pay relationship table. 

Clicking on Save saves the pay relationship under the name Example Pay 

Table 3-10.

Figure 3.10

PWLPay User’s Manual    3‐6
Chapter 3 Setting Up Specs

Importing a Pay Clicking Create the import file for me (see Figure 3.8) brings up the 

Relationship window shown in Figure 3.11, which can be used to create a blank CSV file 

that can then be used to import the pay relationship. 

Figure 3.11 

The Minimum Range and Maximum Range columns are used to input the 

range of sample sizes for which the pay factor in the column will apply. The 

same sample size ranges used in Figure 3.10 have been entered in the 

example window in Figure 3.12. 

Figure 3.12 

Clicking Continue will create a CSV file and, if the Open file after it has 

been created box is checked, the file will be opened in Excel. A portion of 

the CSV file for Figure 3.12 is shown in Figure 3.13. The PWL column runs 

from 100 down to 0, and a corresponding pay factor must be entered for each 

row and column in the spreadsheet. 

PWLPay User’s Manual    3‐7
Chapter 3 Setting Up Specs

Figure 3.13

Pay Equation(s) If Pay Equation(s) is selected in the Add Characteristic window (see Figure

3.7), the Add / Edit Payment Equation(s) window (see Figure 3.14) is opened. 

This allows the user a wide range of options for entering a continuous pay 

relationship based on one to four separate equations, and allows the user to 

input different pay relationships for differing sample sizes. 

Figure 3.14

PWLPay User’s Manual    3‐8
Chapter 3 Setting Up Specs

The user first enters a name for the equation(s). In Figure 3.15, Example 

Equation Figure 3-15 has been entered in the Equation Name: box. The 

Select the number of equations: drop down menu allows the user to select up 

to four equations to represent the continuous pay relationship. Each equation 

can be either linear, a second order curve, or a third order curve. The 

equation at the top of Figure 3.15 defines the input coefficients for the 

equations. 

Figure 3.15

Note that by default the input equation(s) apply for sample sizes For n = 1 to 

200. These values can be changed by entering a new range in the Please 

enter the sample size(s) this equation represents: boxes. If the user wishes to 

enter different pay equations for different sample sizes, this can be 

accomplished by clicking the Add Tab button, which creates a new equation 

input screen and allows the user to enter the range of sample sizes for which 

the equation(s) apply. In Figure 3.16, three sample ranges, 3 to 10, 11 to 30, 

and 31 to 200 have been entered. The sample size range from 31 to 200 has 

one equation for the pay relationship. 

Figure 3.16

Once the number of equations has been selected the coefficients for each 

equation must be entered. Figure 3.17 shows a pay relationship based on a 

single linear equation that applies to sample sizes from 3 to 10. 

Figure 3.17

PWLPay User’s Manual    3‐9
Chapter 3 Setting Up Specs

Note that coefficients can be input for only the selected number of equations. 

For example, in Figure 3.17 only the coefficients for the first equation can be 

input. The other coefficient boxes are inactive. For the example in Figure

3.17, B0 = 55 and B1 = 0.5. Therefore, the pay relationship is represented by 

the equation Pay = 0.5PWL + 55. Also note that with a single equation the 

PWL Range: from 0 to box automatically has a value of 100 and requires no 

input from the user. 

A plot of the pay equation(s) can be shown by clicking the Graph button. 

Clicking on Graph in Figure 3.17 provides the plot shown in Figure 3.18. 

Figure 3.18

Figure 3.19 shows a pay relationship based on two equations–one straight 

line (which is really a constant in this example) and one second order curve. 

Since more than one equation is used, it is necessary to input the range of 

estimated PWL values for which each equation applies. The first equation 

applies for PWL values from 0 to 50, while the second equation applies to 

PWL values from 50 to 100. The program automatically checks to see if the 

equations give consistent values at their points of intersection. If the pay 

factors at the intersection for the two equations differ by more than 0.01, a 

warning message such as the one in Figure 3.20 is displayed. Note that in 

Figure 3.20 the two equations differ in payment by 0.76 at their intersection. 

At this point the higher pay factor would be used. 

The plot for the pay relationship in Figure 3.19 is shown in Figure 3.21. 

PWLPay User’s Manual    3‐10
Chapter 3 Setting Up Specs

Figure 3.19

Figure 3.20

Figure 3.21

PWLPay User’s Manual    3‐11
Chapter 3 Setting Up Specs

Figure 3.22 shows a pay relationship based on four straight line equations. 

The plot for this pay relationship is shown in Figure 3.23. 

Figure 3.22

Figure 3.23

Clicking on Save in either the Add / Edit Payment Equation(s) window or the 

Add Tabular Equation window returns the user to the Add Item Number 

window in Figure 3.24. 

PWLPay User’s Manual    3‐12
Chapter 3 Setting Up Specs

Figure 3.24

Composite Any additional characteristics to be associated with the item number can be 

Pay Factor entered. All of the input characteristics will appear in the Characteristics 

belonging to this item: window. For those characteristics to be used in the 

determination of the lot pay factor, highlighting them and clicking on Add>>

will add them to the Characteristics used in the Composite Pay Factor

window (see Figure 3.25). 

Once the characteristics for the composite pay factor have been entered, the 

user must select one of the options from the Select a Composite Pay 

Equation: drop down menu. 

Figure 3.25

The possible methods for combining the individual characteristics’ pay 

factors into a single composite pay factor for the lot (see Figure 3.25) include 

the following: 

PWLPay User’s Manual    3‐13
Chapter 3 Setting Up Specs

Weighted Average. This method assigns a weighting factor to each 

characteristic and then sums the products of the individual weighting factors 

times the pay factors for the individual characteristics. 

Minimum Pay Factor. This method assigns the lot pay factor as the 

minimum of the pay factors for the individual characteristics. 

Maximum Pay Factor. This method assigns the lot pay factor as the 

maximum of the pay factors for the individual characteristics. 

Multiplied Individual Pay Factors. This method multiplies the pay factors 

(expressed as decimals rather than percents) for the individual characteristics. 

Sum of Price Adjustments. This method uses the arithmetic sum of the 

differences from 100 for each of the individual pay factors. 

Expected Life Pay Factor. This method is based on estimating the expected 

life of the as-built product and using the difference between the design life 

and the expected as-built life to determine the appropriate pay factor for the 

lot. Due to its uniqueness compared with the other methods, the expected life 

approach is covered in a separate section at the end of this chapter. 

Composite Weights If the Minimum, Maximum, Multiplied, or Sum composite pay factor is 

selected, then no additional input is required. However, if the Weighted 

Average method is selected to determine the composite pay factor, then the 

weights must be entered for each of the characteristics in the composite pay 

equation. 

In Figure 3.26, the Weighted Average approach has been selected. Also note 

that the Allow me to fill in the equation information when I click Save box is 

checked by default. When this box is checked, clicking Save will save the 

Add Item Number information and also call up the window in Figure 3.27 to 

allow input of the weighting factors for each characteristic. 

Figure 3.26

PWLPay User’s Manual    3‐14
Chapter 3 Setting Up Specs

In Figure 3.27, weights have been entered for the two characteristics 

associated with this item number. The sum of the weights must be 1.0, and 

the program will give a warning message if the user tries to save a 

combination of weights that does not sum to 1. 

Figure 3.27 

Max / Min The user also has the option of entering maximum and minimum pay factors 

Pay Factors in the OPTIONAL Information: region of the Add Item Number window (see 

Figure 3.26). If entered, these values would override the composite pay factor 

in the event that the pay relationship produced a pay factor that either 

exceeded the value entered for Maximum pay factor (cap):, or was less than 

the value entered for the Minimum pay factor:. 

The Completed Setup Specs Window

When the item number information is saved the user is returned to the Setup 

Specs window in Figure 3.28. Note that when a state or agency is 

highlighted in the Select a State / Agency window, all of the items associated 

with that state or agency are visible in the Item Numbers window. Similarly, 

when one of the items is highlighted, all of the characteristics associated with 

that item are visible in the Characteristics window. Similarly, when one of 

the characteristics is highlighted, the pay relationship associated with that 

characteristic is highlighted in the PWL Equations window and a plot of the 

pay relationship is shown in the window below the PWL Equations window. 

For example, in Figure 3.28, South Carolina has four items associated with it; 

item number P-401 has four characteristics associated with it; the pay 

equation AASHTO is associated with Asphalt Content, and the pay 

relationship is shown as a straight line in the plot. If a different characteristic 

were highlighted a different pay relationship and plot might be highlighted 

and displayed. 

The order in which the characteristics are displayed in the Characteristics

window, which is also the order in which any output regarding the 

PWLPay User’s Manual    3‐15
Chapter 3 Setting Up Specs

characteristics will be displayed, can be changed by clicking on the Up or 

Down buttons under Sort Order. Checking the Auto Sort box sorts the 

characteristics in alphabetical order. 

Figure 3.28

If, for a characteristic, different pay relationships apply for different sample 

sizes (see, for example, Figure 3.16), then there will be multiple lines in the 

Pay Factor vs. PWL plot. For example, in Figure 3.29 there are three pay 

relationships associated with different sample sizes for the characteristic 

Performance Index. 

Figure 3.29

PWLPay User’s Manual    3‐16
Chapter 3 Setting Up Specs

To determine which line applies to which sample size or range of sample 

sizes, the user can hover the mouse pointer over any portion of a line to 

display a popup with information concerning the sample sizes to which the 

line applies as well as the specific values for PWL and Pay Factor where the 

pointer rests on the line. This is seen in Figure 3.30 where it shows that the 

selected line applies to sample sizes in the Range 4 to 6 and that when 

PWL = 80 the Pay Factor = 87. 

Figure 3.30

The Expected Life Approach

In the Select a Composite Pay Equation: drop down menu shown in Figure

3.25 the user can select from six methods for determining the composite pay 

factor for a lot. The first five are discussed previously in this chapter. The 

sixth method, the Expected Life Pay Factor, is quite different in concept from 

the first five. It is discussed in depth below. 

A thorough discussion of the expected life approach for determining payment 

for a lot is well beyond the scope of this manual. It is, therefore, assumed that 

the user of PWLPay has a thorough understanding of this approach. As such, 

this manual covers only the implementation of the expected life approach 

using PWLPay. 

A thorough description and detailed discussion of the expected life approach 

are presented in Optimal Procedures for Quality Assurance Specificatons,

Publication No. FHWA-RD-02-095, April 2003. Reference

PWLPay User’s Manual    3‐17
Chapter 3 Setting Up Specs

Figure 3.31 shows the Setup Specs screen with Item P-501 highlighted in the 

Item Numbers window. Compressive Strength is highlighted in the 

Characteristics window. Note that in the PWL Equations section, for 

Compressive Strength the Equations: box has No Equation highlighted. An 

item for which the pay factor is determined by the expected life approach 

will have no pay equation associated with its characteristics. 

Figure 3.31

Expected Life Figure 3.32 shows the completed Add Item Number input screen for 

Input Procedures Item P-501. Compressive Strength and Thickness have been selected as the 

Characteristics used in Composite Pay Factor. Note that Expected Life Pay

Factor has been selected as the composite pay equation. 

Figure 3.32 

PWLPay User’s Manual    3‐18
Chapter 3 Setting Up Specs

Figure 3.33 shows the completed Add Characteristic input screen for 

Compressive Strength as it was entered for use with an expected life 

approach for pay factor determination. Note that the This characteristic 

requires an equation box is not checked. As a result, the Equations: window 

is grayed out and no equation can be assigned to this characteristic. This is 

the case for any characteristic that is used in determining an expected life pay 

factor. 

Figure 3.33 

When Save is selected in the window in Figure 3.32 the Select An Expected 

Life Equation window shown in Figure 3.34 is displayed. This window 

allows the user to enter the equation that estimates the expected life for the 

pavement based on the PWL values for the various characteristics. 

Figure 3.34 

The general form for the expected life equation, as shown at the top of Figure

3.34, is 

݌ݔܧ݁ܿݐ݁݀ ܮ݂݅݁ ൌ ݁஻బା஻భሺଵ଴଴ି௉ௐ௅భሻೄା஻మሺଵ଴଴ି௉ௐ௅మሻೄାڮା஻ఱሺଵ଴଴ି௉ௐ௅ఱሻೄ

In the above equation, S is a shape factor that defaults to 1 unless the Include

Shape Factor box is checked. 

PWLPay User’s Manual    3‐19
Chapter 3 Setting Up Specs

Pay Adjustment Once the expected life equation has been entered the user can select from one 

Input Procedures of three forms for the pay adjustment equation (see the bottom of Figure

3.34). If the first button is selected as in Figure 3.34, selecting Continue

opens the Pay Adjustment Equation window shown in Figure 3.35 and allows 

the user to input the coefficients for the pay adjustment equation. 

Figure 3.35 

In Figure 3.35, the values for A:, B:, and C:, the coefficients of the pay 

adjustment equation, are entered by the user. PWLPay automatically displays 

the value for E:, which is obtained from the expected life equation when the 

PWL is 100 for all characteristics in the equation. 

Selecting Show Graph displays the plot of the pay adjustment equation 

shown in Figure 3.36. As with all other charts in PWLPay, the user can hover 

the cursor over any point on the line to display the expected life and pay 

adjustment values. The pay adjustment is in dollars per unit, and can be 

either positive or negative. 

Figure 3.36 

PWLPay User’s Manual    3‐20
Chapter 3 Setting Up Specs

If the second button is selected in Figure 3.34, selecting Continue opens the 

Pay Equation window shown in Figure 3.37 and allows the user to input the 

necessary information for the pay adjustment equation. 

Figure 3.37 

The input values for the pay adjustment equation are relatively selfexplanatory. A thorough description and detailed discussion of these values 

are presented in Optimal Procedures for Quality Assurance Specificatons,

Publication No. FHWA-RD-02-095, April 2003. 

The Pay Adjustment Calculation Tool section at the bottom of Figure 3.37

can be used to determine the pay adjustment, in terms of dollars per unit or as 

a percent pay factor, for any value entered into the Expected Life: box. In 

Figure 3.38, an expected life of 9.5 years has been entered and once 

Calculate is selected the corresponding pay adjustment and pay factor values 

are shown. 

Figure 3.38 

Reference

Selecting Show Graph displays the menu shown in Figure 3.39, which allows 

the user to select to Graph Expected Life vs. Pay Factor or Graph Expected 

Life vs. Pay Adjustment. Figure 3.40 shows the plot of expected life vs. pay 

factor for the example shown in Figure 3.37. 

Figure 3.39 

PWLPay User’s Manual    3‐21
Chapter 3 Setting Up Specs

Figure 3.40 

If the third button is selected in Figure 3.34, selecting Continue opens the 

Add / Edit Pay Adjustment Equation(s) window shown in Figure 3.41 and 

allows the user to input the necessary information for the pay adjustment 

equation. The input procedures for this window are similar to those presented 

with the discussion of Figure 3.14, except that (x) is now expected life. 

Figure 3.41 

Selecting Graph in Figure 3.41 displays the pay adjustment equation plot 

shown in Figure 3.42. 

PWLPay User’s Manual    3‐22
Chapter 3 Setting Up Specs

Figure 3.42 

PWLPay User’s Manual    3‐23
Chapter 3 Setting Up Specs

PWLPay User’s Manual    3‐24

(This page is intentionally blank)
Chapter 4 Setting Up Projects

4  Setting Up Projects

The Setup Project Window

Once one or more specifications have been setup, clicking the Setup Project

icon opens the screen shown in Figure 4.1. Any states or agencies that have 

been input using the Setup Specs option are displayed in the Select a State / 

Agency to continue window. If any projects have been setup previously and 

associated with a state or agency they are displayed in the Select the contract 

number window when that state or agency is highlighted. 

Figure 4.1

Adding a Project

The user must first select the state or agency for which the new project 

applies. For example, in Figure 4.1 South Carolina is selected as the agency 

and, since the Select the contract number window is empty, no projects have 

yet been entered for this agency. 

 Contract Number Clicking on Add New Contract Number opens the window shown in Figure

4.2 that allows the user to enter a contract number for the project to be added. 

PWLPay User’s Manual    4‐1
Chapter 4 Setting Up Projects

Figure 4.2 

Project Name When Save is selected the program returns to the Setup Project window as 

and Number shown in Figure 4.3. The new contract number is highlighted and new 

windows are displayed from which project names and numbers can be 

assigned to the highlighted contract number. 

Figure 4.3

The Select the contract number window displays all contract numbers that 

have previously been assigned to the state or agency highlighted in the Select 

a State / Agency to continue window. Similarly, the Select a Project Name / 

Project Number and click the Continue button window displays the project 

name(s) and number(s) that are part of the highlighted contract number in the 

Select the contract number window. Figure 4.4 shows the contract numbers 

associated with the state South Carolina and the project names, Clemson

Bypass and Clemson Bypass Phase II, and numbers, District-3-005-1 and 

District-3-005-2, associated with contract number SC-2008-001. 

PWLPay User’s Manual    4‐2
Chapter 4 Setting Up Projects

Figure 4.4

Clicking Add New Project opens the Project Setup window shown in Figure

4.5. In this figure, the user has entered Clemson Bypass Phase II as the 

Project Name and District-3-005-2 as the Project Number for the new 

project. Also note that all of the item numbers that have been associated with 

South Carolina are displayed in the Items in South Carolina Not Added to this 

Project window. 

Figure 4.5

PWLPay User’s Manual    4‐3
Chapter 4 Setting Up Projects

Adding Project Details

Items The Project Setup window, shown in Figure 4.5, allows the user to input 

project-specific information regarding the type of project, as represented by 

the associated item numbers, and the specification limits for the project. The 

user highlights an item number and then uses the Add >> button to select it 

for the project. Any number of item numbers can be assigned to a given 

project. 

In Figure 4.6, items P-401 and P-501 have been added to the project as 

shown in the Items belonging to this Project window. The characteristics 

associated with the highlighted item are shown in the box in the lower left 

corner of the Project Setup window. Figure 4.6 shows that the characteristics 

Asphalt Content, Air Voids, VMA, and Density are associated with item 

P-401. 

Figure 4.6

Spec Limits The Project Setup window is also used to enter the specification limits for 

the various characteristics associated with each of the item numbers 

associated with the selected project. In Figure 4.6, the Characteristic’s 

Equation Information box displays detailed information regarding the 

specification requirements for the characteristic that is highlighted in the box 

in the lower left corner of the window. This is also where the specification 

limits for the characteristic are entered for the selected project. 

Figure 4.6 shows that the characteristic Asphalt Content uses the pay 

equation named AASHTO, that the pay equation type is Multiple Equation (as 

opposed to using Expected Life), and that the specification has a target value 

with ± tolerance limits. The user must enter the specification limits for all 

PWLPay User’s Manual    4‐4
Chapter 4 Setting Up Projects

characteristics before the project setup information can be saved. If the Save

button is clicked before all limits have been entered a message such as that 

shown in Figure 4.7 is displayed. 

Figure 4.7 

Figure 4.8, which is the bottom portion of the Project Setup window, shows 

that for the characteristic Density, the pay equation is Curve-02 and the 

specification has both a minimum and maximum limit, with a Minimum Limit

of 92.0 and a Maximum Limit of 96.5. 

Figure 4.8 

Once all of the specification limits have been entered, clicking on the Save

button saves the information in the database and returns the user to the Setup

Project window. Note in Figure 4.9 that the newly added project name, 

Clemson Bypass Phase II, and project number, District-3-005-2, are shown in 

the Select a Project Name / Project Number and click the Continue button

window. Clicking on Continue takes the user to the data input screen for the 

selected project so that test results can be entered. 

PWLPay User’s Manual    4‐5
Chapter 4 Setting Up Projects

PWLPay User’s Manual    4‐6

Figure 4.9
Chapter 5 Entering Test Results

5  Entering Test Results

The Add Samples Window

The Add Samples window is used to input all test result data for a selected 

project. Figure 5.1 shows the Add Samples window for a project before any 

test result data have been entered. Project idenification information, 

including project name, project number, specification type, and specification 

limits, is provided at the top of the screen. There are also two drop down 

menus in this region of the window: one for Item Number: and one for 

Characteristic:. 

Figure 5.1

Clicking on the project name, Clemson Bypass in Figure 5.1, opens the 

Project Setup window for that project. Clicking on the Item Number: drop 

down menu displays a selection list of all item numbers associated with this 

project (see Figure 5.2a). Similarly, clicking on the Characteristic: drop 

down menu displays a selection of all characteristics associated with the 

selected item number (see Figure 5.2b). 

Figure 5.2 (a) (b) 

 

PWLPay User’s Manual    5‐1
Chapter 5 Entering Test Results

Opening the There are a number of different ways that the user can open the Add Samples 

Add Samples window to allow input of test result data for a selected project. As shown 

Window and noted in Chapter 4, clicking on Continue in the Setup Project window 

leads to the Add Samples window for the highlighted project (see Figure

4.9). 

The Add Samples window can also be opened two different ways from the 

Home screen. Either left or right clicking on the name (Clemson Bypass in 

Figure 5.3) of one of the Recent Projects opens the dropdown menu shown 

in Figure 5.3. Edit Project opens the Project Setup window. Add Test Results

opens the Add Samples window. Open Project ` Show All Projects or Open

Project ` Show Projects Not on Home Page allows the user to show 

additional projects not shown in the Recent Projects section. 

Figure 5.3 

The Add Samples window can also be opened directly for a project in the 

Recent Projects list by clicking on the Add Test Result button at the end of 

the row in which the project appears (see Figure 5.4). 

Figure 5.4

Entering Test Results

Test result data can be entered directly into the section of the Add Samples

screen that has the appearance of a spreadsheet (see Figure 5.1). Test result 

data can also be imported from a CSV (comma separated variables) file that 

was most likely created using a spreadsheet program. 

 Data Columns Test result data can be entered in the Add Samples screen as they would be 

entered into a typical spreadsheet. Figure 5.1 shows some of the column 

headings for which data can be entered for a project. 

PWLPay User’s Manual    5‐2
Chapter 5 Entering Test Results

The columns for which data can be entered include: 

Lot Number: alpha-numeric lot number to which the test result is 

associated. 

Sample No.: alpha-numeric sample number within the lot. 

Date: can be entered in any typically accepted format, e.g., 2-4-07, or 

02/04/07, or 2/4/2007, etc. The program automatically converts the date to 

a standard format. 

Test Result: test result for the lot/sample number combination. 

Contr/Owner: indicates whether the test was performed by contractor or 

owner personnel.

Lab Name: name of the laboratory where the test was performed. 

Tester ID: alpha-numeric identification number for the technician who 

performed the test. 

Split Sample No.: alpha-numeric identification number for the owner’s 

portion if split samples are used. 

Split Test Result: test result for the owner’s split sample for the lot/sample 

combination. 

SplitLab Name: name of the laboratory where the split sample was tested. 

Split Tester ID: alpha-numeric identification number for the technician who 

performed the split sample test. 

Remarks: whatever comments or additional information the user wishes to 

add. 

Due to the way in which PWLPay and spreadsheet programs sort data, it is 

important that care be taken when assigning and entering lot and sample 

numbers. Since lot and sample numbers may have letters as well as numbers, 

to be sorted properly, they must be recognized as “text” by either PWLPay or 

a spreadsheet program. To ensure that this happens without having to 

remember to declare the cells as text, it is recommended that the user always 

include a letter as well as numbers when assigning lot and sample numbers. 

For example, it is preferable to enter L001, L-001, or Lot-001 rather than 001 

for the lot number. 

Important Note on 

Lot Numbers & 

Sample Numbers 

It is also important that lot and sample numbers are entered with a consistent 

number of places. For example, if lot numbers were entered as L-1, L-2, L-3, 

L-4, L-5, L-6, L-7, L-8, L-9, L-10, L-11, when sorted by lot number the 

order of display would be L-1, L-10, L-11, L-2, L-3, L-4, L-5, L-6, L-7, L-8, 

L-9. This problem can be avoided simply by entering leading zeros so that all 

lots or samples have the same number of values. In the previous example, the 

lots would be displayed in the proper order if the numbers were entered as 

L-01, L-02, L-03, L-04, L-05, L-06, L-07, L-08, L-09, L-10, L-11. 

PWLPay User’s Manual    5‐3
Chapter 5 Entering Test Results

Entering Data Test information and results are entered as in a typical spreadsheet program. 

Figure 5.5 shows the spreadsheet format data entry section of the Add

Samples window before any test result information has been added for the 

characteristic Asphalt Content. 

Figure 5.5

Figure 5.6 shows the same section of the Add Samples screen while data 

entry is underway for the first sample for the project. Once any values are 

input, the Save Table button (see Figure 5.5) changes to the Save Needed

button to remind the user to save the input data after entry (see Figure 5.6). 

Note that the input data will be lost if not saved before exiting from the 

current characteristic. If the user tries to change characteristics before saving 

the input data, a warning window pops up (see Figure 5.7). 

Figure 5.6

Figure 5.7

The Date format in Figure 5.7 has been automatically changed from the value 

10-5-08 (see Figure 5.6) that was input to the standard format used by 

PWLPay.

Figure 5.8

The Contr/Owner column (see Figure 5.8) indicates which party conducted 

the test and has a limited number of input values. As shown in the NOTE

above the spreadsheet section, these allowable values are entered initially by 

clicking on the symbol. This displays the drop down window shown in 

Figure 5.9a. Two possible input values have been entered, Contractor and 

Owner. Clicking Ok sets these as the only two possible values. The cell will 

PWLPay User’s Manual    5‐4
Chapter 5 Entering Test Results

now auto-complete. That is, for example, entering c will cause Contractor to 

be the input value. 

Once the allowable values have been entered, clicking on the drop down 

arrow, , displays the input options for this column, i.e., Contractor or 

Owner in Figure 5.9b. 

Figure 5.9 (a) (b) 

Similarly, the Lab Name and Tester ID columns are limited to values that 

have been input in the same manner. See, for example, Lab Name in Figure

5.10. Names for all labs in the state could be entered all at one time and in 

such a way that allows the auto-complete function to save time. For example, 

if the state had seven district labs, rather than entering them as District-1, 

District-2, etc., it would be better to enter them as 1-District, 2-District, etc. 

In the first case, entering D would not allow auto-completion since seven 

labs begin with D. However, in the second case, entering 1 would autocomplete 1-District (provided there were no other lab names beginning with 

1). 

Figure 5.10

The SplitLab Name and Split Tester ID columns work in similar fashion to 

the Lab Name and Tester ID columns. 

PWLPay User’s Manual    5‐5
Chapter 5 Entering Test Results

Importing Data Test information can also be imported in the form of a CSV file. In this way 

a spreadsheet could be used to maintain the data and then the file could be 

saved as a CSV file and imported into PWLPay. This would also allow 

existing data, with some adjustments in columns and column titles, to be 

quickly and easily entered into PWLPay. 

Right clicking the mouse anywhere in the Add Samples window opens the 

drop down menu shown in Figure 5.11. Only the Import Data option is 

considered here. The other choices on this menu are covered later in this 

chapter. 

Selecting Import Data opens the Import Test Results window shown in Figure

5.12, which allows several options for importing data from an existing CSV 

file. The user can browse to locate the file to be imported. The default for 

Import Data Format: is All Data. The dropdown menu in Figure 5.12 shows 

all of the import options. 

Figure 5.11

PWLPay User’s Manual    5‐6
Chapter 5 Entering Test Results

Figure 5.12 

The Data will be imported for: window, shown in Figure 5.13, indicates that 

All Data allows for input of data from multiple states as well as multiple 

contracts, multiple projects, multiple items, and multiple characteristics for 

each project. This import format probably would not be used often. A 

different import format can be selected by clicking to open the drop down 

menu shown in Figure 5.12. 

Figure 5.13 

Note that the CSV File Input format for selected import type: section lists the 

columns for which import data are needed based on whichever Import Data 

Format: is selected. For example, the column input titles for All Data are 

shown in Figure 5.13. 

PWLPay User’s Manual    5‐7
Chapter 5 Entering Test Results

Figure 5.14 shows the column input titles when Project is selected for the 

Import Data Format: option. Comparing Figures 5.13 and 5.14 shows the 

smaller number of input columns required for the Project import format. The 

Data will be imported for: window shows that no input is necessary for State, 

Contract Number, Project Name, or Project Number since these are 

automatically input as shown based on the Project format that is selected. 

At this point the user can create a CSV file using the column headings shown 

in Figure 5.14, although this can be accomplished more easily by clicking on 

Create a blank file with these columns for me. Doing this opens Excel, if it is 

on the user’s computer, with a CSV file open with the required column 

headings. A sample with the column headings for the CSV file created by 

clicking Create a blank file with these columns for me from Figure 5.14 is 

shown in Figure 5.15. 

Figure 5.14 

Figure 5.15

The user can then either directly input data into the CSV file in Figure 5.15, 

or else import data from an existing spreadsheet into the appropriate columns 

in the CSV file. Figure 5.16 shows a portion of the CSV file from Figure 5.15

after project data have been entered. 

Figure 5.16

PWLPay User’s Manual    5‐8
Chapter 5 Entering Test Results

In Figure 5.16 note that the lot and sample numbers all contain at least one 

letter. This avoids any potential problems with Excel recognizing the input as 

numeric rather than alpha-numeric (i.e., Excel treats the cell as “Text”). As 

noted previously, entering these values as text allows for proper sorting when 

displayed by PWLPay. 

Important Note on 

Cell Formats 

Figure 5.17 shows the Add Samples screen after the CSV file shown in 

Figure 5.16 has been imported. 

Figure 5.17

Working with Test Results

Once the project data have been entered using the Add Samples window they 

can be edited and sorted in a number of ways. 

Sorting the Data Clicking the mouse on any column title causes the data to be sorted based on 

that column. For example, in Figure 5.17, the test data for asphalt content are 

sorted in increasing order by Lot Number. Clicking on the Lot Number

column title will cause the data to be sorted in decreasing order by Lot

Number. 

Similarly, clicking on any other column title causes the data to be sorted by 

increasing (or decreasing for a second click) order based on that column. For 

example, clicking on Split Sample No. would bring all split sample results to 

the top of the spreadsheet so they all could be viewed at once. Similarly, the 

user might click on the Test Result column title to easily identify the lowest 

and highest test values. 

PWLPay User’s Manual    5‐9
Chapter 5 Entering Test Results

PWLPay User’s Manual    5‐10

Editing the Data Figure 5.18 shows the drop down window that opens by right clicking 

anywhere in the Add Samples window (see Figure 5.11). This window 

provides the user a number of options for working with the test data. The 

Delete Row and Delete Cell options are obvious. Clicking on these deletes 

either the highlighted row or cell. 

 Printing the Data Print Preview and Print Grid allow the user to preview a printout and to print 

the spreadsheet for the characteristic that is currently displayed in the Add

Samples window. 

Figure 5.18 

 Analyzing the Data The menu in Figure 5.18 allows the user to directly enter the Analysis

window with the analysis input already set to either Analyze This

Characteristic, e.g., Asphalt Content in Figure 5.17, or Analyze This Item, 

e.g., P-401 in Figure 5.17. The Analysis window is covered in detail in 

Chapter 6. The user can consult Chapter 6 for detailed instructions on the 

various analyses that can be performed on the item or characteristic. 

 Random Sampling The menu in Figure 5.18 allows the user to directly enter the Random 

Sampling window where sampling locations or times can be randomly 

generated. The Random Sampling window is covered in detail in Chapter 7. 

The user can consult Chapter 7 for detailed instructions on how to develop 

random sampling locations or times. 
Chapter 6 Analyzing the Data

6  Analyzing the Data

The Analysis Window

The Analysis window can be reached by clicking on the Analysis icon at the 

top of any screen and selecting New Analysis, or from the right click menu 

on the Add Samples window (see Figures 5.11 and 5.18). The Analysis

window is shown in Figure 6.1. From this window the user can select a wide 

range of projects and initiate any of the analyses of which PWLPay is 

capable. 

Figure 6.1

Selecting Data The user has a wide array of options for selecting the data for analysis. 

for Analysis The user first makes a selection from the Select a State / Agency drop down 

menu. In Figure 6.1, South Carolina has been selected. The Contracts Not

Added window automatically displays all contract numbers associated with 

the selected state / agency. The user can then select Contracts To Analyze by 

highlighting a contract number and clicking Add>>. 

All projects associated with the contract numbers in the Contracts To

Analyze window are automatically displayed in the Projects Not Added

window. Once again, the Add>> button is used to select the Projects To

Analyze. All characteristics associated with the Projects To Analyze and the 

selected item number are automatically displayed in the Characteristics Not 

Added window. As before, clicking on Add>> selects the Characteristics To 

Analyze. The user can select all characteristics for analysis with one click by 

clicking on the Select All> button. 

PWLPay User’s Manual    6‐1
Chapter 6 Analyzing the Data

Figure 6.2 shows that the data selected for analysis are from the state of 

South Carolina, contract number SC-2008-001, project Clemson Bypass, 

item P-401, and for the characteristics Asphalt Content, Air Voids, VMA, and 

Density. Note that the Clemson Bypass Phase II has not been selected and 

therefore appears in the Projects Not Added window. 

Figure 6.2

Types of Analysis

The analyses of which PWLPay is capable are identified in the Select

Analysis Type drop down menu in the Options section. Figure 6.3 shows the 

possible analyses that PWLPay can perform. Each of these analyses is 

described in detail in the following sections. 

Figure 6.3 

PWLPay User’s Manual    6‐2
Chapter 6 Analyzing the Data

Descriptive Statistics

After selecting Descriptive Statistics from the choices in Figure 6.3 the user 

then clicks the Begin Analysis >> button to perform the analysis and display 

the results. 

The Descriptive Statistics window is shown in Figure 6.4. General 

informaton regarding the State / Agency and the Item Number: are displayed 

at the top of the window. Click on the Please select a test category: drop 

down menu to display the options for which tests to analyze. The choices are 

shown in Figure 6.5. 

Figure 6.4

Figure 6.5 

Summary Statistics As shown in Figure 6.4, the Summary Statistics displayed are for contract 

number SC-2008-001 and the Clemson Bypass project. The mean and 

standard deviation for Asphalt Content are shown for each lot on the project. 

In addition, the total mean and standard deviation for all tests are shown in 

the Project Mean and Project Stdev. columns. The Contract Mean and 

Contract Stdev. columns display the results for all data on the contract. The 

user must tab or scroll to the right to see the columns that are to the right of 

the display area. 

Note that summary statistics for Asphalt Content for all project data are also 

shown in the upper right corner of the window in the section labeled Total 

PWLPay User’s Manual    6‐3
Chapter 6 Analyzing the Data

Statistics for characteristic: Asphalt Content. The statistics shown are the 

mean, standard deviation, skewness, and kurtosis. These are common 

measures of probability distributions. These values are calculated using the 

following formulas: 

Mean: 

n

x

x ∑ i =

Standard Deviation: ∑= −

− =

n

i

i

n

xx

s

1

2

1

)(

Skewness: ∑=

−

−− =

n

i

i

s

xx

nn

n

g

1

3

3

1

)(

)2)(1(

Kurtosis: 

)3)(2(

)( )1(3

)3)(2)(1(

)1( 2

1

4

4

2 −−

− −

⎭

⎬

⎫

⎩

⎨

⎧ −

−−−

+ = ∑= nn

n

s

xx

nnn

nn

g

n

i

i

where: x = the sample mean, 

= the individual values in the sample, i x

n = the number of values in the sample, 

s = the sample standard deviation, 

g1 = the sample skewness coefficient, 

g2 = the sample kurtosis coefficient. 

The mean is a measure of the location of the data. The standard deviation is a 

measure of the variability of the data. The skewness coefficient is a measure 

of the degree of asymmetry of the data around the mean. A symmetric data 

set has a skewness coefficient of 0. The kurtosis, which is a measure of the 

“peakedness” or relative amount of data in the upper and lower tails of the 

distribution of the data, is not often used. 

PWLPay automatically checks the skewness and kurtosis values to see if they 

could likely have come from a normal population. This check is conducted at 

the significance level, alpha (α), selected by the user. If the test on either the 

skewness or kurtosis coefficient indicates that a normal population is 

unlikely, the coefficient is shown in bold red print. Otherwise it is shown in 

normal green print. A more detailed discussion of normality tests is presented 

in Chapter 11. 

Detailed Statistics The Detailed Statistics for the same project are shown in Figure 6.6. In this 

case each individual test result is displayed on its own row along with its 

corresponding lot and sample number. The Lot, Project, and Contract means 

and standard deviations are shown as well. Some of these values are in 

columns that are to the right of the display area and therefore the user must 

tab or scroll to the right to see them. 

PWLPay User’s Manual    6‐4
Chapter 6 Analyzing the Data

Figure 6.6

Correlation Matrix The correlation coefficient of two variables x and y, which is always between 

–1 and 1, measures the linear relationship between x and y. 

• If r = 1 (or r = –1), then x and y are strictly positively (or negatively) 

linearly related, i.e., y is precisely some positive (negative) multiple 

of x. 

• If r = 0, then x and y are not linearly related. It does not mean that 

they are not related. They could be related in some non-linear way. 

• If r > 0 (or r < 0), then, when x increases, y tends to increase (or 

decrease). The closer that r is to 1 (or –1), the stronger is the 

correlation, as indicated by less spread in the data. 

To be able to calculate a correlation coefficient between two variables there 

must be a paired correspondence between the sample values. That is, for the 

case of comparing the results of two different characteristics, the test results 

must be obtained from the same sample. 

Important Note on 

Correlation 

between 

Characteristics 

For example, if asphalt content and VMA are both determined from the same 

sample obtained at the plant, then there is a direct relationship between the 

asphalt content and VMA test results for each sample number. Therefore, it 

is appropriate to calculate a correlation coefficient between asphalt content 

and VMA. 

On the other hand, if density is determined by testing cores obtained from the 

finished pavement, it is not possible to calculate a correlation coefficient 

between laboratory-compacted air voids and density since there is no paired 

correspondence between their test results. 

More A detailed discussion of correlation coefficients is presented in Chapter11. Info

PWLPay User’s Manual    6‐5
Chapter 6 Analyzing the Data

Figure 6.7 shows the Correlation Matrix screen for the Descriptive Statistics

analysis window. The second row of tabs now shows all of the possible pairwise comparisons among the characteristics for which test results are 

available. In this case, there are six possible pair-wise comparisons for the 

four characteristics. 

Figure 6.7

Below this row of tabs is a matrix that shows all of the correlation 

coefficients along with small versions of scatter plots of the two 

characteristics showing the best straight line fit between the test results for 

the two characteristics. 

The user can save the correlation matrix to a CSV file by clicking 

Save Correlation Matrix shown in the lower left corner of the screen in Figure

6.7. 

The user can click on one of the tabs or on one of the plots to open a screen 

with a large version of the scatter plot that shows the relationship between 

the two characteristics. Figure 6.8 shows one of these screens for the Asphalt

Content & VMA comparison. 

The user can click on to scroll left or right to access all of the scatter 

plots. 

The best-fit straight line is also shown on the scatter plot. The correlation 

coefficient and the equation for the fitted line are also shown. In Figure 6.8

these values are shown by Correlation Coeff (r): –0.21 and 

Trendline Equation: y = –0.925054 x + 22.864589. 

PWLPay User’s Manual    6‐6
Chapter 6 Analyzing the Data

Figure 6.8

Tests for Normality

Significance Level Before beginning any normality testing the user should select the desired 

level of significance, α, at which the test will be conducted. This can be done 

from any screen by clicking on Tools in the upper left corner of the screen. 

As shown in Figure 6.9, selecting Tools ` Normality Check ` Alpha Value

allows the user to select 0.01, 0.02, 0.03, 0.04, or 0.05 for the level of 

significance. If the user does not select an alpha value, the default value of 

0.05 is used. 

Figure 6.9 

Checking Normality PWLPay can be configured to automatically run a quick check for normality 

in the Background whenever new data are entered. This check is done in the “background” and 

the user is notified only if the check indicates that there is a reason to believe 

that the data are not from a normal population. The user can decide whether 

or not to have PWLPay perform this background check by selecting Tools `

Normality Check. If Show Normality Check is checked, the automatic test is 

performed. If Hide Normality Check is checked, the automatic test is not 

performed. The user can at any time choose to test data for normality by 

using the Analysis window. 

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Chapter 6 Analyzing the Data

The quick background normality check tests at the selected alpha value 

whether or not the skewness and kurtosis coefficients calculated for the test 

data are unlikely to have been obtained from a normal population. When this 

happens the warning window in Figure 6.10 is displayed and asks whether or 

not the user wants more formal normality tests to be performed. 

As shown in Figure 6.10, the warning message informs the user whether the 

skewness and/or kurtosis values are questionable, and indicates at what alpha 

value these coefficients were compared. The user selects whether or not to 

run the normality test by selecting the Run Test or Don’t Run button. 

Figure 6.10 

If Run Test is selected PWLPay performs two different checks for normality. 

The fist is the Anderson-Darling (A-D) goodness of fit test specifically 

modified to test for normality. This is a formal hypothesis test. Since this test 

is quite sensitive to departures from normality, in PWLPay it is always 

conducted at the 0.01 level of significance regardless of what value has been 

selected with the Tools ` Normality Check ` Alpha Value option. 

The second check for normality is to plot the data on a quantile-quantile plot 

(qq-plot). Data that are normally distributed should approximate a straight 

line on the qq-plot. This is, essentially, the same process as plotting the 

cumulative distribution of the data on normal probability paper. 

 Detailed discussions of the Anderson-Darling test and qq-plots are included 

in Chapter 10. More Info

Figure 6.11 shows the normality test results obtained if Run Test is selected 

from the warning message in Figure 6.10. Note that the A-D indicates that 

There is less than 1% chance that the data came from a normal distribution. 

From the qq-plot it is easy to see why the A-D test for normality decalred the 

data to be non-normal. One of the test results has been intentionally entered 

as 20 rather than the correct value of 5.86. 

PWLPay User’s Manual    6‐8
Chapter 6 Analyzing the Data

Figure 6.11 

Tests for Normality After selecting Tests for Normality from the choices shown in Figure 6.3, the 

user then selects the Begin Analysis >> button to perform tests for the 

normality of the data and display the results. The Tests for Normality window 

is shown in Figure 6.12. General informaton regarding the State / Agency

and the Item Number: are displayed at the top of the window. Click on the 

Please select a test category: drop down to display the options for which 

tests to analyze. 

Figure 6.12

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Chapter 6 Analyzing the Data

qq‐Plot Below the general information there is a row of tabs, one for each 

characteristic. The Asphalt Content test results are displayed in the qq-plot in 

Figure 6.12. The user must subjectively decide whether or not the data 

approximate the straight line shown. If the answer is no, then the assumption 

of normality is rejected. 

Anderson‐Darling The descriptive statistics and the results of the A-D test are shown to the 

Test right of the qq-plot. The Count:, Mean:, and Standard Deviation: values are 

obvious. The test statistic from the A-D test is shown in the AndersonDarling: box. This is for information only, and is unlikely to be of direct use 

to the user. The conclusion to be drawn from the A-D test is shown in the 

Anderson-Darling Test Result: box. No significant difference from a normal 

distribution is reported as shown in Figure 6.13a, while a significant 

difference from a normal distribution is reported as shown in Figure 6.13b. 

Figure 6.13 (a) (b) 

Skewness & Kurtosis The Skewness: and Kurtosis: boxes are obvious. As in the Descriptive

Statistics window in Figure 6.8, the skewness and kurtosis coefficients are 

checked at the selected significance level to see if they are different than those 

expected for a normal distribution. No difference is depicted by normal green

print, while a difference is depicted by bold red print. 

Process Control

After selecting Process Control from the choices in shown Figure 6.3, the 

user must then select the types of process control charts to be developed as 

well as how the control limits are to be established. Figure 6.14a shows the 

types of charts that can be prepared and displayed as Output by PWLPay. If 

one of the four control chart output options is selected, then the user must 

also select the method to be used to establish the Control Limits for the 

control charts. The two options are shown in Figure 6.14b. 

Figure 6.14 (a) (b) 

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Chapter 6 Analyzing the Data

Run Charts If Run Chart is selected as the Output, the user then selects the Begin

Analysis >> button to display the run charts for the selected characteristic(s). 

The Run Chart screen is shown in Figure 6.15. A run chart is a chronological 

plot of the individual test results along with the specification limits for the 

characteristic being plotted. 

Figure 6.15

At the top of the window is general information on the State / Agency:, the 

Item Number:, and the Project Number:. There is also a drop down menu to 

select which tests to analyze. Another drop down menu allows the projcect to 

be selected. This choice will need to be made only if there is more than one 

project selected in the Analysis window. 

Below the general information there is a row of tabs, one for each 

characteristic. The Air Voids test results are plotted in the run chart in Figure

6.15. Below the tabs is a description of the type of specification that applies 

for the selected characteristic. PWLPay determines the target value, upper 

specification limit, and lower specification limit based on the item number 

and project that are selected. 

The run chart plots the individual test results, which are identified on the 

horizontal axis by their lot number-sample number combination. The dashed 

line represents the mean of the test results. 

The user can hover the mouse pointer over any of the lines or any of the data 

points to open a bubble with appropriate detailed information. Examples of 

these information bubbles are shown in Figures 6.16a – 6.16d, which 

represent portions of the run chart in Figure 6.15. 

PWLPay User’s Manual    6‐11
Chapter 6 Analyzing the Data

(a) (b) 

Figure 6.16 (c) (d) 

Figures 6.16a and 6.16b show that the target value is 4 and the upper limit is 

5, respectively. Figure 6.16c shows that the data point is for Air Voids and 

that it is #7 of 16 points that are plotted. The x and y coordinates for the point 

are x = 6 (the plotting routine begins with 0, so the x value is one less than 

the data point number) and y = 3.95 (which is the air voids test result). The 

test result is for Sample Number S-03 within Lot Number L-002. Finally, 

Figure 16.6d shows that the mean for all of the plotted test results is 4.56. 

Control Charts If one of the Control Chart outputs is selected (see Figure 6.14a) and 

Calculate Values from Data is selected for establishing control limits, then 

the user selects the Begin Analysis >> button to display the selected control 

chart(s). PWLPay then calculates the target and standard deviation values for 

all of the the selected test results (All Contractor Tests in Figure 6.15) and 

uses these, along with the type of control chart selected, to determine the 

control limits. 

Control Limits The Control Chart window, based on selecting Control Charts for X-Bar and 

 Based on the Data R and selecting Calculate Values from Data to determine the control limits, is 

shown in Figure 6.17. The general information at the top of the window is the 

same as for the Run Chart window. However, now there are two tabs for 

each characteristic—one for the X-bar chart and one for the R chart. The user 

can click on to scroll left or right to access all of the tabs. 

There are not as many points on the control chart since only one point is 

plotted for each lot, as opposed to one point for each test for a run chart or 

for a control chart for individuals. Similar to the Run Chart examples in 

Figures 6.16a – 6.16d, more detailed information can be obtained by 

hovering the mouse pointer over one of the lines or data points. 

PWLPay User’s Manual    6‐12
Chapter 6 Analyzing the Data

Figure 6.17

One difference between a run chart and a control chart is that the run chart 

compares the data points to the specification limits, whereas for a control 

chart the limits are established based on the mean and variability of the 

process, i.e., the population of avaiable test results. Note in Figure 6.17 that 

the control limits are not necessarily horizontal lines. Control limits depend 

upon the sample size for which the mean is determined. If the sample size is 

constant, the control limits will be constant horizontal lines. However, if the 

sample size varies between lots, the control limits will expand for smaller 

sample sizes and shrink for larger sample sizes. 

If individual tests, rather than lot means and ranges, are plotted, then the 

moving range, Rm, is used to control variability. In this case, the first point 

on the moving range chart will lag by one the first point on the control chart 

for individuals. Figure 6.18 shows a sample control chart for individuals. 

While this chart may appear similar to the run chart in Figure 6.15, the 

control limits in Figure 6.18 are determined from the process capability and 

differ considerably from the specification limits plotted in Figure 6.15. 

Control Limits The user can choose to enter the target values and control limits rather than 

Entered by User to have PWLPay calculate the limits from the input data. This might be done 

if the user had a continuous and stable process for which the limits had been 

established from a large amount of historical data. Since these entered 

control limits will be constant, this implies that the sample sizes will be the 

same for all or nearly all lots. 

PWLPay User’s Manual    6‐13
Chapter 6 Analyzing the Data

Figure 6.18

Figure 6.19 shows the selections that would be made to plot control charts for 

individuals and for moving ranges when the target values and control limits 

will be input by the user. At this point the user selects Begin Analysis >> to 

open the window shown in Figure 6.20, which then results in the moving 

range chart shown in Figure 6.21. 

Figure 6.19 

Reference

A thorough description of control chart analysis is provided in the ASTM 

publication Presentation of Data and Control Chart Analysis, Seventh 

Edition, February 2002, which is available at www.ASTM.org. 

PWLPay User’s Manual    6‐14
Chapter 6 Analyzing the Data

Figure 6.20 

Figure 6.21

Verification Tests

Verification tests are performed to compare two sets of test results to 

determine whether or not it is reasonable to assume that they came from 

similar populations. To make this comparison it is necessary to run statistical 

tests to see whether or not it is reasonable to assume that both their means 

and variances (variance is the square of the standard deviation) are equal. 

The t-test is used to compare the sample means of the two sets of test results, 

while the F-test is used to compare their variances. Which form of the t-test 

to use depends upon whether the data to be compared are from split samples 

or independent samples. If the data are from split samples, then the paired ttest is used. If the data are from independent samples, then the two-sample ttest is used. The same F-test is used for both split or independent samples.

PWLPay User’s Manual    6‐15
Chapter 6 Analyzing the Data

Thorough descriptions and discussion of t-tests and F-tests are available from 

numerous statistical texts and websites. The use of these tests for highway 

materials and construction is presented in Optimal Procedures for Quality 

Assurance Specificatons, Publication No. FHWA-RD-02-095, April 2003. 

After selecting Verification Tests from the choices shown in Figure 6.3, the 

user must then select the tests that are to be compared. The Compare drop 

down menu in Figure 6.22 shows the types of comparisons that can be 

performed by PWLPay. The choice of which tests to use for the comparisons 

dictates the options from which the user selects in the Which tests do you 

wish to run? drop down menu since different tests are performed depending 

upon whether the data to be compared are from split samples or independent 

samples. 

Figure 6.22 

Comparing If Split Samples is selected as the Compare option, then the user must select 

Split Samples from the comparison test options shown in Figure 6.23. With split samples 

where the split results make up sample pairs, the paired t-test is preferable to 

the two-sample t-test, and thus only the paired t-test is available for selection. 

Figure 6.23 

Reference

The user can choose to run only the paired t-test to compare sample means, 

only the F-test to compare sample variances, or both tests. The choice will 

obviously affect the calculations performed and the resulting output. After 

making this choice the user selects Begin Analysis >> to perform the 

verification tests. 

Figure 6.24 shows an example of the output if the user has selected to run 

both the paired t-test and the F-test. The t-test output shows the sample 

means, sample standard deviations, and sample sizes along with the P-Value 

for the test statistic. The P-Value is the probability of getting a test statistic 

PWLPay User’s Manual    6‐16
Chapter 6 Analyzing the Data

greater than or equal to the one obtained if the two samples came from the 

same population, i.e., if the two population means were actually equal. 

Therefore, the smaller the P-Value the less likely the sample means are 

equal. PWLPay displays a warning message when the means are statistically 

significantly different at the 0.10, 0.05, or 0.01 levels. This corresponds to PValues less than 0.10, 0.05, or 0.01, respectively. 

Figure 6.24

The F-test output (see Figure 6.24) shows the sample means, sample 

variances, and sample sizes along with the P-Value for the test statistic. As 

with the t-test, PWLPay shows a warning message when the P-Value is less 

than 0.10, 0.05, or 0.01. In Figure 6.24, the variances are shown to be 

Significantly different at the 0.01 level. 

Comparing The user has four different independent sample comparisons in the Compare 

Independent Samples drop down menu shown in Figure 6.22. These comparisons are: 

Contractor’s indpnt vs. owner’s indpnt. This would apply in a situation 

where the contractor was performing acceptance testing for the project, and 

the owner was also obtaining and testing its own samples from different 

random sampling locations than those used by the contractor. 

Contractor’s indpnt vs. owner’s split samples. This would apply in a 

situation where the contractor was performing acceptance testing for the 

project, but a portion of the samples were split and tested by both the 

contractor and the owner. In this case the contractor samples that were not 

split would be independent samples that could be compared with the owner 

tests on the split samples. The contractor’s test results on its portion of the 

split samples would not be included in this comparison. 

PWLPay User’s Manual    6‐17
Chapter 6 Analyzing the Data

All Contractor tests vs. owner’s indpnt. This would apply in the event that 

the contractor was performing the acceptance testing and some of the 

contractor’s samples were split and tested by the owner. However, the owner 

was also obtaining and testing samples from sample locations that were 

independent of those used by the contractor. In this case the owner’s 

independent samples could be compared with all of the contractor’s samples, 

including the contractor’s portion of any split samples. 

Contractor’s indpnt vs. all owner’s. This would apply in the event that the 

contractor was performing the acceptance testing and some of the 

contractor’s samples were split and tested by the owner. However, the owner 

was obtaining and testing samples from sample locations that were 

independent from those used by the contractor. In this case the contractor’s 

independent samples could be compared with all of the owner’s samples, 

including the owner’s portion of any split samples. 

If any of these independent comparison options is selected, then the user 

must select from the comparison test options shown in Figure 6.25. Since the 

samples to be compared are independent, the two-sample t-test is performed 

and the paired t-test is not an option. 

Figure 6.25 

As shown in Figure 6.25, the user can select to run only the t-test. The t-test 

is conducted differently whether or not it is assumed that the sample 

variances are equal, which is why in Figure 6.25 there are two choices for 

running the t-test. The user can also select to run only the F-test. It is most 

likely that the user would select the last option to run the F-test then

appropriate resulting t-test. In this case the user enters the level of 

significance, alpha, at which to run the F-test (see Figure 6.26). PWLPay runs 

the F-test then, depending upon the result, runs either the t-test for equal 

variances or the t-test for unequal variances. After selecting one of the 

options shown in Figure 25, the user selects Begin Analysis >> to perform the 

verification tests. 

Figure 6.26 

PWLPay User’s Manual    6‐18
Chapter 6 Analyzing the Data

Figure 6.27 shows an example of the output if the user had selected the F-test 

then appropriate resulting t-test option for comparing the Contractor’s indpnt 

vs. owner’s split samples for the characteristic Asphalt Content. The F-test 

output shows the sample means, sample variances, and sample sizes along 

with the P-Value for the test statistic. It also shows the entered alpha value 

(alpha = 0.05 in Figure 6.26). PWLPay shows a warning message if the 

variances are significantly different at the entered alpha value. In Figure 6.27, 

the variances are shown to be Significantly different at the 0.05 level. 

The results of the t-test are shown to the right of the F-test results. PWLPay

automatically performs the correct t-test based on the results of the F-test, 

and reports the P-value. In Figure 6.27 the F-test found the variances to be 

unequal at the specified alpha value. So, the results are reported for the t-Test 

Assuming Unequal Variances. 

Figure 6.27

Saving Verification Right clicking anywhere in the results area of the verification test report 

Test Results opens the popup menu shown in Figure 6.27. Selecting Save As saves the 

verification test results as a CSV file in the same format as shown on the 

screen. The user can also select Open with Excel to open the CSV file in 

Excel if it is available on the user’s computer. 

Pay Factors

The Pay Factors option is used to calculate and report the payment factors 

for both the individual characteristics and the composite lot payment factor. 

PWLPay uses the payment relationships that were entered in the Setup Specs

window and the specification limits that were entered in the Setup Project

window to determine the payment factors. 

PWLPay User’s Manual    6‐19
Chapter 6 Analyzing the Data

After selecting Pay Factors from the choices in Figure 6.3 the Output drop 

down menu is used to select whether a detailed payment report or just a 

summary report should be prepared. The Use drop down menu then allows 

the selection of which tests to use to determine the payment factors. The 

options for which tests to use to determine payment are shown in Figure

6.28. 

Figure 6.28 

Which Tests to Use The options for which tests to use include: 

only contractor tests. This might be chosen if the contractor is performing 

the acceptance tests and the tests have been verified by owner tests. 

only independent owner tests. This might be chosen if the owner is 

performing the acceptance tests. 

both contractor and independent owner tests. This might be chosen if the 

contractor is performing the acceptance tests and the tests have been verified 

by independent owner tests. It is not recommended that contractor and owner 

tests be combined even if both their means and variances have not been 

declared different in the verification testing. This is because the verification 

tests do not prove that the two samples are exactly equal, and it is likely that 

combining the two samples will lead to increased variability. 

Important Note on 

Combining Owner 

and Contractor 

Test Results 

only owner split tests. This might be chosen if the contractor is performing 

the acceptance tests and the owner is using split samples for verification 

testing. If the tests do not verify, then the owner may decide to use its tests, 

rather than the contractor’s, to calculate the payment factors. 

only all owner tests. This might be selected if the contractor is performing 

the acceptance tests and the owner is using both split samples and 

independent samples for verification tests. If the tests do not verify, then the 

owner may decide to use all of its tests to calculate the payment factors. 

Rounding Format Since owners use different rounding rules when calculating payment factors, 

PWLPay requires the user to specify the rounding procedures to use before 

payment factor calculations can begin. Rounding can apply to either the 

calculated PWL values that are used in the payment factor calculations, to the 

payment factors that have been calculated, or to both. So, the user must 

PWLPay User’s Manual    6‐20
Chapter 6 Analyzing the Data

choose from the rounding methods shown in Figure 6.29 for both the 

PWL Format and the PF Format. 

Figure 6.29 

The PWL rounding format is applied to the PWL values that are input into 

the payment relationship to calculate the payment factors. The PF rounding 

format is then applied to the calculated payment factors. The rounding format 

options include: 

Do Not Round. This is the default option. No rounding is used for the 

calculated values. 

Round to Nearest Integer. If chosen, the PWL value is rounded to the 

nearest integer before it is input to the payment relationship. If chosen, the 

calculated payment factor is rounded to the nearest integer. 

Always Round Up. If chosen, the PWL value is always rounded up to the 

next higher integer before it is input to the payment relationship. For 

example, a value of 94.02 would be rounded to 95. If chosen, the calculated 

payment factor is always rounded up to the next integer. 

Always Round Down. If chosen, the PWL value is always rounded down to 

the previous lower integer before it is input to the payment relationship. For 

example, a value of 94.82 would be rounded to 94. If chosen, the calculated 

payment factor is always rounded down to the next integer. 

After making this choice the user selects Begin Analysis >> to calculate and 

report the payment factors. 

Detailed Report Figure 6.30 shows an example detailed payment report. The title shows that 

the user had selected a Detailed Report and that payment factors be 

calculated using Only Contractor Tests. The report includes the Project

Name, Project Number, Lot Number, Characteristic, Sample No., Date, and 

the Test Result. The full report does not fit in the viewing window; the user 

must scroll the screen to the right to see the payment information. 

PWLPay User’s Manual    6‐21
Chapter 6 Analyzing the Data

Figure 6.30

Summary Report Figure 6.31 shows an example summary payment report that would 

accompany the detailed report shown in Figure 6.30. The same columns that 

appear in the Detailed Report appear in the Summary Report. However, in 

the Summary Report, only sample sizes, lot means, and lot standard 

deviations are displayed. The individual test results do not appear in the 

report. 

Figure 6.31

PWLPay User’s Manual    6‐22
Chapter 6 Analyzing the Data

Saving the Pay Right clicking anywhere in the spreadsheet area of either payment report 

Factors Report opens the popup menu shown in Figure 6.32. The user can choose to open the 

report with Excel, if it is available on the user’s computer, or to save the 

payment report as a CSV file. The user has three options regarding saving the 

CSV file. Two of the options relate to the format in which the data are stored 

in the CSV file. The third option brings up the message in Figure 6.33, which 

explains the two different formats. 

Figure 6.32

Figure 6.33

As discussed in Chapter 5, always including a letter in lot and sample 

numbers will avoid the potential formatting problems that are explained in 

Figure 6.33. 

Risk Analysis

Risk Analysis allows the user to ask and answer “what-if” questions about the 

potential effects of either making or not making changes in their production 

processes. After selecting Risk Analysis from the choices in Figure 6.3, the 

user must then select one of the two options from the Calculate drop down 

menu shown in Figure 6.34a. Next, the user selects whether to Use “to-date” 

PWLPay User’s Manual    6‐23
Chapter 6 Analyzing the Data

means and standard deviations from existing test results, or to enter directly 

the means and standard deviations to use in the analysis (see Figure 6.34b). 

The “to-date” values would be used if the user wishes to know what will 

happen if the process remains in its current state. The user would choose to 

enter new values if a process change is being considered. 

Figure 6.34 (a) (b) 

Projected Pay The Projected Pay Factor in Figure 6.34 is the payment that the contractor 

Factor would be expected to receive for the project if it achieved the means and 

standard deviations used in the analysis. This value is particularly important 

in cases where the entire project is the payment lot, and the contractor wishes 

to project what its likely payment will be for the total project. 

While likely less important in cases where the payment lot is a day’s 

production, Projected Pay Factor can still be used to determine likely 

individual lot pay factors for selected process means and standard deviations. 

For example, if a contractor has received reduced payments for lots early in a 

project, this analysis could be used to investigate what changes in means 

and/or standard deviations would be necessary to receive full payment or an 

incentive payment for future lots. 

After selecting the option from the Use drop down menu shown in Figure

6.34b, the user selects Begin Analysis >> to calculate and report the projected 

payment factors. What happens next depends upon which selection was made 

in the Use menu. 

Use “to‐date” means & std devs. If the user selected to use means and 

standard deviations calculated from the current test data, the window in 

Figure 6.35 is displayed. The user then enters the projected sample size for 

each characteristic shown. This will usually be the sample size that is 

anticipated at the end of the project. After entering the desired sample sizes, 

the user selects Continue to generate the Projected Pay Factors report shown 

in Figure 6.36. 

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Chapter 6 Analyzing the Data

Figure 6.35 

Figure 6.36

As shown in Figure 6.36, the Projected Pay Factors report shows the 

contract number, SC-2008-001, and item number, P-401, at the top left just 

above the spreadsheet section. Additional project information as well as the 

projected PWL and payment values is presented in the spreadsheet section. 

For example, in Figure 6.36, for Air Voids the mean and standard deviation 

calculated from the “to-date” data are 4.5594 and 0.37941, respectively. 

This information, along with the projected sample size of 40, results in a 

projected Air Voids PWL of 87.77454 with a resulting projected pay factor of 

97.77454. The composite pay factor, based on the Weighted Average

approach, for the lot is projected to be 100.3534. As with other screens in 

PWLPay, right clicking in the spreadsheet window shown in Figure 6.36

allows the user to save the report in CSV file format or open it with Excel. 

The Test category drop down menu shown in the upper right corner of Figure

6.36 allows the user to select which tests should be used to determine the “todate” values for mean and standard deviation. In Figure 6.36, Only

Contractor Tests is selected. Figure 6.37 shows the test categories from 

which the user may choose. 

Figure 6.37 

PWLPay User’s Manual    6‐25
Chapter 6 Analyzing the Data

Input the means & std devs. If the user selects to input hypothetical means 

and standard deviations, the window in Figure 6.38 is displayed. The user 

then enters the projected sample size, mean, and standard deviation for each 

characteristic shown. After entering the values, the user selects Continue to 

generate the Projected Pay Factors report shown in Figure 6.39. 

Figure 6.38 

Figure 6.39

Long Term The Long Term Expected Payment shown in Figure 6.34a is the payment 

Expected Payment that the contractor would be expected to receive on the average for a very 

large number of lots with the stated means and standard deviations for each 

characteristic. In other words, it is the “average payment in the long run.” 

This gives the contractor an idea of the payment it can expect to receive if it 

maintains production at the current, or at a hypothetical, process capability. 

This can help the contractor decide how best to set the target values and how 

much process control is needed for each characteristic that will be measured 

for payment. 

After selecting the Long Term Expected Payment option shown in Figure

6.34a, the user selects Begin Analysis >> to open the window shown in 

Figure 6.40. The user enters the sample size, mean, and standard deviation 

for each characteristic shown. 

PWLPay User’s Manual    6‐26
Chapter 6 Analyzing the Data

Figure 6.40 

Note in Figure 6.40 that the characteristics can be checked as either 

Independent or Correlated. As discussed above in this chapter, to be able to 

calculate a correlation coefficient between two variables there must be a 

paired correspondence between the sample values. That is, for the case of 

comparing the results of different characteristics, the test results must be 

obtained from the same sample. 

Since this is the case, all correlated characteristics checked in Figure 6.40

must have the same sample size. PWLPay checks to see if this is true, and if 

not, issues the warning shown in Figure 6.41. Independent characteristics can 

have different samples sizes per lot. 

Figure 6.41 

Since the Long Term Expected Payment analysis uses computer simulation 

to determine the expected payment, the user also must enter the Number of

iterations (i.e., the number of lots to be simulated) as an integer less than or 

equal to 100,000. Selecting Continue begins the simulation process and, if 

any correlated characteristics are involved, pops up the screen shown in 

Figure 6.42 where the user enters the correlation coefficients for the 

correlated characteristics. 

PWLPay checks the entered coefficients and if they are not consistent with 

one another issues the warning shown in Figure 6.43. 

PWLPay User’s Manual    6‐27
Chapter 6 Analyzing the Data

Figure 6.42 

Figure 6.43 

Figure 6.44 shows the results screen that is displayed when the simulation 

analysis from Figures 6.40 and 6.42 is completed. General information to 

identify the simulation is shown at the top of the display window. 

Figure 6.44

The user can select one of the tabs to display the histogram of the distribution 

of the pay factors for the number of simulated lots (10,000 in Figure 6.44) for 

each characteristic (Asphalt Content is selected in Figure 6.44) or for the 

Composite (Lot) pay factor. The Histogram Bins: section in the lower left 

corner allows the user to select the number of bars (i.e., bins) to use when 

plotting the histogram. The default value is 100, but a value of 50 has been 

entered in Figure 6.44. 

PWLPay User’s Manual    6‐28
Chapter 6 Analyzing the Data

The Statistics section displays a number of descriptive statistics for the 

selected tab. The Number of Iterations:, Mean:, Standard Deviation:, and 

Skewness: values are obvious. The 5th Percentile: shows the pay factor 

value for which 5% of the simulated pay factors were less than or equal. The 

95th Percentile: shows the pay factor value for which 95% of the simulated 

pay factors were less than or equal. 

Figure 6.45 displays the simulation results for the Composite (Lot) pay factor 

for the same simulation shown in Figure 6.44. The Mean: value indicates that 

the contractor can expect a long term average pay factor of 93.73 percent if 

its process has the values shown in Figures 6.40 and 6.42. 

Figure 6.45

Due to the random nature of the computer simulation process that is used in 

the Long Term Expected Payment analysis, the results of each analysis will 

vary slightly for identical input values. This is the case whenever computer 

simulation is used. 

Important Note on 

Computer 

Simulation Results 

Thorough descriptions and discussion of computer simulation for risk 

analysis are well beyond the scope of this user’s manual. Discussion of the 

evaluation of risks for highway materials and construction is presented in 

detail in Optimal Procedures for Quality Assurance Specificatons,

Publication No. FHWA-RD-02-095, April 2003. 

Reference

Saving an Analysis

When working on a project, as new test results become available it is likely 

that the user will perform the same analyses on a routine basis. To speed up 

these repetitive analyses, the user can save an analysis, which allows the user 

PWLPay User’s Manual    6‐29
Chapter 6 Analyzing the Data

to bypass many of the selection menus associated with some types of 

analysis. 

The screen shown in Figure 6.46 shows a Descriptive Statistics analysis that 

is ready to be initiated by selecting Begin Analysis >>. Before beginning the 

analysis the user can click on Save this Analysis to open the window shown 

in Figure 6.47. The user then enters an easily recognizable name that 

describes the analysis (Clemson Bypass – Desc Stat in Figure 6.47). 

Figure 6.46

Figure 6.47 

Working with Any saved analysis will be displayed in the drop down menu that appears 

Saved Analyses by right-clicking on the Analysis icon that is at the top of each PWLPay

screen (see Figure 6.48). The user can then select the analysis that he or she 

wishes to run. If Clemson Bypass – Desc Stat is selected in Figure 6.48, the 

user is taken directly to the completed analysis screen shown in Figure 6.49. 

Figure 6.48 

PWLPay User’s Manual    6‐30
Chapter 6 Analyzing the Data

Figure 6.49

Saved analyses can also be accessed from the Saved Data Analysis drop 

down menu in the Analysis window (see Figure 6.50). 

Figure 6.50

Clicking on Tip shown in Figure 6.50 displays the window shown in Figure

6.51. A saved analysis is opened by selecting it from the drop down menu 

and clicking on Go. 

PWLPay User’s Manual    6‐31
Chapter 6 Analyzing the Data

PWLPay User’s Manual    6‐32

Figure 6.51 

Deleting a A saved analysis can be deleting by clicking on Delete Selected Saved

Saved Analysis Analysis as shown in Figure 6.50. 
Chapter 7 Random Sampling

7  Random Sampling

Background

In most cases locations for samples obtained for testing are selected in some 

random manner. This has traditionally been done by using a table of random 

numbers. However, since effective computer algorithms are available for 

developing lists of random numbers, random sampling locations can easily 

and quickly be obtained from the computer. The PWLPay Random Sampling

window allows the user to generate random samples based on location, 

quantity, or time. 

The Random Sampling Window

The Random Sampling window allows the user to establish random 

locations, quantities, or times for obtaining samples. Clicking on the Random

Sampling icon on any screen opens the Random Sampling window shown in 

Figure 7.1. Identifiers, including State / Agency, Contract Number, Project 

Name, Item Number, and, optionally, Characteristic, are selected from 

dropdown menus on the left side of the window. 

Figure 7.1

Identifying the Sample When the state or agency has been selected from the Please select a State / 

Agency: dropdown menu, the Please select a Contract Number: menu is 

populated with all contract numbers that have previously been assigned to the 

state or agency selected. An example of this is shown in Figure 7.2a. 

Similarly, once the contract number is selected, the Please select a Project 

Name dropdown menu is populated with all projects associated with that 

PWLPay User’s Manual    7‐1
Chapter 7 Random Sampling

contract number. The Project Number is automatically displayed for the 

project name that was selected. An example of this is shown in Figure 7.2b. 

The Please select an Item Number dropdown menu is populated with all 

items that are associated with the selected project (see Figure 7.2c). 

Figure 7.2 (a) (b) (c) 

Based on the chosen item number, the user also has the option of selecting a 

specific characteristic from the Please select a Characteristic dropdown 

menu. If the Include Characteristic box is not checked, the screen appears as 

shown in Figure 7.3a. If the Include Characteristic box is checked, the user 

must select a characteristic from the Please select a Characteristic dropdown 

menu and the screen appears as shown in Figure 7.3b. 

Figure 7.3 (a) (b) 

Output Options As shown in Figures 7.4a and 7.4b, the user can choose to have the random 

numbers Output to screen, Output to file, or to both. If Output to file is 

selected, the user can then select whether or not to have the file open in Excel 

upon completion of the random number calculations. 

Figure 7.4 (a) (b) 

Interval Type The Interval Type section of the Random Sampling window allows the user 

to select the sampling format as well as the sampling interval. As shown in 

Figure 7.5, random samples can be selected By Quantity, By Time

(hours:minutes), or By Roadway Station. 

By Quantity If By Quantity is selected as the Interval Type, the user must enter the Item

units, Item quantity, and Sample interval. For the example shown in Figure

7.6, the units for sampling are tons, the quantity to be sampled is 2500 tons, 

and one sample is to be randomly obtained from each 500-ton increment. 

PWLPay User’s Manual    7‐2
Chapter 7 Random Sampling

Figure 7.5 

Figure 7.6 

Once the unit, quantity, and interval values have been entered, the user 

selects the Execute button to obtain the random sampling locations shown in 

the screen output in Figure 7.7. If the user had also selected the option to 

send the output to a file, then a CSV file also would have been created. The 

CSV output file for the example in Figure 7.7 is shown in Figure 7.8. 

Figure 7.7 

Figure 7.8 

PWLPay User’s Manual    7‐3
Chapter 7 Random Sampling

Once the random numbers have been output, the Execute button changes to 

Start Over. Clicking on this button allows the user to start over with a new 

set of options. 

By Time If By Time is selected as the Interval Type, the user must enter the Begin 

Time, End Time, and Sample interval (hh:mm). For the example shown in 

Figure 7.9, the beginning time is 7:00 AM, the ending time is 5:00 PM, and 

one sample is to be randomly obtained from each 2-hour time interval. 

Figure 7.9 

Once the beginning time, the end time, and sample interval values have been 

entered, the user selects the Execute button to obtain the random sampling 

times shown in the screen output in Figure 7.10. If the user had also selected 

the option to send the output to a file, then a CSV file also would have been 

created. The CSV output file for the example in Figure 7.10 is shown in 

Figure 7.11. 

Figure 7.10 

Figure 7.11 

By Roadway Station If By Roadway Station is selected as the Interval Type, the user must first 

select whether the Units are in hundred feet + feet or kilometers + meters. 

Next, the Beginning Station, Ending Station, Sample interval, and pavement 

Width are entered. The user also has the option to make an entry for Lane

Position. This could be used to enter information such as the lift, e.g., surface 

course, or some descriptor such as Westbound Lane. Finally, the user must 

indicate the reference for the transverse distance for locating the sample. 

That is, the user indicates whether the distance is measured from the Left

Edge, the Right Edge, or the Center Line of the pavement. 

PWLPay User’s Manual    7‐4
Chapter 7 Random Sampling

For the example shown in Figure 7.12, the beginning station is 100+00, the 

ending station is 150+00, and one sample is to be randomly obtained from 

each 500-foot increment. The transverse distance is measured from the Right 

Edge of the 24 foot pavement, and for a Surface course. 

Figure 7.12 

Once all of the information has been entered, the user selects the Execute

button to obtain the random sampling locations shown in the screen output in 

Figure 7.13. If the user had also selected the option to send the output to a 

file, then a CSV file also would have been created. The CSV output file for 

the example in Figure 7.13 is shown in Figure 7.14. 

Figure 7.13 

Figure 7.14 

Partial Intervals For some materials the sample intervals may not divide evenly into the total 

material quantity or time to be sampled. For example, if the sample interval 

is 500 units and the total quantity to be sampled is 2,400 units, then there will 

be 4 full sample intervals of 500 units, with an additional 400 units 

remaining for the last sample interval. 

PWLPay User’s Manual    7‐5
Chapter 7 Random Sampling

PWLPay User’s Manual    7‐6

There are a number of different ways that might be used by different owners 

to handle the situation of uneven sample intervals. Sample intervals are often 

referred to as sublots. Some possibilities for handling unequal sublots 

include: 

• treating the partial sublot as its own sample interval. 

• combining the partial sublot with the last complete sublot to make a 

larger final sublot. 

• dividing the total quantity to be sampled into a desired number of 

complete sample intervals. For example, if the owner wanted a 

sample size of 4, then the 2,400 units could be divided into four 600-

unit sublots. 

There are additional provisions that could be incorporated into the three 

options above. For example, the owner might have a policy to treat the partial 

sublot as a full sublot if it is over some threshold quantity, but to combine it 

with the last complete sublot if the partial sublot is below the threshold 

quantity. 

PWLPay does not attempt to account for all of the possible combinations of 

ways to deal with partial sublots. PWLPay treats any partial sublot as a 

complete sublot when it generates random sampling locations. Therefore, in 

the above example, PWLPay would provide 5 random sampling locations—

one within each of the four 500-unit sublots, and one within the last 400-unit 

sublot. Examples of the input and output for this example case are shown in 

Figures 7.15a and 7.15b, respectively. 

Figure 7.15 (a) (b) 

As shown in Figure 7.15b, PWLPay has developed 5 sample quantities. The 

first 4 are each within one of four complete sample intervals or sublots. That 

is, 460 is between 0 and 500, 835 is between 500 and 1,000, 1155 is between 

1,000 and 1,500, and 1740 is between 1,500 and 2,000. The final sample 

quantity, 2100, is within the last partial interval from 2,000-2,400. 

If necessary, the user can choose to make adjustments in the random sample 

quantities, times, or locations. For instance, if the user desired exactly four 

tests in the previous example, then the sample interval could have been 

entered as 600 (i.e., 2400/4), rather than 500. Or, if the user wished to 

combine the partial sublot with the last full sublot, then the sample could be 

taken at either 1740 or 2100 units. This approach would more likely be 

chosen when the partial sublot was a relatively small quantity in comparison 

to the full sublots. In the above example, it is more likely that the last 400-

unit sublot would be treated as a separate sublot as shown in Figure 7.15b. 
Chapter 8 Navigating PWLPay

8  Navigating PWLPay  

Navigation Options

There are two different methods by which the user can navigate quickly to 

the desired location within PWLPay. These are the Navigation Tree and the 

Navigation pane. The Navigation Tree icon appears at the top of any screen, 

as shown in Figure 8.1. The Navigation pane appears on the right hand edge 

of any screen, and is the shaded region on the right side of Figure 8.1. 

Figure 8.1

The Navigation Tree

The Navigation Tree window allows the user direct access to most locations 

within PWLPay. Clicking on this icon opens the Navigation Tree window 

shown in Figure 8.2a. 

The Display Format The first level shows all states and/or agencies that have been entered. 

Clicking on opens the next level, showing all contract numbers assigned to 

the selected state or agency. The user can continue expanding the tree until 

all item numbers have been displayed. For example, the following selections 

have been made to arrive at the screen shown in Figure 8.2b: 

 State: South Carolina

 Contract Number: SC-2008-001

 Project Name - Number: Clemson Bypass – Distrtict-3-005-1

 Item Number: P-401

 Characteristic: Asphalt Content. 

PWLPay User’s Manual    8‐1
Chapter 8 Navigating PWLPay

Figure 8.2 (a)   (b) 

The user can quickly show all of the navigation tree by selecting Expand All

shown at the bottom of the windows in Figure 8.2. This results in the window 

shown in Figure 8.3. Note that Expand All has been replaced by Collapse All

as an option at the bottom of the window. Selecting Collapse All would 

collapse the tree to the one shown in Figure 8.2a. The user can also collapse 

any individual tree level by clicking on the corresponding box. 

Figure 8.3 

Navigating The user can double click on any level of the navigation tree to go directly to 

a specified location. If the user clicks on either the state/agency level or the 

contract number level, he or she is taken to the Setup Project screen. If the 

user clicks on any level below the contract number, he or she is taken to the 

Add Samples window for the selected project. 

For example, for the tree shown in Figure 8.3, double clicking on a 

state/agency, such as South Carolina, or on a contract number, such as SC2008-001, takes the user to the Setup Project screen shown in Figure 8.4. 

Double clicking at the project level, such as Clemson Bypass – District-3-005-

1, or below, takes the user to the Add Samples window shown in Figure 8.5. 

PWLPay User’s Manual    8‐2
Chapter 8 Navigating PWLPay

Figure 8.4

Figure 8.5

 

The Navigation Pane

The Navigation pane allows the user direct return access to screens that he or 

she has previously visited in PWLPay. All of the visited screens appear as a 

list in chronological order, from top to bottom, on the shaded right edge of 

any PWLPay screen. An example is shown in Figrue 8.6, although, since it 

appears on every screen, the Navigation pane is visible in many of the figures 

in this manual. 

PWLPay User’s Manual    8‐3
Chapter 8 Navigating PWLPay

Figure 8.6

Navigating As can be seen in the list (above the arrow cursor) in the Navigation pane in 

Figure 8.6, the user has visited six different locations in addition to the Home

screen that automatically appears when PWLPay is opened. These include the 

Setup Specs, New Project / Edit Mode, Add Samples (twice), Data Analysis, 

and Random Sampling. The two Add Samples locations could have been for 

different projects or items, or could have been the same screen twice. 

If the user wishes, locations can be removed from the Navigation pane by 

right clicking the mouse on the location and then selecting Close in the 

window that pops up. This process is shown in Figure 8.7a. If desired, the 

user can open the Navigation Tree from the Navigation pane by clicking 

anywhere in the title region at the top of the Navigation pane. As shown in 

Figure 8.7b, this opens a window that allows the user to select to go to the 

Home page or to Open Navigation Tree. 

Figure 8.7 (a)       (b)   

Clicking on a location in the Navigation pane returns the user to the selected 

screen as it appeared when the user left that screen. That is, any selections 

that had been made or data that had been entered will be on the screen when 

it is re-visited. For example, Figure 8.8 shows what the screen might look 

like if Data Analysis is selected from the Navigation pane in Figure 8.6. 

PWLPay User’s Manual    8‐4
Chapter 8 Navigating PWLPay

Figure 8.8

Figure 8.8 shows that a Descriptive Statistics analysis was either set up or 

performed on item P-401 for the Clemson Bypass project on contract SC2008-001 in South Carolina when the user was on this screen. The user can 

now select a different analysis procedure or modify any of the other 

selections before again selecting Begin Analysis. 

Similarly, Figure 8.9 shows what the screen might look like if Random 

Sampling is selected from the Navigation pane in Figure 8.6. 

Figure 8.9

PWLPay User’s Manual    8‐5
Chapter 8 Navigating PWLPay

PWLPay User’s Manual    8‐6

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Chapter 9 Reports

9  Reports

Saving Reports

As seen in Chapter 6, an analysis can be saved so that it can be run with just 

one click of the mouse. This is an important and time saving capability since 

some analyses may be run repeatedly throughout the course of a project. The 

Reports icon extends this one-click capability to multiple analyses. 

 Creating a Report Before a report can be created the user must have saved at least one analysis 

as explained in Chapter 6. The Reports window can be reached by clicking 

on the Reports icon at the top of any screen and selecting New Report from 

the drop down menu as shown in Figure 9.1. 

Figure 9.1

The Reports window is shown in Figure 9.2. Any previously saved reports 

will appear in the Saved Reports drop down menu. In Figure 9.2 this 

window has defaulted to ..Add New Report, signifying that no reports have 

previously been saved. The user can create a new report by entering a Report

Name: and using the Add >> or Select All >> button to move saved analyses 

from the Analyses Not Added window to the Analyses Included in Report

window. 

Figure 9.2

 

PWLPay User’s Manual    9‐1
Chapter 9 Reports

Figure 9.3 shows a report named Clemson Bypass that is ready to be saved 

by clicking on Save. 

Figure 9.3

The user selects Save to save the report under the name appearing in the 

Report Name: box. Saved reports can be accessed directly from the Reports

icon, as shown in Figure 9.4. 

Figure 9.4

Selecting a report in the drop down menu shown in Figure 9.4 causes all of 

the analyses assigned to the report to be executed and to appear in the 

Navigation pane for easy access. For example, if Clemson Bypass is selected 

in Figure 9.4, the four analyses assigned to it in Figure 9.3 will be completed 

and the screen shown in Figure 9.5 will be displayed. 

The four analyses assigned to the Clemson Bypass report (see Figure 9.3) 

have been conducted and they now appear in the Navigation pane. The last 

analysis in the list, Process Control, is active on the screen shown in Figure

9.5. The user can click on any of the analyses in the Navigation pane to 

switch to the selected completed analysis screen. 

PWLPay User’s Manual    9‐2
Chapter 9 Reports

Figure 9.5

Deleting a Saved Report

 A saved report can be deleted by clicking on Delete this saved report as 

shown in Figure 9.3. 

PWLPay User’s Manual    9‐3
Chapter 9 Reports

PWLPay User’s Manual    9‐4

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Chapter 10 Working with Charts

10  Working with Charts

Printing, Saving, and Copying Charts

As seen in previous chapters, there are many charts (plots) that can be 

displayed in PWLPay. Nearly all of these charts can be printed, saved, or 

modified by clicking the right mouse button while the cursor is over any area 

of the plot. When this is done, the menu shown in Figure 10.1 pops up. 

Figure 10.1

 Printing a Chart In the popup menu shown in Figure 10.1, clicking Print Setup or Print Chart

opens the Print menu for the default printer associated with the computer on 

which PWLPay is currently running. 

Saving a Chart In the popup menu shown in Figure 10.1, clicking Save to File opens the 

submenu shown in Figure 10.2. Save Chart to File allows the user to save the 

chart as a picture in an EMF, BMP, GIF, JPG or PNG format. Save Data to 

File allows the user to save the chart data in a CSV file format. 

Figure 10.2 

PWLPay User’s Manual    10‐1
Chapter 10 Working with Charts

Copying a Chart In the popup menu shown in Figure 10.1, clicking Copy to Clipboard opens 

the submenu shown in Figure 10.3, and allows the user to copy the chart to 

the clipboard as a picture in one of the file formats shown. The chart can then 

be pasted from the clipboard into another program. 

Figure 10.3 

Viewing Chart Data In the popup menu shown in Figure 10.1, clicking Show Data opens the 

Graph Data window shown in Figure 10.4. This window allows the user to 

see the values of the data points that are plotted in the chart. 

Figure 10.4 

Modifying Chart Properties

In the popup menu shown in Figure 10.1, clicking Properties allows the user 

to change the properties of the chart with respect to colors, scales, titles, etc. 

Selecting Properties opens the Chart Properties window shown in Figure

10.5. 

The charting module in PWLPay was licensed from ComponentOne (hence 

the in the lower left corner of the Chart Properties window) and 

provides powerful charting capabilities. Most of these capabilities are not 

needed for PWLPay, but they are still available from the Chart Properties

window. 

PWLPay User’s Manual    10‐2
Chapter 10 Working with Charts

Figure 10.5 

Chart Type Gallery is highlighted in the upper left corner of the Chart Properties window 

shown in Figure 10.5. From Gallery the user can select from several simple 

and complex types of charts (e.g., Scatter, Step, Column, Bar, Area, etc. in 

Figure 10.5). The Line chart type is the one of most interest to the user since 

this chart is used nearly exclusively in PWLPay. The user might switch to a 

Scatter plot, for example, if they did not want lines connecting the points on 

a control chart. However, this manual limits its discussion to the properties of 

Line charts since they predominate in PWLPay. 

Chart Properties Note in Figure 10.5 that a small preview of the chart is always available in 

the lower left corner of the Chart Properties window. Clicking the Preview

tab displays a larger preview of the chart (see Figure 10.6). 

Selecting Data in the upper left window (see Figure 10.5) allows the user to 

change the line style, thickness, and color of the lines and/or symbols in the 

plot. These options are shown in Figure 10.7. 

PWLPay User’s Manual    10‐3
Chapter 10 Working with Charts

Figure 10.6 

Figure 10.7 

Similarly, selecting AxisX or AxisY in the upper left window allows the user 

to change the axis title, font style, font size, color, and location. The user can 

select Gridlines to add them to or remove them from either axis. The scale of 

either axis can be changed by selecting the Scale tab. Some of these options 

are shown in Figure 10.8. 

Chart Appearance Selecting Appearance in the upper left window (see Figure 10.9) allows the 

user to add or to modify the appearance of a Header, Footer, or Legend, or to 

modify the appearance of the ChartArea or PlotArea. 

PWLPay User’s Manual    10‐4
Chapter 10 Working with Charts

Figure 10.8 

Figure 10.9 

PWLPay User’s Manual    10‐5
Chapter 10 Working with Charts

PWLPay User’s Manual    10‐6

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Chapter 11 Computational Methods

11 Computational

Methods

Overview

PWLPay obviously incorporates a number of computational methods in its 

various analysis modules. Some of these are obvious, while others may not 

be as obvious to the user of the program. This chapter is not intended to 

explain in detail the theory that underlies these methods. It does, however, 

expand on some computational methods that may not be familiar to the 

typical PWLPay user. 

Tools is one of the options that appear at the top of every PWLPay screen. 

The drop down menu under Tools has Computational Routines as one of its 

options (see Figure 11.1). This chapter explains the various options that are 

available under Computational Routines. The chapter also includes a 

discussion of the various normality testing methods used by PWLPay. 

Figure 11.1

Distribution Calculator

PWLPay uses several probability distributions while performing the various 

analyses of which it is capable. Selecting Tools ` Computational Routines ` 

Distribution Calculator opens the Distribution Calculator window shown in 

Figure 11.2. This window allows the user to calculate areas under the t-, F-, 

and Beta distributions. The critical values corresponding to any selected level 

of significance can also be calculated for each of these distributions. 

Figure 11.2 

PWLPay User’s Manual    11‐1
Chapter 11 Computational Methods

There is no need to use these calculations when using PWLPay pay. The 

appropriate calculations are automatically performed by the program. It is 

included only to allow the user to verify the PWLPay calculations if desired. 

The values calculated using the Distribution Calculator can be checked 

against those calculated by other programs such as Excel. 

If desired, the user could use this window as a separate calculation tool when 

working with these distributions in settings other than using PWLPay. 

 T DIST Selecting the T DIST button in Figure 11.2 allows the user to calculate the 

t-distribution and its inverse. PWLPay uses these calculations when 

performing t-tests as part of the verification analysis option. The tdistribution and t-tests are explained in detail in numerous statistical tests, 

and are covered in this manual only for illustrative purposes. 

The t-distribution and the values calculated with T DIST are illustrated in 

Figure 11.3. 

Figure 11.3 (a) (b) 

 Example 1. Suppose that the t-statistic from a one-tailed t-test is determined 

to be 1.42 and there are 10 degrees of freedom. The Result: in Figure 11.4

shows that the probability of receiving a t-statistic greater than or equal to 

1.42 (i.e., the non-shaded area in Figure 11.3a) is 0.093. 

Figure 11.4 

 Example 2. Suppose that in the previous example the user wishes to know 

the critical t-value that corresponds to a level of significance, α = 0.05. In 

this case the user checks the inverse box and enters 0.05 in the X box. The 

Result: in Figure 11.5 shows that the critical value is 2.228. 

Figure 11.5 

PWLPay User’s Manual    11‐2
Chapter 11 Computational Methods

 F DIST Selecting the F DIST button in Figure 11.2 allows the user to calculate the 

F-distribution and its inverse. PWLPay uses these calculations when 

performing F-tests as part of the verification analysis option. The Fdistribution is based on the ratio of variances, and is characterized by the 

degrees of freedom in the numerator and degrees of freedom in the 

denominator of this ratio. 

In Figure 11.6, Degrees of Freedom 1 is for the numerator, while Degrees of

Freedom 2 is for the denominator. It is important to have these values correct 

since the degrees of freedom are not interchangeable. That is, for example, 

the F-distribution with 10 degrees of freedom in the numerator and 12 in the 

denominator is NOT the same as one with 12 degrees of freedom in the 

numerator and 10 in the denominator. 

Figure 11.6 

The F-distribution and the values calculated with F DIST are illustrated in 

Figure 11.7. 

Figure 11.7 (a) (b) 

 Example 1. Suppose that the F-statistic from an F-test is determined to be 

2.906, there are 20 degrees of freedom in the numerator, and 5 degrees of 

freedom in the denominator. The Result: in Figure 11.8 shows that the 

probability of receiving an F-statistic greater than or equal to 2.906 (i.e., the 

non-shaded area in Figure 11.7a) is 0.120. 

Figure 11.8 

 Example 2. Suppose that in the previous example the user wishes to know 

the critical F-value that corresponds to a level of significance, α = 0.05. In 

this case the user checks the inverse box and enters 0.05 in the X box. The 

Result: in Figure 11.9 shows that the critical value is 4.558. 

PWLPay User’s Manual    11‐3
Chapter 11 Computational Methods

Figure 11.9 

 Beta Selecting the Beta button in Figure 11.2 allows the user to calculate the 

Beta distribution and its inverse (see Figure 11.10). PWLPay uses these 

calculations when determining PWL values in the payment factor and risk 

analysis options. The Beta distribution is particularly useful when dealing 

with percentages since it always ranges between 0 and 1, and because it can 

take on a wide variety of shapes depending upon the value of its two 

parameters, Alpha (α) and Beta (β). Figure 11.11 illustrates this versatility of 

shapes. 

Figure 11.10 

Figure 11.11 

A Beta distribution illustrating the values calculated with Beta is illustrated 

in Figure 11.12. 

Figure 11.12 (a) (b) 

PWLPay User’s Manual    11‐4
Chapter 11 Computational Methods

 Example 1. Suppose that we have a Beta distribution with α = β = 2. The 

Result: in Figure 11.13 shows that the percentage of the distribution with 

values less than or equal to 0.70 (i.e., the non-shaded area in Figure 11.12a) 

is 0.784. 

Figure 11.13 

 Example 2. Suppose that in the previous example the user wishes to know 

the value for which 60 percent of the distribution is less than. In this case the 

user checks the inverse box and enters 0.60 in the X box. The Result: in 

Figure 11.14 shows that the value is 0.567. 

Figure 11.14 

Correlation

PWLPay can perform analyses for independent or correlated characteristics. 

Selecting Tools ` Computational Routines ` Correlation ` provides the user 

with the two options shown in Figure 11.15. Each of these is discussed in a 

separate section below. 

Figure 11.15 

Graphical Graphical Illustration provides the user visual examples of various levels of 

Illustration correlation between two variables. It also illustrates the effectiveness of the 

random number generating routines used by PWLPay. Selecting Graphical

Illustration opens the Correlated Illustration window shown in Figure 11.16. 

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Chapter 11 Computational Methods

Figure 11.16 

As shown in Figure 11.16, the Choose the number of characteristics for the 

example drop down menu allows the user to select two, three, or four 

characteristics for which random numbers will be generated. Also note that 

in the Mode section the user can choose from Random Normal Generator or 

Histogram (1,000 times). These are discussed in separate sections below. 

Random Normal Random Normal Generator results in the computer generation of up to four 

Generator random normal characteristics with a desired correlation structure that is 

input by the user. The number of values generated for each characteristic is 

that shown in the Iterations box. The number of Iterations is theoretically 

unlimited. However, for illustration purposes the default value of 10,000

should be quite sufficient in most cases. 

Once the user selects the number of characteristics, a correlation matrix (see 

Figure 11.17) is displayed and the user then enters the correlation 

relationships between each pair of characteristics. 

Figure 11.17

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Chapter 11 Computational Methods

PWLPay automatically checks and issues a warning message in the event that 

there are inconsistencies in the correlation coefficients entered. For example, 

if Characteristic 1 is positively correlated with both Characteristic 2 and 

Characteristic 3, then Characteristic 2 and Characteristic 3 could not be 

negatively correlated. So, if the correlation coefficients shown in Figure

11.18 were entered, when Continue was selected, the warning message in 

Figure 11.18 would be displayed. 

Figure 11.18 

The user can use the default values, mean = 0 and standard deviation = 1, or 

can enter a desired mean and standard deviation. If a consistent correlation 

matrix has been entered, selecting Continue begins the random number 

generating process and then presents the results, including a plot of the 

values generated. 

Example. Suppose the user has entered the values shown in Figure 11.19. 

Selecting Continue results in the screen shown in Figure 11.20. 

Figure 11.19 

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Chapter 11 Computational Methods

Figure 11.20 

Note from Figure 11.19 that for each characteristic the simulated mean was 0

and standard deviation was 1. Also, the correlation coefficient between 

Characteristics 1 and 2 was 0.75. Figure 11.20 shows that the means for the 

10,000 random numbers generated (i.e., 0.01448 and –0.00177) were close to 

the population mean of 0, and that the standard deviations of the generated 

numbers (i.e., 0.99124 and 1.00493) were close to the population value of 1. 

Finally, the correlation coefficient for the generated numbers (i.e., 0.75114) 

was close to the input value of 0.75. 

The positive slope of the plot of the generated numbers shows that the two 

characteristics are positively correlated. The amount of spread around the 

line that fits the data is consistent with the correlation coefficient of 0.75. 

The results in Figure 11.20 are for the correlation between Characteristics 1 

and 2. This is shown by the Results for 1 & 2 tab that is highlighted at the top 

left of the window. The user can select the other tabs to see the 

Results for 1 & 3 or Results for 2 & 3. 

Figure 11.21 shows the plots for the generation of two variables with various 

correlation coefficients. 

Histogram The output from Random Normal Generator (see, for example, Figure 11.20) 

(1,000 times) includes the mean, standard deviation, skewness coefficient, and correlation 

coefficient for the generated values. Selecting Histogram (1,000 times)

causes the Random Normal Generator function to be repeated 1,000 times. It 

then plots histograms of the 1,000 means, standard deviations, skewness 

coefficients, and correlation coefficients. Note that when Histogram is 

selected the analysis is always performed using only Characteristic 1 and 

Characteristic 2, regardless of the number of characteristics showing in the 

Choose the number of characteristics for the example window. 

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Chapter 11 Computational Methods

(a) (b) 

Figure 11.21 (c) (d) 

Since Histogram repeats the Random Normal Generator function 1,000

times, the user should consider carefully the value selected in the Iterations

window. A large value for Iterations can lead to a long wait when repeated 

1,000 times. For this reason, the default value for Iterations is 10 when 

Histogram is selected. However, the number of iterations can be set to 

whatever value is desired. 

Important Note on 

Number of 

Iterations 

The Histogram option allows the user another method to verify that 

PWLPay’s random number generating routine is operating properly. This is 

illustrated and discussed in the following examples. 

Example 1. Suppose the user has entered the values shown in Figure 11.22. 

The correlation coefficient has been entered as 0.0, which would be the case 

if the two characteristics are independent of one another. The default Mean, 

Std dev, and Iterations values have been used. 

Figure 11.22

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Chapter 11 Computational Methods

Distribution of Sample Means. Selecting Continue in Figure 11.22 results in 

a screen similar to the one shown in Figure 11.23. The histogram shows the 

distribution of the 1,000 sample means of size n = 10 that were generated 

from a normal distribution with mean = 0 and standard deviation = 1 (see

Figure 11.22). A normal distribution should have a skewness coefficient = 0. 

The simulated values in Figure 11.23 (i.e., Mean = –0.00879, Std dev =

0.3179, Skewness = 0.02502) compare quite closely with those predicted by 

theory. 

Figure 11.23 

The Central Limit Theorem states that the distribution of sample means 

should be normally distributed with mean equal to the population mean, i.e., 

0, and standard deviation equal to the population standard deviation divided 

by the square root of n, i.e., 1⁄√10 ൌ 0.316. The shape of the histogram and 

the Skewness coefficient of 0.02502 indicate that it is reasonable to assume 

the values are normally distributed. 

Note that the Histogram Bins box shows a value of 30. This is the number of 

bars in the histogram plot. This value can be set to any number between 1 

and 100. However, around 30 bins is usually a good choice for a sample of 

1,000 values. 

Distribution of Sample Standard Deviations. Selecting the Standard 

Deviation tab in Figure 11.23 results in the screen shown in Figure 11.24. 

The histogram shows the distribution of the 1,000 sample standard deviations 

of size n = 10 that were generated from a normal distribution with mean = 0

and standard deviation = 1. 

The average sample standard deviation is known to be a biased estimator for 

the population standard deviation. It can be shown theoretically that the 

average standard deviation for samples of size n = 10 taken from a 

population with standard deviation = 1 should equal 0.9727. This compares 

favorably with the simulated value of 0.97481. 

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Chapter 11 Computational Methods

Figure 11.24 

Unlike the distribution of sample means, the distribution of sample standard 

deviations is not normally distributed, but is positively skewed, with the 

degree of skewness decreasing as the sample size, n = 10 here, increases. The 

skewness of 0.21638 for the simulated values is consistent with what would 

be expected. 

Distribution of Sample Skewness Coefficients. Selecting the Skewness tab 

in Figure 11.23 results in the screen shown in Figure 11.25. The histogram 

shows the distribution of the 1,000 sample skewness coefficients of size n =

10 that were generated from a normal distribution with mean = 0 and 

standard deviation = 1. 

Since the values are simulated from a normal distribution, the average of the 

1,000 skewness coefficients should be approximately 0, i.e., the skewness of 

a normal distribution. In Figure 11.25, the average skewness coefficient of 

-0.03692 compares favorably with the theoretical value of 0. It can be shown 

theoretically that the standard deviation of the sample skewness coefficients 

for samples of size n = 10 should be 0.687. This compares reasonably well 

with the simulated value of 0.65425. 

Distribution of Sample Correlation Coefficients. Selecting the Correlation 

tab in Figure 11.23 results in the screen shown in Figure 11.26. The 

histogram shows the distribution of the 1,000 sample correlation coefficients 

of size n = 10 that were generated from a normal distribution with mean = 0

and standard deviation = 1. 

Since the values are simulated from two normal variables with 0 correlation, 

the average of the 1,000 correlation coefficients should be approximately 0. 

In Figure 11.26, the average skewness coefficient of –0.00977 compares 

favorably with the theoretical value of 0. It can be shown theoretically that 

the standard deviation of the sample correlation coefficients for samples of 

size n = 10 should be approximately 0.333. This compares favorably with the 

simulated value of 0.33844. 

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Chapter 11 Computational Methods

Figure 11.25 

Figure 11.26 

In Figure 11.26, the histogram is approximately symmetric as evidenced by 

the skewness coefficient relatively close to 0. This is a unique case that 

applies when the two populations that are being simulated have 0 correlation. 

If there is either a positive or negative correlation between the two 

populations, then the histogram of the sample correlation coefficients will be 

skewed. 

Figure 11.27 shows the input screen for the case where the two 

characteristics to be simulated have a correlation of +0.5 between them. The 

resulting histogram of the correlation coefficients is shown in Figure 11.28. 

Note that the histogram is negatively skewed. The maximum possible value 

for any individual sample correlation coefficient is +1.0. Therefore, the 

sample cannot over estimate the population correlation by more than 0.5. 

However, it is possible that the sample could under estimate the population 

by a much greater amount (-1.5 in this problem). These facts lead to the 

distribution shape shown in the histogram in Figure 11.28. 

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Chapter 11 Computational Methods

Figure 11.27 

Figure 11.28 

If the two populations were negatively correlated, then the histogram of the 

sample correlation coefficients would be positively correlated for the same 

reasons. Figure 11.29 shows the sampling distributions for populations with 

several different population correlation values. 

Correlated Correlated Random Numbers (see Figure 11.15) provides the user with a 

Random Numbers check of how well PWLPay is able to generate random values with the 

correlation structure that is desired. It is another way to illustrate the 

effectiveness of the random number generating routines used by PWLPay. 

Selecting Correlated Random Numbers in Figure 11.15 opens the screen 

shown in Figure 11.30. 

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Chapter 11 Computational Methods

(a) (b) 

Figure 11.29 (c) (d) 

Figure 11.30

The Select the number of Correlations drop down menu allows the user to 

select 2, 3, 4, or 5 characteristics for which to generate correlated random 

values. Figure 11.31 shows the screen after the correlation coefficients for 5 

correlated characteristics have been entered and Calculate has been selected. 

As can be seen, the simulated values have similar correlation coefficients to 

those desired. 

PWLPay User’s Manual    11‐14
Chapter 11 Computational Methods

Figure 11.31

Normality Testing

As discussed in Chapter 6, PWLPay can be configured to automatically run a 

quick check for normality whenever new data are entered. PWLPay can also 

perform two more detailed procedures to check for normality. These 

procedures are illustrated in Chapter 6 and are discussed in more detail in the 

following sections. 

Anderson‐Darling There are a large number of statistical goodness of fit (GOF) tests. The 

Test typical PWLPay user may be familiar with GOF tests such as the Chi-Square 

or Kolmogorov-Smirnov (K-S) tests. These are general purpose GOF tests 

that are not particularly powerful for detecting departures from normality. 

PWLPay uses the Anderson-Darling GOF test to identify data that do not 

appear to have come from a normal population. The Anderson-Darling test, 

like the K-S test, is based on differences between the cumulative distribution 

of the observed data and that of the theoretical distribution against which the 

data are being compared. However, unlike the K-S test statistic, which is 

based on only the maximum individual difference, the AndersonDarling test statistic, A2

, includes all of the sample values in its 

calculation.

PWLPay uses a form of the Anderson-Darling test that has been 

modified to apply specifically to testing for normality when the true 

population mean and standard deviation are unknown and are 

estimated from the sample mean and standard deviation values. The 

procedure for this test is valid for sample sizes ≥ 8.

Important Note on 

Sample Size for 

Anderson-Darling 

Test 

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Chapter 11 Computational Methods

Test Procedure. PWLPay performs all necessary calculations and notifies the 

user whether or not the normality assumption is reasonable. As such, it is not 

necessary for the user to understand the steps involved in the AndersonDarling GOF test. The steps are presented only briefly here for users who 

may be interested. A more thorough description and explanation of the 

Anderson-Darling test can be found in statistics texts. 

Step 1: Arrange the n sample values in ascending order such that 

nn ≤ ≤ ≤ ≤ xxxx 21 ... −1

Step 2: For all xi, calculate the standardized values ( )/ i i y xx = − s . 

Step 3: For each sample value, i = 1, ... , n, calculate Pi, which is the area 

under a standard normal curve that is less than or equal to the 

standard score yi. This is the cumulative probability of obtaining a 

value less than or equal to yi. 

Step 4: Calculate the Anderson-Darling test statistic, A2

: 

∑[ ] { } =

−+−−= −+ −

n

i

i in A nnPPi

1

1

2 /)1ln(ln)12(

Step 5: Calculate the modified test statistic, A* (modified specifically for 

testing for normality when μ and σ are unknown): 

⎥

⎦

⎤ ⎢

⎣

⎡ ++= 2

2 25.275.0 0.1*

n n

AA

Step 6: Reject the null hypothesis of normality if A* exceeds the critical 

value for the selected level of significance, α, in Table 11.1. 

α 0.15 0.10 0.05 0.025 0.01

Table 11.1 Critical A* 0.561 0.631 0.752 0.873 1.035

Since the Anderson-Darling test is quite sensitive to departures from 

normality, in PWLPay it is always conducted at the 0.01 level of significance 

regardless of what value has been selected with the Tools ` Normality Check

` Alpha Value option. 

Important Note 

Regarding Level of 

Significance 

qq‐Plot The quantile-quantile (qq-) plot is a general method for determining if two 

samples come from the same population. The qq-plot can also be used to 

determine if one sample comes from a specific theoretical distribution 

(simply replace the other sample by the target distribution), in which case it 

is sometimes called a probability plot, or, particularly, a normal probability 

plot if the target distribution is normal. PWLPay can develop and display 

normal probability plots to evaluate whether or not a set of test data 

reasonably approximate a normal distribution. 

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Chapter 11 Computational Methods

The normal qq-plot plots the sample quantiles against the theoretical 

quantiles (sometimes called the expected quantiles) of the standard normal 

distribution. Suppose the sample is of size n. First, rank order the sample 

values from lowest to highest. Take any observation x and record its rank k. 

Then x is the p(k)-quantile of the sample, where 

= − /)5.0()( nkkp

In other words, the probability of an observation being below x (or the k-th 

order statistic) is p(k). This probability is usually taken to be simply k/n, but 

that would be biased for our purpose (see comments below). The expected 

quantile associated with x is then the p(k)-quantile of the standard normal 

distribution. For example, the 0.05 quantile of the standard normal 

distribution is –1.645, i.e., a normal distribution has 0.05 of its area less than 

or equal to a z value of –1.645. The p(k)-quantile values could be looked up 

from the standard normal table, however, PWLPay automatically calculates 

these values for the user. 

While PWLPay uses the method shown above, other methods for determining 

p(k) are also used. The most natual choice is obviously p(k) = k/n, which is 

obviously biased when k = n since for a standard normal distribution only 

positive infinity has a quantile of 1.0. Like the equation above, other versions 

of p(k) attempt in some way to correct this bias. 

Important Note 

Regarding 

Determining p(k) 

Example 1: Computing the Expected Quantiles. Suppose the sample size is 

n = 50. After rank ordering the sample the 4th smallest value is x = 10 (i.e., 

the rank for the observation x = 10 is 4). Then x = 10 is the 

[(4 – 0.5) / 50 = 0.07] quantile of the sample. Thus the expected quantile 

associated with x = 10 is the 0.07-quantile of the standard normal 

distribution, which is –1.476. 

Drawing Conclusions. If the sample comes from the standard normal 

distribution, the sample quantiles and theoretical quantiles should be very 

close and thus the displayed points should fall around the straight line 

passing through the origin with slope 1. If the sample comes from any 

normal distribution, the displayed points should approximately fall in a 

straight line (because any normal distribution is a linear transformation of the 

standard normal distribution and thus the plot shifts and rotates the 

previously described straight line, but it remains straight). Thus, a straight 

line for the qq-plot indicates the sample comes from a normal distribution, 

while any significant departure from linearity shows evidence against 

normality. 

Example 2: Interpreting qq‐Plots. Figure 11.32 shows four probability 

density functions, one for the standard normal distribution (Figure

11.32a), one for the normal distribution with mean 3 and standard 

deviation 2 (Figure 11.32b), one for the t-distribution with degrees of 

freedom 3 (Figure 11.32c), and one for the -distribution with 

degrees of freedom 5 (Figure 11.32d). 

2 χ

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Chapter 11 Computational Methods

PWLPay User’s Manual    11‐18

(a) ‐10 0 10 (b) 

N(μ = 0, σ = 1)

‐10 0 10

N(μ = 3, σ = 2)

Figure 11.32 (c) ‐10 0 10 (d)

t(df = 10)

0 10

χ2(df = 5)

20

Figure 11.33 shows qq-plots for samples of size n = 200 drawn from the 

populations shown in Figure 11.32. Clearly Figures 11.33a and 11.33b

indicate normality, while Figures 11.33c and 11.33d do not. It should be 

noted that it is difficult to distinguish a t-distribution from the normal 

using histograms. But the qq-plot is able to detect it. We know that tdistributions are flatter and thus have heavier tails than the standard 

normal distribution. Accordingly, the quantiles for the t-distributions 

depart from those of the standard normal distribution at the lower end 

and at the upper end, as can be seen from Figure 11.33c.

(a) (b) 

‐4

‐3

‐2

‐1

0

1

2

3

4

‐4 ‐3 ‐2 ‐10 1 2 3 4

Z‐Value

Sample Result

N(μ = 0, σ = 1)

‐4

‐3

‐2

‐1

0

1

2

3

4

‐4 ‐20 2 4 6 8

Z‐Value

Sample Result

N(μ = 3, σ = 2)

10

Figure 11.33 (c) (d) 

‐5

‐4

‐3

‐2

‐1

0

1

2

3

4

‐8 ‐6 ‐4 ‐20 2 4 6 8

Z‐Value

Sample Result

t(df = 3)

‐4

‐3

‐2

‐1

0

1

2

3

4

0 5 10 15

Z‐Value

Sample Result

Chi‐Sq(df = 5)