Z-1

Z.0 PRECISION AND BIAS

Z.1 PAIRED SAMPLING TRAIN PRECISION

Z.1.1 What Is Precision and Why Is It Important?

Precision is a measure of the agreement between two or more independent observations of the

same physical quantity obtained under the same conditions. When paired sampling trains are

used, assessing the precision of the two data points collected for a given test run is important in

identifying possible problems with the sampling train or sample collection method. Although

the PM concentrations measured by the two sampling trains should be the same for a test run,

there invariably are differences in the two measurements. Typically, the differences in emissions

between the two sampling trains are insignificant. In some cases, it is unclear whether the two

data points are in relative agreement or if there are significant differences, which could indicate a

problem with the sampling instruments, collection methods, or analytical procedures. 

Z.1.2 Does PS-11 Require Me To Evaluate the Precision of Paired Sampling Train Data?

Evaluating the precision of paired sampling train data is not required under PS-11. However, in

Section 8.6(1)(i) of PS-11, it is recommended that you use paired sampling trains and screen the

data for precision when conducting a correlation test or an RCA Likewise, the procedures

described here for evaluating paired sampling train precision are recommendations and not

requirements. Section 8.6(3) of PS-11 specifies that when conducting a correlation test, you may

reject the results of up to five test runs without explanation; additional test runs can be rejected

under certain conditions. The same provisions apply to RCAs. The procedures described in the

following paragraphs can help in deciding which test runs, if any, you should consider rejecting.

Z.1.3 What Methods Can I Use To Determine the Precision of Paired Sampling Train Data?

This section discusses two methods for evaluating the precision of paired sampling train data. 

Section Z.1.4 describes a method that is based on the determination of the relative standard

deviation (RSD). Section Z.1.5 presents an alternative method, which involves calculating the

regression residuals for the data set and comparing them with the standardized residuals for the

test run. Section Z.1.6 compares the two methods and presents recommendations on which

procedure to use and how to use the results of evaluations of data precision for initial correlation

tests and RCAs.
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Z.1.4 How Do I Evaluate Precision Using the Relative Standard Deviation?

Determining the RSD is a relatively simple and straightforward procedure for evaluating the

precision of paired sampling data. This procedure, which was included in earlier proposals of

PS-11, currently appears in Section 12.2 of Method 5i. 

The RSD presented in this discussion and in Method 5i is not the true RSD but is actually a

measure of the standardized half range. For two data points, the standardized half range differs

from the true RSD by a factor of (1//2) or 0.707; that is, the RSD (as presented here) equals the

true RSD × 0.707.

Evaluating the precision of paired sampling train data using the RSD entails the following steps:

1. Calculate the RSD for each test run;

2. Determine the average PM concentration for each run;

3. Determine the recommended maximum RSD for each run, based on the average PM

concentration for the run; and

4. Compare the calculated RSD with the recommended maximum RSD for each run and

flag those test runs for which the calculated RSD exceeds the recommended

maximum RSD value. 

The following paragraphs describe each of these steps in more detail. Example calculations are

also presented.

Step 1: Calculate RSD

The RSD is calculated for each test run using Equation Z.1-1:

(Equation Z.1-1) ( )

( ) RSD

C C

C C

a b

a b

= × −

+

100%

where

RSD = relative standard deviation, percentage

Ca = concentration measured using Train A, units of concentration

Cb = concentration measured using Train B, same units as Ca

|Ca-Cb| = absolute value of the difference between Ca and Cb, same units of

concentration.

The concentration units for the variables Ca and Cb do not matter, provided the same units are

used for both concentrations (e.g., both concentrations could be in units of mg/dscm or both

could be in units of mg/acm). 
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In the following example calculation, assume the concentration for Train A was 9.2 mg/acm, and

the concentration for Train B during the same test run was 7.6 mg/acm.

( )

( ) RSD = × −

+

100% = ×=

92 76

92 76

16

168 100 9 5% . .. .

.

. .

Step 2: Determine Average PM Concentration

The recommended maximum RSD (RSDr) is based on the average PM concentration (Cave) for

the test run. For the previous example, the average PM concentration for the run is

C

C C ave

a b = + = + = 2

9.2 7.6

2 8.4

Step 3: Determine Recommended Maximum RSD

The recommended maximum RSD is a function of the average PM concentration for the two

data points, as follows:

# For average PM concentrations that are at least 10 mg/dscm (or at least 10 mg/acm),

the recommended maximum RSD is 10 percent;

# For average PM concentrations that are less than or equal to 1.0 mg/dscm (or less

than or equal to 1.0 mg/acm), the recommended maximum RSD is 25 percent; and

# For average PM concentrations between 1.0 and 10 mg/dscm (or between 1.0 and

10 mg/acm), the recommended maximum RSD is determined using Equation Z.1-2,

which is simply a linear interpolation between 10 and 25 percent:

RSDr = (26.67 - 1.67Cave) (Equation Z.1-2)

where

RSDr = recommended maximum RSD for average PM concentrations between 1.0 and

10, percent

Cave = average PM concentration for the two sampling trains, units of concentration.

For the above example, the average PM concentration is greater than 1.0 mg/dscm and less than

10 mg/dscm; therefore, Equation Z.1-2 is used to determine the recommended maximum RSD as

follows:

RSDr = 26.67 - 1.67 × Cave = 26.67 - 1.67 × 8.4 = 12.6%
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Step 4: Compare Calculated RSD to Recommended Maximum RSD

The calculated RSD is 9.5 percent, which is less than the recommended maximum RSD of 12.6

percent. Therefore, the test run satisfies the criterion for RSD.

Example Problem: How Do I Evaluate the Precision of Paired Sampling Train Data

Using the RSD?

For the initial correlation test of a PM CEMS, 20 test runs are conducted using paired sampling

trains. The results of the test are summarized in Table Z.1-1. The table also shows for each test

run the average PM concentration, the RSD calculated using Equation Z.1-1, and the

recommended maximum RSD. Finally, the table indicates whether each test run met the RSD 

criterion.

Table Z.1-1. Example RSD Calculations – Results of Initial Correlation Test

Run

PM Concentration, mg/dscm Calculated

RSD, %

Recommended

maximum RSD,

%

Does the test run

satisfy the RSD

Train A Train B Average criterion?

1 2.6 4.2 3.4 23.5 21 No

2 3.9 4.4 4.2 6 19.7 Yes

3 4.2 2.8 3.5 20 20.8 Yes

4 8.6 10.3 9.5 9 10.9 Yes

5 15.1 14.5 14.8 2 10 Yes

6 18.3 19.7 19.0 3.7 10 Yes

7 22.4 21.3 21.9 2.5 10 Yes

8 24.5 20.2 22.4 9.6 10 Yes

9 24.3 36.9 30.6 20.6 10 No

10 32.9 32.2 32.6 1.1 10 Yes

11 6.4 7.4 6.9 7.2 15.1 Yes

12 9.5 14.2 11.9 19.8 10 No

13 12.4 15.1 13.8 9.8 10 Yes

14 14.2 11.2 12.7 11.8 10 No

15 7.8 7.1 7.5 4.7 14.2 Yes

16 21.3 18.3 19.8 7.6 10 Yes

17 4.6 13.5 9.1 49.2 11.6 No

18 28.7 24.2 26.5 8.5 10 Yes

19 12.6 14.8 13.7 8 10 Yes

20 10.5 11.3 10.9 3.7 10 Yes
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As shown in the table, Runs 1, 9, 12, 14, and 17 did not meet the precision criterion for RSD. 

For Run 1, the average PM concentration was 3.4 percent, and the RSD was 23.5 percent. Since

the average PM concentration is between 1.0 and 10 mg/dscm, Equation Z.1-2 is used to

determine the recommended maximum RSD for this run. Using that equation, the recommended

maximum RSD for Run 1 is calculated to be 21 percent (26.67 - 1.67 × 3.4 = 21.0). For Run 9,

the RSD (20.6 percent) exceeded the recommended maximum RSD of 10 percent. For Runs 12

and 14, the RSDs were 19.8 percent and 11.8 percent, respectively, both of which exceed the

recommended maximum of 10 percent. Finally, the RSD for Run 17 was 49.2 percent, which

exceeds the recommended maximum RSD of 11.6 percent for that run, as determined by

Equation Z.1-2.

Section 8.5(3) of PS-11 requires a minimum of 15 test runs for the initial correlation test, and

paragraph 8.5(3)(iii) allows up to 5 test runs to be discarded without explanation. In the above

example, the facility might consider discarding the 5 test runs for which the RSD exceeded the

recommended maximum RSD; if it did so, there still would be a sufficient number of test runs to

develop the initial correlation. However, the facility would not be required to reject any of the

data; it could decide to keep the results of any or all of the test runs that did not meet the RSD

criterion.

Z.1.5 How Do I Evaluate the Precision of Paired Sampling Train Data Using Residuals?

A residual is defined as the difference between an observed value and the fitted value that is

determined using a regression equation. Using residuals to evaluate the precision of paired

sampling train data is an iterative process, which consists of the following steps:

1. Determine the linear regression equation for the data;

2. Calculate the residual and standardized residual for each test run;

3. Eliminate any run for which the absolute value of the standardized residual exceeds

3.0; 

4. Recalculate the residuals and standardized residuals for each test run; and

5. Repeat Steps 3 and 4 until the absolute value of the standardized residual for each run

is less than or equal to 3.0.

The following paragraphs describe these steps in more detail. An example problem to

illustrate all of the calculations for evaluating paired sampling data precision using residuals also

is presented.

Step 1: Determine Regression Equation

Before the residuals can be calculated, the regression equation must be determined. For paired

sampling train data, a linear regression model is used. The procedure for determining the linear

regression equation is identical to that specified in Section 12.3(1) of PS-11 for developing a

linear correlation model. The data from either sampling train can be used for the dependent
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variable (y) or independent variable (x). For this discussion, the data from Train B represent the

dependent variable, and the data from Train A represent the independent variable. The

regression equation is of the form presented in Equation Z.1-3:

y b bx $ = + (Equation Z.1-3) 0 1

where

í = predicted (fitted) PM concentration for Train B

b0 = intercept of the regression line, as calculated using Equation Z.1-4

b1 = slope of the regression line, as calculated using Equation Z.1-6

x = PM concentration measured by Train A.

The y intercept (b0) of the regression line is determined using Equation Z.1-4:

b ybx (Equation Z.1-4) 0 1 =−⋅

where

Gx = mean of the Train A data, as calculated using Equation Z.1-5

Gy = mean of the Train B data, as calculated using Equation Z.1-5.

x (Equation Z.1-5)

n

x y

n

y i

i

n

i

i

n

= = = =

∑ ∑ 1 1

1 1

,

where

xi = PM concentration measured using Train A for run i

yi = PM concentration measured using Train B for run i

n = number of data pairs.

The slope (b1) of the regression line is determined using Equation Z.1-6:

b (Equation Z.1-6)

S

S

xy

xx

1 =

where

Sxx, Sxy = as calculated using Equation Z.1-7:

S x x , S x x y y ( ) ( )( ) (Equation Z.1-7) xx i

i

n

xy i i

i

n

= − = −− = =

∑ ∑ 2

1 1

where 

xi

, xG, yi, and yG = as defined above for Equations Z.1-4 and Z.1-5.
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For the data listed in Table Z.1-1, the calculated values are as follows:

x

i

= + ++ = =

∑1

20 2 6 39 105 14 24

1

20

( . . ... . ) .

y

i

= + ++ = =

∑1

20 4 2 4 4 113 1518

1

20

( . . .... . ) .

S ( . . ) ( . . ) ... ( . . ) , ( ) xx

i

= − + − ++ − = =

∑ 2 6 14 24 39 14 24 105 14 24 1511 22 2

1

20

S ( . - . )( . - . )+...+( . - . )( . - . ) , ( ) xy

i

= = =

∑ 2 6 14 24 4 2 1518 105 14 24 113 1518 1348

1

20

b1

1 348

1511 = = 0892 ,, .

b0 =− × = 1518 0892 14 24 2 47 . (. . ) .

The regression equation is then expressed as

y x $ = + 2 47 0892 . .

Step 2: Calculate Residuals and Standardized Residuals

For a paired sampling train test run, the residual is simply the difference between the

concentration (y) measured by one of the trains (e.g., Train B) and the concentration determined

using the regression equation (í), as shown in Equation Z.1-8:

e = y ! í (Equation Z.1-8)

where

e = residual

y = concentration measured by Sampling Train B

í = predicted (fitted) concentration for Sampling Train B calculated using the

regression equation.

The standardized residual for a test run is a function of the residual for the run and the residual

variance, as calculated using Equation Z.1-9:

(Equation Z.1-9)

( )

e

e

Var e

S

i =
Z-8

where

es = standardized residual

ei = residual for Run i

Var(e) = residual variance, as determined using Equation Z.1-10.

(Equation Z.1-10) Var e

ei

i

n

( ) = −

=

∑ 2

1

(n ) 1

where

Var(e), ei

, and n = as defined previously.

Based on the data presented in Table Z.1-1, the fitted y values for Runs 1 and 2 are

( )

( )

Run y

Run y

1 2 47 0892 2 6 4 79

2 2 47 0892 39 595

: $ . . ..

: $ . . ..

= + ×=

= + ×=

From Equation Z.1-8, the residuals for Runs 1 and 2 are

Run e

Run e

1 4 2 4 79 059

2 4 4 595 155

1

2

: .. .

: .. .

= − =−

= − =−

From Equation Z.1-10, the residual variance for the data in Table Z.1-1 is

( )

( ) Var e( )

. ( . ) ... ( .

= .

− + − + +−

− = =

∑ 059 155 054

20 1

1619

22 2

1 1

20

From Equation Z.1-9, the standardized residual for Run 1 is

es =

− = − 059

1619

0147 .

. .

For this run, the absolute value of the standardized residual is 0.147, which is less than 3.0. 

Therefore, Run 1 meets the criterion for precision based on the residuals.
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Example Problem: How Do I Evaluate the Precision of Paired Sampling Train Data

Using Residuals?

This example uses the same initial correlation test data presented in Table Z.1-1. A total of

20 test runs were conducted using paired sampling trains. The first step is to perform a linear

regression using the data from sampling Train A as the independent variable (x values) and the

data for Train B as the dependent variable (y values). Table Z.1-2 summarizes the results of the

regression analysis. The resulting regression equation is í = 2.473 + 0.892 y.

Table Z.1-2. Summary of Regression Calculations

Variable Calculated value Equation no.

xG 14.24 Z.1-5

Gy 15.18 Z.1-5

n 20 Z.1-5

Sxx 1.511 Z.1-7

Sxy 1348 Z.1-7

b0 2.473 Z.1-4

b1 0.892 Z.1-6

The residuals are calculated using Equation Z.1-8, and the standardized residuals are calculated

using Equations Z.1-9 and Z.1-10. Table Z.1-3 shows the predicted (fitted) y value, the

residual, and the standardized residual for each test run. As shown in the table, all runs meet the

precision criterion except Run 9. For that run, the absolute value of the standardized residual is

3.167, which exceeds 3.0. 

Table Z.1-3. Summary of Residuals for Example Problem – First Iteration

Run

PM concentration, mg/dscm Predicted PM

concentration(í) Residual (ei)

Standardized

residual (es Train A Train B )

1 2.6 4.2 4.79 -0.593 -0.147

2 3.9 4.4 5.95 -1.553 -0.386

3 4.2 2.8 6.22 -3.421 -0.850

4 8.6 10.3 10.15 0.153 0.038

5 15.1 14.5 15.95 -1.447 -0.360

6 18.3 19.7 18.80 0.897 0.223

7 22.4 21.3 22.46 -1.161 -0.289

8 24.5 20.2 24.34 -4.135 -1.028

(continued)
Table Z.1-3. (continued)

Run

PM concentration, mg/dscm Predicted PM

concentration(í) Residual (ei)

Standardized

residual (es Train A Train B )

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9 24.3 36.9 24.16 12.743 3.167

10 32.9 32.2 31.83 0.369 0.092

11 6.4 7.4 8.18 -0.784 -0.195

12 9.5 14.2 10.95 3.250 0.808

13 12.4 15.1 13.54 1.562 0.388

14 14.2 11.2 15.14 -3.944 -0.980

15 7.8 7.1 9.43 -2.333 -0.580

16 21.3 18.3 21.48 -3.180 -0.790

17 4.6 13.5 6.58 6.922 1.720

18 28.7 24.2 28.08 -3.883 -0.965

19 12.6 14.8 13.72 1.083 0.269

20 10.5 11.3 11.84 -0.543 -0.135

The next step is to remove Run 9 from the data set and recalculate the regression equation and

standardized residuals to determine if any of the data points exceed 3.0 after the outlier (Run 9)

has been removed. Table Z.1-4 summarizes the regression parameters.

Table Z.1-4. Summary of Regression Calculations

 for Example Problem with Run 9 Removed

Variable Calculated value

xG 13.71

yG 14.04

n 19

Sxx 1404

Sxy 1118

b0 3.12

b1 0.796

The residuals and standard residuals are recalculated. The revised calculations, summarized in

Table Z.1-5, show that the absolute value of the standardized residual is less than 3.0 for all runs. 

Therefore, the evaluation is complete with only one outlier (Run 9) identified. The facility can

decide to reject that run, but it is not required to do so.
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Table Z.1-5. Revised Calculation of Residuals after Removal of Run 9

Run

PM concentration, mg/dscm Predicted PM

concentration(í) Residual (ei

)

Standardized

residual (es Train A Train B )

1 2.6 4.2 5.19 -0.99 -0.378

2 3.9 4.4 6.23 -1.825 -0.696

3 4.2 2.8 6.46 -3.664 -1.398

4 8.6 10.3 9.97 0.332 0.127

5 15.1 14.5 15.14 -0.643 -0.245

6 18.3 19.7 17.69 2.009 0.766

7 22.4 21.3 20.96 0.344 0.131

8 24.5 20.2 22.63 -2.428 -0.926

10 32.9 32.2 29.32 2.884 1.1

11 6.4 7.4 8.22 -0.816 -0.311

12 9.5 14.2 10.68 3.516 1.341

13 12.4 15.1 12.99 2.107 0.804

14 14.2 11.2 14.43 -3.227 -1.231

15 7.8 7.1 9.33 -2.231 -0.851

16 21.3 18.3 20.08 -1.780 -0.679

17 4.6 13.5 6.78 6.717 2.563

18 28.7 24.2 25.97 -1.772 -0.676

19 12.6 14.8 13.15 1.647 0.629

20 10.5 11.3 11.48 -0.180 -0.069

Z.1.6 Which Procedure Should I Use for Evaluating Paired Sampling Train Data Precision?

The methods presented in Sections Z.1.3 and Z.1.4 for evaluating precision differ in terms of

simplicity and results. The method based on residuals analysis is a relatively complex, iterative

procedure. The RSD method is much simpler and requires a single round of calculations. 

However, as indicated by the results of the example problem, the RSD method is more stringent

in terms of identifying outliers. In the example problem, the results of five test runs were

identified as possible outliers based on RSD, but the residuals analysis of the same data set

identified only one test run as a potential outlier. Although either method can be used to

evaluate paired sampling train precision, a recommended procedure is to use the RSD method

initially to evaluate data precision. If the RSD analysis shows that the data for all test runs are

within the recommended maximum RSD, no further assessment of data precision should be

necessary because the RSD method is the more stringent of the two methods presented here. If

the RSD values for one or more test runs indicates that the data set contains outliers, an analysis

of the residuals is recommended.
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Regardless of which method is used, any potential outliers should be investigated to determine

possible causes for the lack of precision. If problems are identified in how the samples were

collected or analyzed, the suspect data pairs should be discarded. If no problems can be

identified, it may be that the apparent lack of precision is simply evidence of the natural

variations in the data. In any case, the facility has the option of discarding the results of up to

five test runs without explanation, provided there remains a minimum number of test runs to

satisfy PS-11 (i.e., 15 runs for developing a correlation or 12 runs for an RCA). The results of

additional test runs can be rejected if the basis for rejecting the runs is specified in PS-11, the test

method, or the quality assurance plan for the facility. As stated above, a minimum of 15 test

runs is needed to develop a correlation, and a minimum of 12 test runs is required for an RCA.

Z.2 PAIRED SAMPLING TRAIN BIAS

Z.2.1 What Is Bias and Why Is It Important?

Bias is a systematic error in data. Bias can arise from differences in parameter measurement

methods, data collection methods, or data reduction methods. In paired sampling trains, bias can

occur when the instruments used to measure emissions differ from one train to the other, or when

the test crew operates the two sampling trains differently. For example, if two sampling trains

include heated filters, and the filter temperatures are significantly different, more PM can

condense on the filter with the lower temperature, resulting in biased data. Because it may not

be apparent that there are differences between the sampling trains or data collection methods, it

is important to evaluate the data for bias. When developing the correlation equation for PS-11,

bias can introduce more scatter in the data, skew the correlation equation, or result in failure to

meet the correlation acceptance criteria specified in Section 13.2 of PS-11. 

Z.2.2 How Do I Determine Bias in Paired Sampling Train Data?

There are several methods that can be used to identify bias in paired sampling train data. The

method described in the following paragraphs consists of three steps:

1. Determine the linear regression equation for the data;

2. Calculate the confidence interval for the slope of the regression line; and

3. Calculate the confidence interval for the intercept of the regression line.

The following paragraphs describe these steps in more detail.

Step 1: Determine Regression Equation

The procedure for determining the linear regression equation is the same as is described in

Section Z.1.2.1 for evaluating residuals for precision. The data from either of the paired

sampling trains can be used for the dependent variable or the independent variable, and the

resulting linear regression equation has the form indicated by Equation Z.1-3.

y b bx $ = + 0 1
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Step 2: Determine Confidence Interval for Slope

The next step is to determine the 95 percent confidence interval for the slope of the regression

line (b1). The criterion for the slope of the regression line is that the value 1.0 falls within the

confidence interval. The 95 percent confidence interval for the slope is calculated using

Equation Z.2-1.

CI b t SE b( ) (Equation Z.2-1) b1 1 95% 1 =± ×

where

CIb1 = 95 percent confidence interval for the slope

b1 = slope of the regression line

t95% = t(n-2,0.975) is the 100(1-0.005/2) percentage point of a Student’s tdistribution with n-2 degrees of freedom (see Appendix A)

SE(b1) = standard error of the slope.

The standard error of the slope is calculated using Equation Z.2-2:

(Equation Z.2-2) ( )

( )

SE b

S

x x i

i

n 1

2

1

=

− =

∑

where

xi and = as defined above for Equations Z.1-4 and Z.1-5 x

S = root mean square error.

The root mean square error (S) is calculated using Equation Z.2-3:

(Equation Z.2-3)

( ) S

y y

n

i i i

n

=

− ⎛

⎝

⎜ ⎞

⎠

⎟

−

∧

=

∑1

2

2

where

yi

, íi, and n = as defined above for Equations Z.1-3 through Z.1-5.

For the data presented in Table Z.1-1, the regression equation was previously determined to be

í = 2.47 + 0.892x. Using that equation and the Train B data from Table Z.1-1, the root mean

square error is calculated using Equation Z.2-3 as follows:
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( ) S i =

− + − ++ −

− = =

∑ ( . . ) ( . . ) ... ( . . )

.

4 2 4 79 4 4 595 113 1184

20 2 413

2

1

20

2 2

The mean x value was previously determined to be 14.24. When the value for S and the data for

Train A from Table Z.1-1 are inserted in Equation Z.2-2, the standard error of the slope is

calculated as follows:

( )

( )

SE b

i

1

2

1

20

2 2

413

2 6 14 24 39 14 24 105 14 24

= 0106

− + − ++ −

=

=

∑

.

( . . ( . . ) ... ( . . )

.

For 18 degrees of freedom (n - 2 = 20 - 2 = 18), the value for the t statistic (refer to Appendix A)

is 2.101. Inserting that value and the results of the above calculations for root mean square error

and the standard error of the slope into Equation Z.2-1, the confidence interval for the slope is as

follows:

CIb1 =± × =± 0892 2101 0106 0892 0 223 . (. . ) . .

This can also be written as

0.669 #b1 #1.116 with 95% confidence

Since this interval includes the value 1.0, the data satisfy the bias criterion for slope.

Step 3: Determine Confidence Interval for Intercept

The final step is to determine the 95 percent confidence interval for the intercept of the

regression line (b0). If the value 0.0 falls within the confidence interval for the intercept, the data

satisfy the bias criterion and are acceptable. The 95 percent confidence interval for the intercept

is calculated using Equation Z.2-4:

CI b t SE b( ) (Equation Z.2-4) b0 0 95% 0 =± ×

where

CIb0 = 95 percent confidence interval for the intercept

b0 = intercept of the regression line

t95% = as defined above for Equation Z.2-1

SE(b0) = standard error of the intercept.
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The standard error of the intercept is calculated using Equation Z.2-5:

( ) (Equation Z.2-5)

( )

SE b S

x

n xx

i

i

n

i

i

0 n

2

1

1

=

−

=

=

∑

∑

where all variables are as defined above in Equations Z.1-3 to Z.1-5 and Z.2-4.

For the data presented in Table Z.1-1, the parameter SE(b0), which is determined from

Equation Z.2-5, is as follows:

( ) ( )

(( ) ( ) ( )) SE b0

22 2

413

2 6 39 105

20 2 6 14 24 39 14 24 105 14 24 = 1774

+ ++

− + − ++ − = ∑

∑ . .. .

.. .. .. . K

K

Substituting into Equation Z.2-4, the confidence interval for the intercept is as follows:

CIb0 =± × =± 2 47 210 177 2 47 373 . (. . ) . .

This can also be written as

-1.25 # b0 # 6.20 with 95 percent confidence

Since this interval includes the value 0.0, the data satisfy the bias criterion for intercept.

Example Problem A: How Do I Evaluate Paired Sampling Train Data for Bias?

In this example, 15 test runs were performed using paired sampling trains. Table Z.2-1

summarizes the data for the two sampling trains.
Z-16

Table Z.2-1. Example A – Summary of Test Results

Run

PM Concentration

Train A (x) Train B (y)

1 2.3 2.5

2 3.9 4.4

3 4.2 4.1

4 8.6 10.3

5 14.2 13.1

6 15.1 14.5

7 18.3 19

8 22.4 21.3

9 24.5 22.8

10 28.1 28.2

11 32.9 32.2

12 6.4 6.9

13 7.2 6.9

14 12.6 12.4

15 10.5 11.3

Table Z.2-2 summarizes the results of the regression calculations.

Table Z.2-2. Summary of Regression Calculations

Variable Calculated value Equation No.

xG 14.08 Z.1-5

Gy 13.99 Z.1-5

n 15 Z.1-5

Sxx 1239 Z.1-7

Sxy 1182 Z.1-7

b0 0.556 Z.1-4

b1 0.954 Z.1-6

S 0.793 Z.2-3

t95% 2.16 Appendix A

SE(b1) 0.0225 Z.2-2

SE(b0) 0.378 Z.2-5

CIb1 0.906< b1 < 1.003 Z.2-1

CIb0 -0.260 < b0 < 1.372 Z.2-4
Z-17

The resulting regression equation is

y x $ = + 0556 0 954 . .

The confidence interval for the slope is determined from Equation Z.2-1 to be 0.906 < b1

< 1.003. Because the confidence interval includes the value 1.0, the data are considered to be

unbiased with respect to regression line slope. The confidence interval for the intercept is

determined from Equation Z.2-4 to be !0.260 < b0 < 1.372. Because the confidence interval

includes the value 0.0, the data are considered to be unbiased with respect to regression line

intercept. Therefore, the data are considered to be unbiased.

Example Problem B: How Do I Evaluate Paired Sampling Train Data for Bias?

In this example, 18 test runs were performed using paired sampling trains. Table Z.2-3

summarizes the data for the two sampling trains.

Table Z.2-3. Example B – Summary of Test Results

Run

PM Concentration

Train A (x) Train B (y)

1 2.3 2.5

2 3.9 4.4

3 4.2 4.8

4 8.6 10.3

5 14.2 14.1

6 15.1 16.2

7 18.3 19.0

8 22.4 22.5

9 24.5 26.8

10 28.1 28.2

11 5.2 7.7

12 6.4 6.9

13 7.2 8.5

14 12.6 13.9

15 10.5 11.3

16 22.4 24.3

17 29.3 29.1

18 26.9 26.1
Z-18

The resulting regression equation is

y x $ = 118 0 974 . . +

Table Z.2-4 summarizes the results of the regression calculations. The confidence interval for

the slope is determined as 0.924 < b1 < 1.024. Because the confidence interval includes the

value 1.0, the data are considered to be unbiased with respect to regression line slope. The

confidence interval for the intercept is calculated to be 0.325 < b0 < 2.036. In this case, the

confidence interval does not include the value 0.0. Consequently, the data are considered to be

biased. A closer look at the data shows that the PM concentration measured by Train B was

higher than the PM concentration for the same run measured by Train A in 15 of the 18 test runs. 

The test does not reveal which sampling train provided the more accurate measurements, so both

trains should be checked for possible sources of bias. If the source of the bias can be

determined, only the data from the train that provided the unbiased results should be used to

develop the correlation equation. If the source of the bias cannot be determined, the owner or

operator can decide to use the data or repeat the correlation test. However, if paired sampling

data are required by regulation, the emission test should be repeated.

Table Z.2-4. Summary of Regression Calculations

Variable Calculated value Equation No.

xG 14.56 Z.2-3

Gy 15.37 Z.2-3

Sxx 1425 Z.2-5

Sxy 1388 Z.2-5

b0 1.18 Z.2-2

b1 0.974 Z.2-4

S 0.893 Z.2-8

t95% 2.12 Appendix A

SE(b1) 0.0236 Z.2-7

SE(b0) 0.404 Z.2-10

CIb1 0.924 < b1 < 1.024 Z.2-6

CIb0 0.325 < b0 < 2.036 Z.2-9

Z.2.5 Using the Bias Check Spreadsheet

This section provides instructions for using the Bias Check Spreadsheet. The spreadsheet can be

used to check paired sampling train data for bias using the procedure described in the previous

sections.

The spreadsheet includes three worksheets. The worksheets appear as tabs at the bottom of the

screen. You can move from one worksheet to another by clicking on the appropriate worksheet
Z-19

Figure Z.2-1. Example of Test Data worksheet for Bias Check Spreadsheet

(Example Problem A).

tab. The initial worksheet (Test Data) is used for data entry. The Calculations worksheet shows

the intermediate calculations that are needed for checking bias. The Summary of Results

worksheet presents the results of the bias check

The Calculations and Summary worksheets are completely locked, which means that the user

can open the worksheet and view the results of the calculations but cannot modify any of the

cells in the worksheet. The Test Data worksheet is partially locked. The user enters the test data

for one sampling train (Train A) in cells A6 through A65 and the data for the other sampling

train (Train B) in cells B6 to B65. It makes no difference in the analysis which train is

considered Train A and which is considered Train B. Cells F3 to F6 are used to enter the facility

name, location, emission unit, and test dates. All other cells in the Test Data worksheet are

locked. Figure Z.2-1 shows the Test Data worksheet for Example Problem A described

previously.
Z-20

Figure Z.2-2. Example of Calculations worksheet for Bias Check Spreadsheet

(Example Problem A).

The Calculations worksheet consists of several columns that display the results of the various

calculations needed for the linear regression. All cells are locked in this worksheet. 

Figure Z.2-2 shows the Calculations worksheet for Example Problem A.

The Summary of Results worksheet includes the information about the facility, emission unit,

and the emission test date that were entered on the Data Entry worksheet, followed by a table

displaying the values for the statistical parameters that must be calculated to evaluate the data for

bias. Finally, the summary of the results is displayed. This summary includes the upper and

lower bounds for the confidence intervals for the slope and intercept of the regression line. The

spreadsheet also indicates whether the data satisfy the bias criteria for slope and intercept. 

Figure Z.2-3 shows the Summary of Results worksheet for Example Problem A.
Z-21

Figure Z.2-3. Example of Summary of Results worksheet for Bias Check Spreadsheet

(Example Problem A).

Z.2.6 Using Excel Data Analysis Tools to Evaluate Bias

The data analysis tools included in Microsoft Excel also can be used to perform the bias check.

# Key in the data for the two sampling trains in separate columns.

# Under the Tools menu, select Data Analysis.

# Scroll down the list of options to Regression, then click OK.

# Click on the icon in the Input Y Range cell, then block off the data for sampling Train

B and press enter.

# Click on the icon in the Input X Range cell, then block off the data for sampling Train

A and press enter.
Z-22

Figure Z.2-4. Output of Excel Regression Analysis for Example Problem A.

# Check that the Confidence Level is indicated as “95%.” If it is not, click on

Confidence Level, then key in 95 in the adjacent cell.

# Choose where you want the results to be displayed. If you click on the icon in the

Output Range cell, you must select a cell in the spreadsheet where you want the first

cell of the results to appear and press enter. If you click on New Worksheet Ply, a

new worksheet will be inserted with the results displayed. If you click on New

Workbook, a new workbook will be created with the results displayed. Selecting the

other options, such as Labels and Residuals, provides additional results, but is not

necessary for checking the data for bias.

# After deciding where you want the results displayed, click OK.

The results will then be displayed in the selected location (starting in the cell you indicated for

Output Range, new worksheet, or new workbook). Figure Z.2-4 shows the output for the

Example Problem A. The last two lines of the results pertain to the regression line intercept and

the slope (labeled as “X Variable 1”). The lower and upper limits of the 95 percent confidence

interval are displayed several columns to the right of the intercept and slope. If the confidence

interval for the intercept includes the value 0, and the confidence interval for the slope includes

the value 1, the data satisfy the bias criteria.
A-1

Appendix A 

Student’s T-Distribution
A-2

Student’s T-Distribution: One-Sided at 95% Confidence Interval

Number of pairs (n) Degrees of freedom Student’s t95% value

8 6 2.447

9 7 2.365

10 8 2.306

11 9 2.262

12 10 2.228

13 11 2.201

14 12 2.179

15 13 2.160

16 14 2.145

17 15 2.131

18 16 2.120

19 17 2.110

20 18 2.101

21 19 2.093

22 20 2.086

23 21 2.080

24 22 2.074

25 23 2.069

26 24 2.064

27 25 2.060

28 26 2.056

29 27 2.052

30 28 2.048

31 29 2.045

32 30 2.042

33 31 2.040

34 32 2.037

35 33 2.035

36 34 2.032

37 35 2.030

38 36 2.028

39 37 2.026

40 38 2.024

41 39 2.023

42 40 2.021

43 41 2.020

44 42 2.018

(continued)
Student’s T-Distribution (continued)

Number of pairs (n) Degrees of freedom Student’s t95% value

A-3

45 43 2.017

46 44 2.015

47 45 2.014

48 46 2.013

49 47 2.012

50 48 2.011

51 49 2.010

52 50 2.009

53 51 2.008

54 52 2.007

55 53 2.006

56 54 2.005

57 55 2.004

58 56 2.003

59 57 2.002

60 58 2.002

61 59 2.001

62 60 2.000

63 61 2.000

64 62 1.999

65 63 1.998

66 64 1.998

67 65 1.997

68 66 1.997

69 67 1.996

70 68 1.995

71 69 1.995

72 70 1.994

73 71 1.994

74 72 1.993

75 73 1.993

76 74 1.993

77 75 1.992

78 76 1.992

79 77 1.991

80 78 1.991