I II MINI Illllll PUBLICATIONS

A111D7 D1707M

NIST Special Publication 960-18

NIST Recommended Practice Guide:

Computing Uncertainty for Charpy

Impact Machine Test Results

J. D. Splett

C. N. McCowan

H. K. Iyer

C.-M. Wang

QC

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'At Boulder, CO 80305 Some elements at Boulder, CO
NIST Special Publication 960-18

NIST Recommended Practice Guide:

Computing Uncertainty for Charpy

Impact Machine Test Results

J. D. Splett

H. K. Iyer

C.-M. Wang

Statistical Engineering Division

Information Technology Laboratory

C. N. McCowan

Materials Reliability Division

Materials Science and Engineering Laboratory

National Institute ofStandards and Technology

Boulder, Colorado 80305

September 2007

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

National Institute of Standards and Technology

James M. Turner, Acting Director
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept adequately. Such

identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the

entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Special Publication 960-18

Natl. Inst. Stand. Technol. Spec. Publ. 960-18, 36 pages (September 2007)

CODEN: NSPUE2

U. S. GOVERNMENT PRINTING OFFICE

Washington: 2007

For sale by the Superintendent of Documents, U. S. Government Printing Office

Internet: bookstore.gpo.gov - Phone: (202) 512-1800 - Fax: (202) 512-2250

Mail: Stop SSOP, Washington, DC 20402-0001

ii
Contents

1. Introduction

2. Uncertainty of a Test Result

3. Type B Uncertainty Evaluation

4. Machine Bias

5. Direct Verification

6. Temperature

7. Expanded Uncertainty

8. Examples

8.1 Both <?systematlc and £systematlc and Their Uncertainties Are Negligible

8 - 2 hematic Has One Component

9. Closing Remarks

10. References

Appendix A. Uncertainty Details

A.l Test Material

A.2 Indirect Verification Test

A.3 NIST Reference Value

A.4 Estimating Machine Bias

Appendix B. Direct Verification Components of Uncertainty

B. l Anvils and Supports, A

\^

B.2 Height of Pendulum Fall, h

B.3 Potential Energy, E

B.4 Impact Velocity, v

B.5 Center of Percussion, L

B.6 Friction Loss, D

B.7 Scale Accuracy, r

Appendix C. /-Table

Appendix D. Glossary of Terms

in
\
NIST Recommended Practice Guide:

Computing Uncertainty for Charpy Impact Machine Test Results

J. D. Splett,

1 C. N. McCowan,2 H. K. Iyer,1 and C.-M. Wang1

National Institute of Standards and Technology

325 Broadway

Boulder, CO 80305

This recommended practice guide demonstrates how to determine the uncertainty associated with

mean absorbed energy of specimens tested on a Charpy impact machine. We assume that the

Charpy machine has successfully met the requirements for both direct and indirect verification as described in the ASTM E 23, Standard Test Methods for Notched Bar Impact Testing of Metallic

Materials. We follow the recommendations and procedures in the "Guide to the Expression of

Uncertainty in Measurement" for computing uncertainty. We assume the reader is somewhat

familiar with the Charpy machine verification program administered by the National Institute of

Standards and Technology.

Keywords: absorbed energy; Charpy V-notch; impact test; pendulum impact test; uncertainty;

verification testing

1. Introduction

X

The absorbed energy of a test material, measured using a Charpy impact machine, is often

reported as the mean absorbed energy of a set of specimens. However, the sample mean does

not account for known sources of bias, including machine bias, which can be substantial. We

address the estimation of a test result for the case in which the test result is corrected for known

biases and the case in which it is not. It is left to the user's discretion whether or not to correct a

test result.

Computing the reported test result is straightforward; however, computing the uncertainty

associated with the test result requires more consideration. The purpose of this document is to clarify the concept of uncertainty and to provide Charpy laboratories with a procedure for

computing the uncertainty of a test result.

1 Statistical Engineering Division, Information Technology Laboratory

2 Materials Reliability Division, Materials Science and Engineering Laboratory

1
Before valid Charpy measurements can be made in the laboratory, the machine needs to pass

both direct and indirect verification tests as specified in ASTM E 23 [1]. Even if a Charpy

machine has passed the indirect verification test, it is likely that results for the verification

specimens differ from the certified value. This difference can be used to quantify machine bias.

Thus, the indirect verification results and the certified reference value (along with their

uncertainties) play key roles in the calculation of uncertainty of a test result. We express uncertainty according to the accepted criteria described in the "Guide to the

Expression of Uncertainty in Measurement," or GUM [2], taking into consideration both random

and systematic sources of error. The procedure we recommend for computing uncertainty is very general and can accommodate any number of random or systematic error sources including

the following:

Anvils and supports

Center of percussion

Center of strike

Friction loss

Height of pendulum fall Impact velocity

Material inhomogeneity

Operator

Potential energy

Repeatability

Scale accuracy

Test temperature

The uncertainty contributions from individual error sources can be estimated if they are

identified as significant, but generally these errors are assumed to be minimized by adjustments

made to the machine during direct verification and by following the standard test procedure. As

will become apparent, the calculation of uncertainty is greatly (and often) simplified by assuming

that direct verification contributions are zero, and only contributions from indirect verification are considered. This is a widely accepted approach to the calculation of uncertainty for Charpy

impact tests, and is used in standards such as ISO 148-1 [3]. We present more detail here,

because understanding the individual contributions to uncertainty, and how to quantify them,

leads to better control of the test. We encourage the users to consider these, and other relevant

details.

We present an example in Section 2 that provides instructions for calculating the uncertainty of a

test result. Section 3 provides details regarding the Type B evaluation of errors, Section 4

addresses the computation of machine bias, Section 5 discusses direct verification sources of

error, Section 6 addresses temperature measurement errors, Section 7 provides some information

about expanded uncertainty, and Section 8 gives some example uncertainty calculations.

Complete details regarding the justification of the uncertainty procedures are given in Appendix

A.

2. Uncertainty of a Test Result

In this section, we provide details for computing the uncertainty of a test result within the context

of an example. A Charpy laboratory will typically compute the sample mean and sample

standard deviation of n specimens of the test material using the following two equations:

2
(1)

S = n-\

(2)

The degrees of freedom (df) associated with the sample standard deviation, s , are n - 1 . It is important to note that s includes all sources of random error, including machine variability,

material variability, and the typical variations expected when following the standard test procedure. The individual components of the random error cannot be estimated separately in the case of destructive impact testing (multiple measurements on the same specimen are needed to do this). In addition, random errors (unlike systematic errors) do not remain constant during the

measurement of n specimens, so these errors do not result in a bias.

The data given in Table 1 are used to illustrate the calculations needed to assess the uncertainty

associated with the result for our example. Table 1 lists test results and summary statistics for n = 5 observations of a particular test material measured at 80 °C.

Suppose we are also given the values in Table 2. (We provide details for computing the

quantities in Table 2 in subsequent sections.) Our best estimate of the machine bias is b , which

is defined as the difference between the verification result for the test machine and the certified

value of the verification specimens. Systematic errors due to all other factors that are not already

included in the machine bias are denoted by <?systematic . The values u{b) and w(<?systematjc ) are the

uncertainties associated with b and esvstematjc , while dfb and dfe , represent degrees of freedom for

u(b) andw(e.systematic )

Table 1. Measurement results for a test material.

Observed data, J Summary statistics

58.0 n = 5

62.0 7 = 57.6 J 54.0 5" = 3.6 J 54.0 df=4

60.0

Table 2. Example quantities required to compute uncertainty of a test result. Machine bias Systematic error

Estimate

Uncertainly

Degrees of freedom

b = -4.2 J

u(b) = 2.S J dfb = 84

systematic

^(^systematic) dfe = 2

3.0 J = 0.6 J
To compute the uncertainty of the test result, we first compute a mean of the test material that is corrected for machine bias and all other systematic effects:

^corrected V ^systematic ' (3)

Substituting the values from Tables 1 and 2 into Eq. (3) gives

^.ected =57.6 J -(-4.2 J) -(3.0 J) = 58.8 J.

Next, we calculate the uncertainty of the corrected mean, ~

corTected . Assuming all of the terms in

^corrected are independent, the combined standard uncertainty of ~

C0ITected is

= J— + u

z (b) + uz (^ystematlc )

.

n

Substituting the appropriate values from Tables 1 and 2 into Eq. (4) gives

^corrected) = + (2-8 J)

2 + (0.6 J)2 = 3.3 J

Typically, the standard uncertainty is multiplied by a coverage factor that expands the

uncertainty to form an "uncertainty" interval about the measurement result. The interval is expected to encompass a large fraction of possible values of the result. Thus, the expanded

uncertainty is defined as the combined standard uncertainty multiplied by a coverage factor. The

coverage factor is often set equal to two for simplicity, but this approximation can be

problematic, so it is recommended that the degrees of freedom be used to obtain the appropriate

coverage factor.

An uncertainty interval with 100(1 - a) % coverage probability (a is 0.05 for 95 % coverage) is given by

^corrected — ^\-<*/

2 ,dfcfT

' ^(^corrected ) ' (5)

where t

x_a, df ^ is found in a stable (see Appendix C). The degrees of freedom associated with

W ( ^corrected) '

Jf ^ (^corrected ) (f.\ dfUJ dfb " dfe

are determined from the Welch-Satterthwaite approximation as described in the GUM. [2]

4
Substituting appropriate values from Tables 1 and 2 into Eq. (6) gives

dfeff =

(3-3)

4

^(3.6)

2 Y [ (2.8)4

| (0.6)

4 = 47.9,

J 84

which rounds down to 47. Using a /-table we get a coverage factor of t0975A1 = 2.012. Thus, a 95 % interval for our example is

58.8 J + ?o.975,47 3.3 J

58.8 J±2.012-3.3 J

58.8 J±6.6 J.

The expanded uncertainty U is 6.6 J, indicating that 95 % of possible measurement results lie within the uncertainty interval (52.2 J, 65.4 J). If we compute the interval based on the

uncorrected value, we can express our uncertainly interval as

(y±U)-(b + esystematic )

(57.6 J + 6.6 J) -(-4.2 J + 3.0 J)

(51.0J, 64.2 J) + 1.2

J

and just report the uncorrected interval (5 1 .0 J, 64.2 J) along with the correction ( 1 .2 J). The

decision to report a corrected test result is left to the user. However, if the corrected test result is reported, we recommend that the report clearly state how the correction was computed and

include pertinent information such as the magnitude and sign of the correction, the test standard

used, and the source of the indirect verification specimens tested\

The remainder of this document is dedicated to providing additional details regarding the

computation of individual components needed to compute the uncertainty of a Charpy test result.

3. Type B Uncertainty Evaluation

Typically, direct estimates of systematic errors based on actual measurements are difficult to obtain and even harder to quantify because the required data are not generally available. In such

cases, uncertainties due to systematic errors are estimated based on past experience, engineering

knowledge, information from published literature, and so on. An uncertainty evaluation that does

not involve actual measurements is called a Type B uncertainly evaluation. Type A uncertainty

evaluations are based on data obtained under repeatability conditions. Type B uncertainty

5
evaluations can be associated with either random or systematic errors, but are most commonly

used with systematic errors.

Type B uncertainty evaluations utilize assumptions regarding distributions of errors. For

example, instrument manufacturer's specifications can be thought of as limits to a rectangular

distribution. From this, the standard uncertainty associated with measurements by that instrument can be deduced. These types of uncertainties can be highly subjective, but are sometimes useful.

The following example (also shown in Section B.7) illustrates how to use a manufacturer's

specification for a Type B uncertainty evaluation. Suppose r is the random error in the Charpy

machine scale mechanism and ± Ar represents the manufacturer's specified error bounds of the

measurement instrument. Assuming that the error can be anywhere within the ± Ar bounds, a rectangular distribution is used to describe the distribution of possible biases, and in this case bounds are already expressed in the proper units (joules). The standard uncertainty of r is

w(r)= 7T

A rectangular distribution is often used in the absence of specific information about the error

distribution; however, other distributions can be used if more is known about the errors. (See

Reference [2] for details regarding Type B uncertainty evaluations.) It is also necessary to provide an estimate of degrees of freedom for each uncertainty component. We will assume

dfr = oo , which implies that we know «(r) exactly. The GUM provides a method for assigning a df value to Type B estimates of uncertainty, which will be demonstrated shortly.

In the previous scale-error example, the distribution of possible errors was defined by the

interval (-r, r) , which is centered on zero. Sometimes the distribution of a systematic error is centered on a value other than zero, resulting in a nonzero systematic error estimate. For

example, an operator might be consistently reading the scale too high, so that the distribution of

errors is described by a rectangular distribution defined by (a,b), where a and b are both

greater than zero ( < a < b ). In this case, the estimated systematic error is (a + b)/2 and the

associated standard uncertainty is (b-a)/2y[?> .

There are also systematic errors associated with the test procedures that can be approximated

using a Type B uncertainty evaluation. Suppose an operator notices that the lengths of fractured

specimen halves are uneven and determines that the specimens were all impacted off-center

(striker impact is not aligned with the notch). In addition, the operator knows that the 1 mm to 2

mm offsets observed for the broken specimens result in an increase in the absorbed energy

between 2 J and 4 J based on extensive experience with this particular material. To estimate the

systematic error and its uncertainty, we assume that the 2 J and 4 J limits to error represent

bounds of a rectangular distribution so that

2J+4J 4J-2J

esys.emat1C =

^ = 3 J alld ^systemaj = TJT = 6 J

6
To determine the degrees of freedom associated with w(<?systematlc )

, we employ a useful

relationship from the GUM (Eq. (G.3)). In general,

, c

1 TAw

n

" at = —

2

where the quantity in square brackets represents the relative uncertainty, or the uncertainty of the

uncertainty. In our example, we judge the uncertainty of w(esystematjc ) to be 50 % or 0.50, so that

dfe =i[0.50]"2 =2.

In general, the degrees of freedom provide information regarding the quality of the uncertainty

estimate. For Type A uncertainty evaluations, the degrees of freedom provide an objective measure of quality, while degrees of freedom associated with Type B uncertainty evaluations

provide a subjective measure of quality.

We can also combine several sources of systematic error to determine esystematic and its

uncertainty. For example, suppose we would like to combine three independent sources of

systematic error: friction loss, potential energy, and impact velocity, so that

Systematic = D + E + V.

Then the combined standard uncertainty of esystematic is

"(Systematic) = ^

(D) + U

2 (E) + u\v) ,

with effective degrees of freedom from the Welch-Satterthwaite approximation,

u

4 (e )

\ systematic^ df =

e u\D)

| u\E)| u\v)

dfD dfE dfv

This type of procedure can be applied to any number of independent systematic errors.

7
4. Machine Bias

To estimate the machine bias, we assume that the machine bias for the material under test is the same as the machine bias based on the indirect verification. This is an important assumption that

allows us to estimate machine bias for all test materials. We use the results of an indirect

verification test and the associated reference value for our best estimate of machine bias,

b = V-SsysXematiC -R, (7)

where

V = -^—

(8) nv

is the sample mean absorbed energy from the indirect verification test based on nx

. = 5 test

results, ^systematic represents errors due to all systematic effects associated with indirect

verification test, and R represents the certified reference value for the batch of verification

specimens.

To illustrate the computation of machine bias and its uncertainty, we will return to the example

from Section 2. Table 3 lists quantities provided by the National Institute of Standards and

Technology (NIST) with the high-energy verification test specimens that were used for the most

recent high-energy indirect verification of the Charpy machine of interest.

We use the high-energy indirect verification test results because the nominal value of the

absorbed energy of the test material is closest to the high-energy verification material. The

uncertainty associated with the certified verification specimens ( u(R) ) is provided by NIST with

the results of the indirect verification test (or by request). Table 4 displays the indirect

verification data that were observed when the verification set was broken on the machine of

interest.

Table 3. Information provided by NIST for high energy verification specimens. Reference value, R 109.9 J Reference value standard uncertainty, u(R) 2.6 J Degrees of freedom, dfR 102

8
Table 4. High energy indirect verification test results.

Vpi*ifii*sitiftn Kf*t Hatft ouiiiiiiiii y 3in ii silt

3

108.0 w = 5

i A/i n V = 106.2 J 109.0 SK = 2.3 J 106.0 dfv = 4

104.0

Table 5. Systematic error associated with the indirect verification.

Estimate, Systematic °- 5 J

Standard uncertainty, u(S) 0-2 J Degrees of freedom, dfs 10

The "V" subscript is used to distinguish the indirect verification results from the test material

results. The sample standard deviation associated with the indirect verification specimens ( Sv )

is calculated as s was calculated previously in Section 2,

(9) nv -\

As was the case for s , Sv also includes all sources of random error related to both machine

variability and material variability, and the individual contribution of errors cannot be

determined.

Suppose we are given #systematic , its associated uncertainty, and degrees of freedom, as shown in

Table 5. We will not elaborate on the origin of the systematic eYror in Table 5; however, the

same general procedures used to estimate <? systematlc , discussed in detail in Section 3, can also be

used to estimate ^systematlc .

Although ^systematIC = 0.5 J in this illustrative example, typically <^systematic is assumed to be zero because errors that are well understood and could be corrected for are minimized during direct

verification of the machine. So, neglecting contributions to the bias from ^systematic , the estimated

machine bias is calculated as the difference between the mean of the specimen tested in the

indirect verification test and the certified value of the specimens tested. For our example, in

which ^systematlc is not assumed to be zero, the machine bias is

b = 106.2 J - 0.5 J - 109.9 J = -4.2 J

.

Assuming independent input quantities, the standard uncertainty of the machine bias is

9
systematic ) + u\R). (10)

Substituting the appropriate values from Tables 3 through 5 into Eq. ( 1 0) provides the following

estimate of the standard uncertainty of the machine bias:

u(b) = J

(23J) + (0.2 J)

2 + (2.6 J)2 = 2.8 J

The degrees of freedom associated with the uncertainty estimate,

dfb =

dfv

u\b)

\ nvJ

| "Vsystematic)

| u\R)

(11)

df df„

are determined from the Welch- Satterthwaite approximation. In our example, the degrees of

freedom are

dfb =

(2.8)

4

+

(0.2)

4

, (2.6)

10

= 84.4

,

+

102

which rounds down to 84.

In the examples presented here, the "bias compared to what?" issues are clear. Machines

verifying to ASTM E 23 requirements are all compared with a single target for impact energy,

defined by ASTM E 23. However, when considering the performance of an ASTM E 23

machine to machines not tested under ASTM E 23 requirements, the comparison is less direct

because bias can exist between the various verification systems used around the world (multiple

certified values for absorbed energy). We encourage the users to understand this issue, and how

it might affect them. Users should also know that the various national measurement institutes

distributing impact verification specimens are working to minimize biases among them, and

make the quantification of bias for impact testing more transparent to users around the world.

10
5. Direct Verification

Direct-verification uncertainty sources are related to physical properties of the Charpy machine

including: anvil and supports, center of strike, potential energy, impact velocity, center of

percussion, friction loss, and scale accuracy. With the possible exception of friction loss, all direct verification sources of uncertainty are Type B evaluations. We provide information

regarding calculation of the individual sources of direct verification uncertainty in Appendix B.

While it is relatively easy to compute each individual source of uncertainty, it is difficult to quantify the uncertainty components in terms of the effect on Charpy measurements in joules.

The recognized sources of uncertainty for our problem are minimized during the direct

verification of an impact machine and by following the standard test procedure. So, it is general

practice to estimate the uncertainty of impact tests from the results of indirect verifications and

the variations associated with repeat measurements on the material being tested. However, it is also of interest, and part of the exercise in calculating uncertainty, to better understand your

machine and process so that it might be better controlled and quantified. It is left up to individual laboratories to identify and include the appropriate uncertainty sources.

Although it is common for laboratories to ignore the uncertainty due to direct-verification bias, it

is important to acknowledge the potential for error due to these sources. Thus, it is informative

for laboratories to document their reasons for either including or excluding direct verification sources of error. If possible, the uncertainty associated with direct verification should be re- examined each time the machine is verified directly.

6. Temperature

Although systematic error due to temperature probably exists to\some extent for all Charpy

measurements, it is difficult to quantify the sign (direction) and magnitude of the error. Thus, we

typically assume the estimated error is zero, but there is some uncertainty associated with the

estimate. This section outlines a procedure that can be used to estimate the uncertainty due to systematic temperature errors.

The uncertainty due to temperature does not depend on machine properties; however, it is highly

dependent on the material being tested. For example, steels undergo a transition in fracture

behavior from brittle to ductile with increasing temperatures. Supplemental data can be collected

for a particular steel of interest, and used to estimate the uncertainty associated with temperature.

If later measurements are taken in stable regions defined by the lower shelf or upper shelf

(Figure 1), then the uncertainty associated with temperature is probably negligible. However, the

uncertainty due to temperature can be significant if measurements are being taken in the

transition region of the curve.

11
E

Upper Shelf

Lower Shelf

Temperature

Figure 1. A temperature transition curve.

100-r

80-

35 37 39 41 43 45 47 49

Temperature, C

Figure 2. Mapping of temperature error into energy error in the temperature transition region.

Assuming we have data for a particular material that have been collected across a range of

temperatures, we can fit a straight line to the data within the temperature transition region

(ignoring the shelf data). The information from the regression fit can be used to quantify the

effect of the temperature error on impact energy (Figure 2) for future samples of the same

material.

For example, suppose we are interested in collecting some new data at 43 °C, but our

temperature can be measured only to within ±2 °C. The true temperature could be anywhere in

12
the range of 41 °C to 45 °C. Using the regression equation, we can compute the value of impact

energy for both 41 °C and 45 °C, thus providing a range of potential impact energy values AE

.

Assuming the true impact energy has a rectangular distribution within AE , we can use the range

of impact energy to compute the uncertainty as follows:

u.(0

2V3

The degrees of freedom are dft = n - 2 , where n is the number of observations used in the

regression fit.

Optionally, multiple measurements could be made at each temperature (which is how the original curve is obtained) and define the uncertainty as the maximum uncertainty observed in the region.

This procedure can also be applied in cases where upper or lower shelf regions have more

gradual slopes.

7. Expanded Uncertainty

Sometimes we need to calculate an expanded uncertainty, U , which is just the combined

standard uncertainty multiplied by a constant, or coverage factor, so that ^ = ^-«(37

corrected)-

The coverage factor k is determined by looking up the appropriate value in a Mable (Appendix

C) based on the degrees of freedom associated with w(jycorrected) • The expanded uncertainty

associated with a 95 % interval is

^95 = ^95

' W( ^corrected) ~ ?

0.975;dfeff

' "(.^corrected ) (^)

The expanded uncertainty is interpreted as an uncertainty interval encompassing a large fraction

of possible measurement results.

The degrees of freedom can be difficult to determine if there are many sources of uncertainty

within w(_ycorrected ) . Fortunately, we can compute the effective degrees offreedom from the

Welch- Satterthwaite approximation [2]

AC ^ (^corrected) (] lis I

| u\b)| u (<?systematlc ) df dfb dfe

13
where df = n-\ and dfe are from the Type B uncertainty evaluation (see Section 3). We will

also need to calculate dfb from

dfb= — — ; ; . (14)

=£==

dfv

i

\ nvj df8 dfR

where dfv = nv - 1 , dfRis provided by NIST with the indirect verification specimens, and

df5 is from Type B uncertainty evaluation (see Section 3).

In general, an uncertainty interval for yconected is

^corrected ~ ^\-a or

^corrected — A-<^,dfefr

* U( ^corrected )

1 5)

Typically a is 0.05, which corresponds to a 95 % interval. If a Charpy lab does not report

results corrected for machine bias and systematic errors, they may want to indicate the

magnitude of the estimated biases for informational purposes,

(y-b- Systematic ) ± U,_a OT (y ±U{ _a ) ~ (b + £systemiitK ) . (16)

Thus, the interval would be shifted by b + <?systematlc if the laboratory wished to report the corrected

mean absorbed energy; however, the expanded uncertainty would not be affected by the machine

bias and systematic error corrections.

In practice, k = 2 is often used to compute the expanded uncertainty to approximate a 95 %

interval, and the effective degrees of freedom are never calculated. However, if the effective

degrees of freedom are small, then the level of confidence is thought to be less than 95 %.

8. Examples

All examples utilize the data displayed in Tables 1, 3, and 4.

14
8.1 Both esystematlc and £systematlc and Their Uncertainties Are Negligible

In the general case,

-^corrected 3^ systematic '

but if the systematic error associated with the material variation esystematic is assumed to be

negligible, then

ycorrected - y -b

.

If the systematic error for the test machine variation £systematlc is assumed to be negligible,

b = V- £systematlc - R = V - R = 106.2 J - 109.9 J = -3.7 J

and

ycorrected = = 57.6 J-(-3.7 J) = 61.3 J.

The combined standard uncertainty of b is

u(b)=K + u

2 (Ssy^tic ) + u2 (R)

= \^- + u<{R)

v nv

'(2.3 J)

2 + (2.6 J)

2

= 2.8J,

with effective degrees of freedom

u\b) (2.8y

+

u\R) I dfR 4

(2.3)

2 +

102

= 84.5,

which rounds down to 84. Thus, the uncertainty of the corrected mean value is

15
"(^corrected) = j^" + (*) +

«

' (^systematic

)

+ w

2 (6)

(3.6 J)'

= 3.2J,

+ (2.8jy

with effective degrees of freedom

dfeff =

U (^corrected) (3.2)<

df \ n J +

u\b) 1 df, 4

(3.6)

: +

(2.8)'

84

= 43.5,

which rounds down to 43. A 95 % interval for ycomctei is

-^corrected — ^l-^,dfeS

' W (^con-ected) 61.3 J±?0975 43

- 3 . 2 J

61.3 J ± 2.017 -3.2 J

61.3 J±6.5 J.

The expanded uncertainty, associated with a 95 % level of confidence is 6.5 J. The 95 %

uncertainty interval is (54.8 J, 67.8 J).

If the value reported is not corrected for machine bias, we can express our interval as

(y±U)-b

(57.6 J ± 6.5 J) - (-3.7 J)

(51. 1J, 64.1J) + 3.7J.

The Charpy laboratory may or may not wish to disclose the estimated machine bias, however the

information is available if needed. Notice that the interval for the uncorrected parameter is shifted just by the amount of the correction and the expanded uncertainty is the same regardless

of whether or not the reported value is corrected.

16
8 2 esystematlc Has One Component

Suppose esystematic contains the error due to temperature so that <?systematic = i . The temperature

error is systematic because it is likely to be in the same direction (always warmer or always

cooler than the target temperature) for a single set of measurements. However, we do not

typically estimate the temperature error, so we will we assume the value of <? systematlc is zero. The

uncertainty associated with esystematic is

W (^systematl c) = W (0-

The procedure outlined in Section 5 will be used to determine the uncertainty due to temperature

u(t ) . Figure 3 displays temperature data for the material of interest along with the regression

line fit to the data in the transition region (ignoring the data on the "shelves").

Suppose our test specimens from Table 1 were broken using a temperature of 80 °C, which is within the temperature transition region. A regression line was fit to the 2 1 data points in the

transition region, resulting in the following equation:

E(J) = -0.03973(J) + 0.74084(J/°C) • T(°C) .

-20 20 40 60 80 100 120 140 160

Temperature, C

Figure 3. Temperature data for the test material. The straight line in the plot represents a regression fit to the data in the transition region only. We ignore data on the "shelves" at - 10 °C

and 141 °C.

17
If our temperature can be measured to within ±1 °C, then the true temperature is between 79 °C

and 81 °C, and the energy range is defined by the following:

E(J) = -0.03973(J) + 0.74084(J/°C) • 79(°C) = 58.5 J

E(J) = - 0.03973(J) + 0.74084(J/°C) • 8 1 (°C) = 60.0 J A£ = 60.0J-58.5 J = 1.5 J .

Next, the energy range is converted to a standard uncertainty based on a rectangular distribution,

with degrees of freedom dft = w-2 = 21-2 = 19. Thus the uncertainty of esystematlc is

W(esystemat1 c) = - 4J 5

with 19 degrees of freedom.

The mean absorbed energy corrected for machine bias and other systematic effects is

JWcted = y~b- £systematic = 57.6 J - (-3.7 J) - J = 61 .3 J

,

where b , its uncertainty, and degrees of freedom have not changed from example 7.1. The

combined standard uncertainty is

"(JWted) = J- + u\b) + u

2 (esystematlc ) = + < 2 " 8 J )

2 + (°- 4 J)2 = 3.3 J ,

with degrees of freedom

Jf U ( ^con-ected ) (3-3)

Ui eff ~ . / 2 y - - - yy -^ ' J_

df

, «*(P)

| »

4 (gsystematic ) lf(3.6)

2 ^ , (2.8)4

, (0.4)'

dfb dfe V 5 j

+

84 19

which rounds down to 49. A 95 % uncertainty interval for >>corrected is

^corrected — ^l-«^,dfeff

' U(^corrected )

61.3 J ± ^o.975,49 '3-3 J 61.3J + 2.010-3.3J

61.3 J±6.6 J.

18
The expanded uncertainty is 6.6 J, and the uncertainty interval encompassing 95 % of possible

measurement results is (54.7 J, 67.9 J). If the value reported is not corrected for bias, we can

express our interval as

(y±U)-(b + esystemm)

(57.6 J ±6.6 J) -(-3.7 J + J)

(51.0J, 64.2J) + 3.7J.

9. Closing Remarks

We have developed a procedure for estimating the uncertainty associated with a reported mean

absorbed energy from a Charpy test. The procedure is flexible enough to account for several

systematic error sources, if necessary, and allows the user the choice of correcting the reported

mean or not. The uncertainty procedure in this document applies to measurements completed in a Charpy laboratory.

Occasionally, there is some confusion about the NIST reference value, the reference value

uncertainty, and Charpy verification limits with respect to results obtained in a Charpy

laboratory. The reference value is the measured mean absorbed energy of a batch of reference

specimens. The reference value uncertainty describes the variability of the reference value and

includes material, system, and machine variability. The reference value uncertainty does not

describe the variability of a single verification specimen or the variability in the verification

specimens (specimen variation cannot be estimated separately from machine variation). In the

Charpy laboratory, the reference value and its uncertainty are used only to estimate the bias of a Charpy machine and the uncertainty of the bias; they provide no information regarding Charpy

measurements for other materials. It is also important to remember that the reference value

uncertainty is associated with a specific measurement result, while the verification limits

describe the acceptable variation among means for a test method. These two items are not

necessarily related.

19
10. References

[1] ASTM E 23-02a, "Standard Test Methods for Notched Bar Impact Testing of Metallic

Materials," ASTM International, West Conshohocken, PA.

[2] International Organization for Standardization, Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Geneva Switzerland, 1 993

(corrected and reprinted 1995).

[3] ISO 148-1, "Metallic materials - Charpy pendulum impact, Part 1: Test method " ISO,

Geneva, Switzerland.

[4] Yamaguchi, Y., Takagi, S., and Nakano, H., "Effects of Anvil Configurations on Absorbed

Energy," Pendulum Impact Testing: A Century of Progress, STP 1380 , T. A. Siewert and M. P. Manahan, Sr., Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000

[5] NIST/SEMA TECH e-Handbook ofStatistical Methods,

http://www.itl.nist.gov/div898/handbook/, July 18, 2006.

20
Appendix A. Uncertainty Details

The following information is provided for completeness and to document the justification for the

recommended uncertainty procedures.

A.1 Test Material

We define a single measurement for a test material measured in a Charpy laboratory as

V = U + h + £ •+<? +£? + £ si r*Y Y /, inhomogeneity i, repeatability i, other random systematic

for i = 1,2, ...,n measurements. Terms on the right side of the equation having the "z" subscript

denote random errors that change from measurement to measurement.

HY represents the true mean breaking energy of the test material ifthe material could have been

tested on the three NIST reference machines.

bY represents the true machine bias for the test material. This term includes all machine

differences that are constant for the duration of the set of n measurements.

^mhomogeneny represents the material inhomogeneity.

ei, repeatability represents the machine repeatability.

ei, other random represents all other sources of error due to random effects.

Systematic represents errors due to all other systematic effects that are not already included in the

machine bias (for example, operator error). Systematic errors remain constant for the duration of

the set of n measurements.

The mean of n measurements of the test material is

TJ =//+/)+ g -\-£ 4- f> -\- p J r^Y Y inhomogeneity repeatability other random systematic '

and the true variance of y is

2 2 2 Var(_y ) = lnhom°geneity +

repeatability ^other random n n n

s

2 which is estimated by —, with df = n-\ degrees of freedom. The three random errors cannot

n be estimated separately. The corrected value is

21
^corrected ~~ ^ e:systematic '

The indirect verification results will be used to estimate bY and its uncertainty, and we will assume that esystematic is zero. There is uncertainty associated with each of the estimated

systematic errors. The combined standard uncertainty of corrected value is

"(^corrected) = y— + W ( bY ) + U (^systematic)

The effective degrees of freedom based on the Welch-Satterthwaite approximation are

df«• =

^ (^corrected

)

J f -2 \

2

( S

] u (bY )| u (esystematic ) dfb dfe df

The expanded uncertainty associated with yconeaed is

^ = ?l-^,dfe ,T

' U(^corrected)

The corrected value reported by the Charpy laboratory has the form ycomcted ± U . If a Charpy

laboratory does not report results corrected for machine bias, they may want to indicate the

magnitude of the estimated bias for informational purposes as

(y-by- esystematic ) ± U or (y ± U) - (bY + esystematic )

.

A.2 Indirect Verification Test

The Charpy laboratory's indirect verification test will be used to estimate machine bias in

conjunction with the associated NIST reference value. A single measurement in the indirect

verification test is defined as

i f^Z V /'.inhomogeneity /.repeatability i. other random systematic '

where i = 1,2,..., nv measurements (nv

is usually five). The "/" subscript denotes errors that

change from measurement to measurement.

Hz represents the true mean breaking energy of the reference material if the material could have

been tested on the NIST reference machines.

22
bv represents the machine bias for the reference material. This term includes all machine

differences that are constant for the duration of the set of nv measurements.

8i inhomogeneity represents the reference material inhomogeneity.

^.repeatability represents the machine repeatability.

5i other random represents all other sources of error due to random effects.

Systematic represents errors due to all other systematic effects that are not already included in the

machine bias. Systematic errors remain constant for the duration of the set of nv measurements.

The mean of nv measurements is

^ ~ ftz ^ by + ^inhomogeneity ^repeatability ^other random ^systematic '

and the variance of V ,

2 2 Var(K ) =

inhom°geneity _|_ repeatabiUty ^pther random

s

2 is estimated by —, with dfv -nv -\ degrees of freedom. The three random errors cannot be

nv estimated separately.

A.3 NIST Reference Value

The NIST reference value will be used to estimate machine bias in conjunction with the

customer's associated verification test.

According to ASTM E 23-06, the reference value of Charpy indirect verification specimens is established using three master machines maintained by NIST.

In the NIST Charpy verification program, the reference value and its associated uncertainty are based on two sets of measurements. The first set of measurements involves breaking 75

verification specimens (25 on each master machine) from a "pilot" lot to determine if the

material meets the rigid specifications of the verification program. If the material is acceptable,

the remaining verification specimens in the lot are machined and a second set of measurements

are performed from the full "production" (25 on each master machine). Assuming the

production lot has not changed significantly from the original pilot lot, the material is sold to the

23
public in sets of five specimens as a Standard Reference Material. The reference value R is established using the 75 verification lot and 75 production lot specimens.

We make the following assumptions when determining the reference value and its uncertainty.

1 . The reference value is defined to be the "truth," so there is no bias associated with the

reference value.

2. There is no difference between pilot lot specimens and production lot specimens.

(Differences are evaluated using a Mest for means and an F-test for variances.) In the event

that the verification lot and production lot have significantly different means and/or

variances, the reference value will be based solely on the production lot data.

A.3.1 Reference Machine

We define a single measurement taken on a NIST reference machine as

^\k Yk, inhomogeneiry Yk, repeatability Yk, other random Ysystematic '

where k = 1,2,...,«, measurements («, is usually 50). The subscripts on the right hand side

of the equation denote errors that change from measurement to measurement.

represents the true mean breaking energy of the reference material as measured by the NIST

reference machine.

7/t,.nhomogeneity represents the reference material inhomogeneity.

Yk, repeatability represents the machine repeatability.

Yk other random represents all other sources of errors due to random effects.

Systematic represents the errors due to all systematic effects. Systematic errors remain constant for

the duration of the set of nx measurements. Although we assume /systematic is zero, it does have

some uncertainty.

The mean of nl measurements taken on a NIST reference machine is

^\ /^l ^inhomogeneity ^repeatability Mother random Ysystematic '

and the variance associated with the mean,

24
2 2 2

, ,o W 7 \ — '"homogeneity repeatability ^"other random var^z , ; — i

i , «j «j

is estimated by —, with df, -nx -\ degrees of freedom. The three random errors cannot be

n\ estimated separately.

The corrected value for the NIST reference machine is

Z1, corrected =Z1 -r/ systematic'

The combined standard uncertainty of the corrected value is

corrected) = J W (^systematic) '

which has effective degrees of freedom

U (-^1, corrected) dfz =

W systematic^

based on the Welch-Satterthwaite approximation.

The procedure for computing Z, corrected , «(Z, corrected )

, and dfz for one reference machine also

applies to the remaining two NIST reference machines so that we obtain Z2 corrected , w(Z2 corrected )

,

and dfZj for the second reference machine, and Z3 corrected , w(Z3

xcorrected )

, and dfZj for the third

reference machine. The results from all three reference machines are needed to compute the

NIST reference value, as we discuss below.

A.3.2 NIST Reference Value

The NIST reference value based on data observed for the three reference machines is defined as

where jux , ju2

, and ju

3 denote the respective true mean breaking energies for each of the three

reference machines. The NIST reference value defines the true breaking energy of the material.

25
We estimate the NIST reference value using

^\ .corrected ^2, corrected ^3, corrected

which has combined standard uncertainty

U( R) = ^k(^l.corrected) + ^(^2. corrected ) + ^(^.corrected)] >

and effective degrees of freedom

u\R)

df,

.corrected . corrected )

| (I) M (Z3.corrected) df7 df7 df7

^1 ^2 *-3

based on the Welch-Satterthwaite approximation. The reference value expanded uncertainty is ^=u* »(*>

A.4 Estimating Machine Bias

Assume the machine bias is the same for the new material ( bY

) and the reference material ( bv ), so that bY =bv = b . The best estimate of the machine bias b is

b — V — ^systernatj c — R j

which has combined standard uncertainty

u{b) = ]^- + u

2 (SsystenUitic ) + u\R)

nv

The effective degrees of freedom based on the Welch-Satterthwaite approximation are

u\b)

df =

dfv

| U (Systematic)| u\R)

\ nvJ df, dfR

26
Appendix B. Direct Verification Components of Uncertainty

B.l Anvils and Supports, A

A paper by Yamaguchi, Takagi, and Nakano [4] provides some information regarding the

uncertainty associated with anvil configurations. Assuming that other Charpy machines behave

similarly to the machine tested in the paper, we can use the uncertainties listed in the paper

(Table 9) as ballpark estimates. Table B.l lists the uncertainty estimates for low, high, and

super-high energies.

Table B.l. Estimated uncertainties due to the anvil and support bias.

Standard uncertainty Low energy High energy Super-high energy

u(A) 0.05 J 0.29 J 0.77 J

Since degrees of freedom are not provided in the paper, we will also assume that dfA = go , which

implies that we know u{A) exactly.

B.2 Height of Pendulum Fall, h

The height of the pendulum fall is

h = S-(\-cosj3),

where S is the measured length of the pendulum, and P is the measured fall angle. Letting AS

and A/? denote the manufacturer's stated error bounds, respectively, and assuming a rectangular

distribution bounded by ± AS and ± AJ3 , the uncertainties for S and J3 are \

f

" AS Afi

u(S) = —j= and u(p) = —j=.

V3 V3

Then the uncertainty of h is

u\h) =

dh_

K dSj

2 ..2

u\S) +

dh_ u\P) + 2dh_

dS

u(S,fi),

c

ls u\S) + c2

/} u

1 {P) + 2cs c

p u(S, P) ,

where

dh_

dS

dh

Cn = —= 1 - cos/? and cR = —= S (sin B).

p dp

27
If ( S, ft ) are independent, then only the first two terms are needed to determine the uncertainty.

We can assume that dfs = oo and df

p = oo , which implies that we know u(S) and w(/?) exactly.

(See the ISO-GUM, G.4.3 [2] for details.) The effective degrees of freedom associated with

u{h) are

dfh =

c

4s u\S)

| c>W

dfs df

p

based on the Welch-Satterthwaite approximation.

B.3 Potential Energy, E

The potential energy is

E = h-F

,

where F is the measured supporting force exerted by the pendulum in horizontal position, and

h is the height of the pendulum fall defined in Section B.2. Letting AF denote the

manufacturer's stated error bound of the measurement instrument, and assuming a rectangular

distribution bounded by ± AF , the uncertainty of F is

u(F) = AF

Then the uncertainty of E is

u

2 (E) = fdE^2

K dhj

.2 2,

u\h) +

dE_

K dFj

u\F) + 2

^dE}

K dhjydFj

= c^u\h) + c\ u

2 (F) + 2ch cFu(h, F),

where

c, = —= F and c r_ —= h

.

dh dF

The uncertainty associated with h is defined in Section B.2. lf(h, F) are independent, then

only the first two terms are needed for the uncertainty. We can assume dfF = oo , which implies

that we know u(F) exactly. The effective degrees of freedom associated with u(E) are

28
clu\h)

| c}u\F)

dfh dfF

based on the Welch-Satterthwaite approximation.

B.4 Impact Velocity, v

The impact velocity is

where g is the local acceleration of gravity, and h is the height of the pendulum fall defined in

Section B.2. Letting Ag denote the manufacturer's stated error bound of the measurement

instrument (0.001 m/s according to ASTM E 23), and assuming a rectangular distribution

bounded by ± Ag , the uncertainty for g is

The uncertainty of V is

u{g) =

Ag

u\v) =

,2 . 2

w

2 (g) + u2 (h)

= c'u\g) + c

2

h u

2 (h),

where

dv h42

dg ljg~h

and ch =

dv gV2

a/? 2,/g^

The uncertainty associated with h is defined in Section B.2. We can assume df

g = oo , which

implies that we know u(g) exactly. The effective degrees of freedom associated with u(v) are

df =

u\v)

df df

based on the Welch-Satterthwaite approximation.

29
B.5 Center of Percussion, L

The center of percussion is

L = S'P

4tt

2

where g is the local acceleration of gravity defined in Section B.4, and p is the mean period of

the swing of the pendulum from three measurements for 100 swings. (There may be some

systematic error associated with p that should taken into account.) The standard deviation of

three p measurements is s , so the uncertainty of the mean period is

u(p)= fr

with df =3-1 = 2 degrees of freedom.

The uncertainty of L is

where

u

2 (L) =

\ dSj

u\g) +

\ dPj

u\p),

= c

2s u2 (g) + c2 u2 (p),

dL p dL g-p

c = —= -£—r and c =

dg Ax2 dp 2nl

The uncertainty associated with g is defined in Section B.4. From the Welch-Satterthwaite

approximation, the effective degrees of freedom associated with u(L) are

u\L)

u\g)

+ clu\p)

df df

B.6 Friction Loss, D

The friction loss is D = E -El ,

30
where E is the potential energy due to the combined indicator and pendulum, and Ex is the

potential energy due to the pendulum. The uncertainty of D is

u

2 (D) = u2 (E ) + u2 (El ) + 2u{EQ ,El ) .

Assuming perfect correlation between E and Ex , a conservative estimate of the covariance

u(E ,E{ ) is

u(E ,E{ ) = yju

2 (E ) + u2 (E^ .

The effective degrees of freedom associated with u(D) are

df = *M

u\E )

| u\Ex )

'

dfE dfE,

based on the Welch-Satterthwaite approximation.

B.7 Scale Accuracy, r

Let r represent the bias in the scale mechanism and ± Ar be the specified error bounds of the

measurement instrument. Assuming a rectangular distribution, the uncertainty of r is

t \ Ar

V3 \

We will assume dfr = oo , which implies that we know u(r) exactly.

31
Appendix C. f-Table

The following stable values were taken from NIST/SEMA TECH e-Handbook ofStatistical

Methods [5].

df 900 950 975 \j • 990 y y \j \jm 995 y y *j \j • 999 y y y

11 .J1 . 078 Vy / O 6 114 121 — . 706 / Vy ly .J111 . 821 OZ. I 61UJ . 657 VJ.J / J118lOJ1111 ^y 9Z» 1 886 2Z, . 920 J7Z.V/ 4 101 6 965 9 925 Z.Z*..yZ. 99 197/ x 11 . 618 \J .J O 9 151 1 . 189 1 OZ, 4 541 .J5 . O"8411 10i yj . z.915i j

4 1 511 Z.9 119 1JZ. Z.9 . 776 1 1 \J 1 747 4 604 7 1 71

.J 1 476 Z.vlJ 9 015 2 571 .J1 . JU165J 4 019 J.07J 5 891

A 1 440 1 941 9 447 1 141 1 707 J5 .z.908\JO

7 1 41 5 11 . 895 O .7 J Z.9 JU165J 2 998 j1 ,*yy 499 4 789 / OZ- 8o 1 197 1i . 860 o vj\j 2 106 z,2 . 896 o y yj 1 155 4" 499y y Qy 11 ..J181O.J 11 . 811 0«y J 2 262 Z.2 . 891 OZ. 1 .y1 .Z.950J Vy 4 996

10 1 172 1i . 812 O I _ 2 228 z*. 9 764 / *y1 . 169 1 Vy ^ 4 141

1

1

1 1 11 .«y161Vy J 1 796 2 201 Z.2 . 718 / 1 O 1 106 4". 094 Vy Z," 12 1 156 11 . 782 / OZ. Z.2 . 179 1 / J7 2 681 1 055 1j . 999 y y

1

1

1 150 11 . 771 / / 1 9 160 Z.UJ 2 650vy 1 019 -y1 . 859 o*yZ. 14 1 145 11. 761 / Ul Z.9 . 145 1 ^.J 2Z. . 624 V/ZrT^ Z,9 . 977/ / .y1 . 787 / O /

1 5 1 J 1 141 1 751 Z,9 1J1111 Z.9 . 609 Uv/Z. Z.9 . 947/ .y1 . 711 / J J

16 1 117 1 746 9 190 9 581 Z.9 . 991 7L 1 1 686

1 7 1 111 11 . 740 / ^Vy 9 110 9 567 L.070 9 898 1 646

11 8O 11 .JJV/ 110 1 714 Z.9 . 1101v/ 1 Z,9 . J559JZ 9 878 J1610 .U 1 Vy 1Q 1 198 1 799 9 091 Z.JJ7 9 519 9 861 1 579

90 1 195 11 . 795 / Z, J 9 086 Z.9 JZ.O 598 9_ . 845 O"J J1 .J559JZ

91 1 191 11 . 791 / Z 1 9 080 Z*..J 9 5181 o Z.O 9 811J l 1 597

zvz, 11.JZ.1 191 1717 9 074 9 508 9 819 J1 . JVy 505

J

91 i HQ 1 714 9 069 9 500 9 807 1 485

94 1 118 1711 9 064 9 499 Z.9 797 (7/ 1 467

z,95j 1 116 1 708 9 060 9 485 z.9 787 / o / 1 450

96zu 1115 1 706 9 056 9Z.^T479/ 1/ z.9 779 / / y 1 415

Z971

1 114 1 701 9 0S9 Z.t 9 471/ J z.9 771 / / 1 J1 .tz 4911 98ZO 1111 1 701 9 048 9 467 9 761 1 408

Ly90 1111 1 6QQ 9 04S 9 469 Z.. 9 7S6 / JO J1 . jyv 1Q6

10 1 110 1 697 Z.9 . 049 Vy^Z, Z.9 . 457/ 9 750 J.JOJ 1 185

40 1.303 1.684 2.021 2.423 2.704 3.307

50 1.299 1.676 2.009 2.403 2.678 3.261

60 1.296 1.671 2.000 2.390 2.660 3.232

70 1.294 1.667 1.994 2.381 2.648 3.211

80 1.292 1.664 1.990 2.374 2.639 3.195

90 1.291 1.662 1.987 2.368 2.632 3.183

100 1.290 1.660 1.984 2.364 2.626 3.174

00 1.282 1.645 1.960 2.326 2.576 3.090

32
Appendix D. Glossary of Terms

ny

sdfnvV

Sy

dfvR

u(R)

dfR

^systematic u(e , » ) V systematic / dfe

^systematic

^( ^systematic ) df5

-^corrected ^(^corrected)

dfeff U

Number of test material samples measured

Mean absorbed energy of test material samples

Standard deviation of test material samples

Degrees of freedom for test material standard deviation

Number of indirect verification samples measured

Mean absorbed energy of verification samples

Standard deviation of verification samples

Degrees of freedom for verification material standard deviation

NIST reference value

Standard uncertainty of NIST reference value

Degrees of freedom for reference value standard uncertainty

Systematic error estimate associated with test material

Standard uncertainty of test material systematic error Degrees of freedom for standard uncertainty of test material systematic error Systematic error estimate associated with verification material

Standard uncertainty of verification material systematic error

Degrees of freedom for standard uncertainty of verification material systematic

error Corrected test result

Combined standard uncertainty of corrected test result

Degrees of freedom for combined standard uncertainty of corrected test result

Expanded uncertainty of corrected test result

33

GX> U.S. GOVERNMENT PRINTING OFFICE: 2008 — 760-976 / 85011 Region No. 8



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serve as a vehicle for final reports of work performed at NIST under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The standards establish nationally recognized requirements for products, and provide all concerned interests with a basis for common understanding of

the characteristics of the products. NIST administers this program in support of the efforts of private-sector

standardizing organizations.

Order the following NISTpublications—FIPS and NISTIRs—from the National Technical Information Service,

Springfield, VA 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively

constitute the Federal Information Processing Standards Register. The Register serves as the official source of

information in the Federal Government regarding standards issued by NIST pursuant to the Federal Property and

Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1 127), and as implemented by

Executive Order 1 1717 (38 FR 12315, dated May 1 1,1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

NIST Interagency or Internal Reports (NISTIR)—The series includes interim or final reports on work performed

by NIST for outside sponsors (both government and nongovernment). In general, initial distribution is handled by

the sponsor; public distribution is handled by sales through the National Technical Information Service, Springfield,

VA 22161, in hard copy, electronic media, or microfiche form. NISTIR's may also report results of NIST projects of

transitory or limited interest, including those that will be published subsequently in more comprehensive form.

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