NIST Recommended Practice Guide

Special 

Publication

960-18

Computing Uncertainty

for Charpy Impact 

Machine Test Results

J. D. Splett

H. K. Iyer

C.-M. Wang

C. N. McCowanPrecision

Accuracy

October 2008

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

National Institute of Standards and Technology

Patrick D. Gallagher, Deputy Director

NIST Recommended Practice Guide

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 of Standards and Technology

Boulder, Colorado 80305
U. S. GOVERNMENT PRINTING OFFICE

Washington: 2008 

------------------------------------------------------------------------------------------------------------------

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

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

Mail: Stop SSOP, Washington, DC 20402-0001

National Institute of Standards and Technology Special Publication 960-18

Natl. Inst. Stand. Technol. Spec. Publ. 960-18, 34 pages (October 2008)

CODEN: NSPUE2

Certain commercial entities, equipment, or materials may be identifi ed in this document 

in order to describe an experimental procedure or concept adequately. Such identifi cation 

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.
iii

 Machine RContents

 1. Introduction 1

 2. Uncertainty of a Test Result 2

 3. Type B Uncertainty Evaluation 6

 4. Machine Bias 8

 5. Direct Verifi cation 11

 6. Temperature 12

 7. Expanded Uncertainty 14

 8. Examples 15

 8.1 Both systematic eˆ and systematic

ˆ and Their Uncertainties

 are Negligible 15

 8.2 systematic eˆ has One Component 17

 9. Closing Remarks 19

10. References 20

Appendix A. Uncertainty Details 21

A.1 Test Material 21

A.2 Indirect Verifi cation Test 22

A.3 NIST Reference Value 23

A.4 Estimating Machine Bias 26

Appendix B. Direct Verifi cation Components of Uncertainty 27

B.1 Anvils and Supports, A 27

B.2 Height of Pendulum Fall, h 27

B.3 Potential Energy, E 28

B.4 Impact Velocity, 29

B.5 Center of Percussion, L 30

B.6 Friction Loss, D 31

B.7 Scale Accuracy, r 32

Appendix C. t-Table 33

Appendix D. Glossary of Terms 34

Table of Contents 

1

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 verifi cation 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 verifi cation program administered by the National Institute of Standards and 

Technology. 

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

uncertainty; verifi cation testing

1. Introduction

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

Introduction 
2

Computing UnCertainty for Charpy Impact Machine Test Results

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

to pass both direct and indirect verifi cation tests as specifi ed in ASTM E 23 [1]. Even 

if a Charpy machine has passed the indirect verifi cation test, it is likely that results for 

the verifi cation specimens differ from the certifi ed value. This difference can be used to 

quantify machine bias. Thus, the indirect verifi cation results and the certifi ed 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 Material inhomogeneity

Center of percussion Operator

Center of strike Potential energy 

Friction loss Repeatability

Height of pendulum fall Scale accuracy

Impact velocity Test temperature 

 The uncertainty contributions from individual error sources can be estimated if 

they are identifi ed as signifi cant, but generally these errors are assumed to be minimized 

by adjustments made to the machine during direct verifi cation and by following the 

standard test procedure. As will become apparent, the calculation of uncertainty is 

greatly (and often) simplifi ed by assuming that direct verifi cation contributions are 

zero, and only contributions from indirect verifi cation 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 verifi cation 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 justifi cation 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 
3

sample standard deviation of n specimens of the test material using the following two 

equations: 

 1

n

i

i

y

y n

 



 , (1)

 

2

1

( )

1

n

i

i

y y

n s 



 



 . (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 

ˆ

b , which is defi ned as the difference between the verifi cation result for the test machine 

and the certifi ed value of the verifi cation specimens. Systematic errors due to all other 

factors that are not already included in the machine bias are denoted by systematic eˆ . The 

values ˆ u b( ) and systematic u e( ) ˆ are the uncertainties associated with systematic ˆ

b e and ˆ , while dfb

and dfe

, represent degrees of freedom for systematic ˆ ub ue ( ) and ( ) ˆ . 

Table 1. Measurement results for a test material.

Observed data, J Summary statistics

58.0 n  5

62.0 y  57.6 J

54.0 s  3.6 J

54.0 df = 4

60.0

 Uncertainty of a Test Result 
4

Computing UnCertainty for Charpy Impact Machine Test Results

Table 2. Example quantities required to compute uncertainty of a test result.

Machine bias Systematic error

 

Estimate ˆ

b  4.2 J systematic eˆ  3.0 J 

Uncertainty ˆ u b( ) 2.8  J systematic u e( ) 0.6 ˆ  J

Degrees of freedom dfb

 = 84 dfe = 2

To compute the uncertainty of the test result, we fi rst compute a mean of the test material 

that is corrected for machine bias and all other systematic effects:

 corrected systematic ˆ y ybe   ˆ . 

(3)

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

corrected y     57.6 J ( 4.2 J) (3.0 J) 58.8 J .

Next, we calculate the uncertainty of the corrected mean, corrected y . Assuming all of the 

terms in corrected y are independent, the combined standard uncertainty of corrected y is

 2

2 2

corrected systematic ˆ ( ) () ( ) ˆ s uy u b u e n   . (4)

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

2

2 2

corrected

(3.6 J) ( ) (2.8 J) (0.6 J) 3.3 J 5 u y    .

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 defi ned 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 ) D % coverage probability (D is 0.05 for 95 % 

coverage) is given by 
5

 Uncertainty of a Test Result 

 2 eff corrected corrected 1 ,df y t uy a ( )    , (5)

where 2 eff 1 ,df t a is found in a t-table (see Appendix C). The degrees of freedom associated 

with corrected u y( ) , 

 

 

4

corrected

eff 2 4 2 4 systematic

b e

( ) df

1 ()ˆ ( ) ˆ

df df df

u y

s ub u e

n



     

 

 (6)

 

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

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

4

eff 2 2 44

(3.3) df 47.9

1 (3.6) (2.8) (0.6)

4 5 84 2

 

     

 

,

which rounds down to 47. Using a t-table we get a coverage factor of 0.975,47 t = 2.012. 

Thus, a 95 % interval for our example is

0.975,47 58.8 J 3.3 J

58.8 J 2.012 3.3 J

58.8 J 6.6 J.

  t

 



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 uncertainty interval as 

systematic ˆ ( )( ) ˆ

(57.6 J 6.6 J) ( 4.2 J 3.0 J)

(51.0 J, 64.2 J) 1.2 J

yU be  

  



and just report the uncorrected interval (51.0 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 verifi cation specimens 

tested. 
6

Computing UnCertainty for Charpy Impact Machine Test Results

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 

diffi cult 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 uncertainty evaluation. Type A uncertainty evaluations are based on data obtained 

under repeatability conditions. Type B uncertainty 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 specifi cations 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 specifi cation for a Type B uncertainty evaluation. Suppose r is 

the random error in the Charpy machine scale mechanism and ± r represents the 

manufacturer’s specifi ed error bounds of the measurement instrument. Assuming that 

the error can be anywhere within the ± r 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

( ) 3

r u r  .

A rectangular distribution is often used in the absence of specifi c 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 r df   , which implies that we know u 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 defi ned 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 
7

Uncertainty Evaluation 

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 defi ned by 

(,) a b , where a and b are both greater than zero ( 0   a b ). In this case, the estimated 

systematic error is ( )2 a b  and the associated standard uncertainty is ( )23 b a  .

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

systematic

2 J 4 J ˆ 3 J

2

e

   and systematic4 J 2 J ( ) 0.6 J ˆ 2 3

u e    .

To determine the degrees of freedom associated with systematic u e( ) ˆ , we employ a useful 

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

2

1 df

2

u

u



        ,

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

of the uncertainty. In our example, we judge the uncertainty of systematic u e( ) ˆ to be 50 % or 

0.50, so that

  2

e

1 df 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 systematic eˆ 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 ˆ ˆ e DEv ˆ ˆ   .
8

Computing UnCertainty for Charpy Impact Machine Test Results

Then the combined standard uncertainty of systematic eˆ is

2 22

systematic ˆ ˆ ue u D u E u v ( ) ( ) ( ) () ˆ ˆ   ,

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

4

systematic

e 4 44

D Ev

( ) ˆ df ˆ ˆ ( ) ( ) ()ˆ

df df df

u e

uD uE uv 

 

.

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

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 verifi cation. This is an important 

assumption that allows us to estimate machine bias for all test materials. We use the 

results of an indirect verifi cation test and the associated reference value for our best 

estimate of machine bias,

 systematic ˆ ˆ bV R   , (7)

 

where 

 1

Vn

i

i

V

V

V n

 



 (8)

is the sample mean absorbed energy from the indirect verifi cation test based on 5 Vn 

test results, represents errors due to all systematic effects associated with 

indirect verifi cation test, and R represents the certifi ed reference value for the batch of 

verifi cation 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 verifi cation test specimens that 

were used for the most recent high-energy indirect verifi cation of the Charpy machine of 

interest.

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

systematic

ˆG
9

Machine Bias 

absorbed energy of the test material is closest to the high-energy verifi cation material. 

The uncertainty associated with the certifi ed verifi cation specimens ( u R( ) ) is provided by 

NIST with the results of the indirect verifi cation test (or by request). Table 4 displays the 

indirect verifi cation data that were observed when the verifi cation set was broken on the 

machine of interest.

Table 3. Information provided by NIST for high energy verifi cation specimens.

Reference value, R 109.9 J

Reference value standard uncertainty, u R( ) 2.6 J

Degrees of freedom, dfR 102

Table 4. High energy indirect verifi cation test results.

Verifi cation set data, J Summary statistics

108.0 5 Vn 

104.0 V  106.2 J

109.0 2.3 V S  J

106.0 dfV = 4

104.0

Table 5. Systematic error associated with the indirect verifi cation. 

Estimate, systematic

ˆG 0.5 J

Standard uncertainty, ˆ u( ) G 0.2 J

Degrees of freedom, df 10

The “V” subscript is used to distinguish the indirect verifi cation results from the test 

material results. The sample standard deviation associated with the indirect verifi cation 

specimens ( V S ) is calculated as s was calculated previously in Section 2,

 

2

1

( )

1

Vn

i

i

V

V

V V

S n





 



 . (9)

As was the case for s, V S also includes all sources of random error related to both 
10

Computing UnCertainty for Charpy Impact Machine Test Results

machine variability and material variability, and the individual contribution of errors 

cannot be determined. 

Suppose we are given , systematic

ˆG its associated uncertainty, and degrees of freedom, as 

shown in Table 5. We will not elaborate on the origin of the systematic error in Table 

5; however, the same general procedures used to estimate systematic eˆ , discussed in detail in 

Section 3, can also be used to estimate systemati ˆG .

Although systematic

ˆG = 0.5 J in this illustrative example, typically 

is assumed to be zero because errors that are well understood and could be corrected for 

are minimized during direct verifi cation of the machine. So, neglecting contributions to 

the bias from systematic

ˆG , the estimated machine bias is calculated as the difference between 

the mean of the specimen tested in the indirect verifi cation test and the certifi ed value of 

the specimens tested. For our example, in which systematic

ˆG is not assumed to be zero, the 

machine bias is 

ˆ

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

 

 

 . (10)

Substituting the appropriate values from Tables 3 through 5 into Eq. (10) provides the 

following estimate of the standard uncertainty of the machine bias:

2

2 2 (2.3 J) ˆ ( ) (0.2 J) (2.6 J) 2.8 J 5

u b    .

The degrees of freedom associated with the uncertainty estimate, 

 

 , (11) 

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

degrees of freedom are 

2

2 2

systematic ˆ ˆ () ( ) ( ) V

V

S ub u u R n   G

4

b 2 2 4 4 systematic

V į R

ˆ ( ) df ˆ 1 () ( )

df df df

V

V

u b

S u u R

n

G 

§ ·  ̈  ̧  

© 1
11

 Direct Verifi cation 

 

4

b 2 2 44

(2.8) df 84.4

1 (2.3) (0.2) (2.6)

4 5 10 102

 

     

 

,

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, defi ned 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 verifi cation systems 

used around the world (multiple certifi ed 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 verifi cation specimens are 

working to minimize biases among them, and make the quantifi cation of bias for impact 

testing more transparent to users around the world. 

5. Direct Verifi cation

Direct-verifi cation 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 verifi cation sources of uncertainty are Type B evaluations. We 

provide information regarding calculation of the individual sources of direct verifi cation 

uncertainty in Appendix B. While it is relatively easy to compute each individual source 

of uncertainty, it is diffi cult 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 

verifi cation 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 

verifi cations 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 quantifi ed. 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-verifi cation 

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 
12

Computing UnCertainty for Charpy Impact Machine Test Results

direct verifi cation sources of error. If possible, the uncertainty associated with direct 

verifi cation should be re-examined each time the machine is verifi ed directly.

6. Temperature 

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

Charpy measurements, it is diffi cult 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 defi ned 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 signifi cant if measurements are being taken in the transition region of 

the curve.

Figure 1. A temperature transition curve.
13

Temperature 

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 fi t a straight line to the data within the temperature transition 

region (ignoring the shelf data). The information from the regression fi t 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 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 E . Assuming the true impact energy has a rectangular 

distribution within E , we can use the range of impact energy to compute the 

uncertainty as follows:

| | ( ) 2 3E

u t

  .

The degrees of freedom are t df 2  n , where n is the number of observations used in 

the regression fi t. 

Optionally, multiple measurements could be made at each temperature (which is how 

the original curve is obtained) and defi ne 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.
14

Computing UnCertainty for Charpy Impact Machine Test Results

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

corrected U kuy   ( ) .

The coverage factor k is determined by looking up the appropriate value in a t-table 

(Appendix C) based on the degrees of freedom associated with corrected u y( ) . The expanded 

uncertainty associated with a 95 % interval is

eff 95 95 corrected 0.975;df corrected U k uy t uy    () () . (12)

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

fraction of possible measurement results. 

The degrees of freedom can be diffi cult to determine if there are many sources of 

uncertainty within corrected u y( ) . Fortunately, we can compute the effective degrees of 

freedom from the Welch-Satterthwaite approximation [2]

 

4

corrected

eff 2 4 2 4 systematic

b e

( ) df

1 ()ˆ ( ) ˆ

df df df

u y

s ub u e

n



     

 

, (13)

where df 1  n and e df are from the Type B uncertainty evaluation (see Section 3). We 

will also need to calculate b df from

 

4

b 2 2 4 4 systematic

V į R

ˆ ( ) df ˆ 1 () ( )

df df df

V

V

u b

S u u R

n

G 

§ ·  ̈  ̧  

© 1

 , (14)

where V df 1 V   n , R df is provided by NIST with the indirect verifi cation specimens, and

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

In general, an uncertainty interval for corrected y is

corrected 1 y U a

or 
15

 Expanded Uncertainty 

 2 eff corrected corrected 1 ,df y t uy a ( )    . (15)

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,

systematic 1 1 systematic ˆ ˆ ( ) or (y ) ( ) ybe U U be ˆ ˆ        a a . (16)

Thus, the interval would be shifted by systematic ˆ

b e  ˆ 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 confi dence is thought to be less 

than 95 %.

8. Examples

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

8.1 Both systematic eˆ and systematic

ˆG and Their Uncertainties Are Negligible 

In the general case, 

 corrected systematic ˆ y ybe   ˆ ,

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

negligible, then

corrected ˆ y yb   .

 If the systematic error for the test machine variation systematic

ˆG is assumed to be 

negligible, 

systematic ˆ ˆ bV RV R     G 106.2 J 109.9 J

and
16

Computing UnCertainty for Charpy Impact Machine Test Results

corrected ˆ y yb      57.6 J ( 3.7 J) 61.3 J .

The combined standard uncertainty of ˆ

b is

 

 

2

2 2

systematic

2

2

2

2

ˆ ˆ () ( ) ( )

( )

(2.3 J) (2.6 J) 5

2.8 J ,

V

V

V

V

S ub u u R n

S u R n

   G

 

 

 

with effective degrees of freedom

4 4

b 2 2 2 24 4

V R

ˆ ( ) (2.8) df 84.5

1 () 1 (2.3) (2.6)

df df 4 5 102

V

V

u b

S u R

n

 

       

   

,

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

 

2

2 2

corrected systematic

2

2

2

2

ˆ ( ) () ( ) ˆ

ˆ ( )

(3.6 J) (2.8 J) 5

3.2 J ,

s uy u b u e n

s u b n

 

 

 



with effective degrees of freedom

 

4 4

corrected

eff 2 2 24 2 4

b

( ) (3.2) df 43.5 ˆ 1 ( ) 1 (3.6) (2.8)

df df 4 5 84

u y

s ub

n

 

       

  

 ,
17

 Examples 

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

2 eff corrected corrected 1 ,df

0.975,43

( )

61.3 J 3.2 J

61.3 J 2.017 3.2 J

61.3 J 6.5 J.

y t uy

t

  a

 

 



The expanded uncertainty, associated with a 95 % level of confi dence 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 

ˆ (y )

(57.6 J 6.5 J) ( 3.7 J)

(51.1 J, 64.1 J) 3.7 J.

  U b

 

 

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.

8.2 systematic eˆ Has One Component

Suppose systematic eˆ contains the error due to temperature so that systematic e t ˆ  ˆ . 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 systematic eˆ is zero. The uncertainty associated with systematic eˆ is

systematic ue ut ( ) () ˆ  ˆ .

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

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

with the regression line fi t 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, 
18

Computing UnCertainty for Charpy Impact Machine Test Results

which is within the temperature transition region. A regression line was fi t to the 21 data 

points in the transition region, resulting in the following equation:

E(J) 0.03973(J) 0.74084(J/ C) T( C)      .

Figure 3. Temperature data for the test material. The straight line in the plot represents a 

regression fi t to the data in the transition region only. We ignore data on the “shelves” at -10 °C 

and 141 °C.

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 defi ned 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) 81( C) 60.0 J

E 60.0 J 58.5 J 1.5 J .

    

    

  

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

distribution,

1.5 J ( ) 0.4 ˆ

2 3

u t   J ,

with degrees of freedom t df 2 21 2 19   n . Thus the uncertainty of systematic eˆ is

systematic u e( ) 0.4 J ˆ  ,

with 19 degrees of freedom.
19

 Examples 

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

corrected systematic ˆ y ybe        ˆ 57.6 J ( 3.7 J) 0 J 61.3 J ,

where ˆ

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

The combined standard uncertainty is

2 2

22 2 2

corrected systematic

(3.6 J) ˆ ( ) ( ) ( ) (2.8 J) (0.4 J) 3.3 J ˆ 5s uy u b u e n       ,

with degrees of freedom

 

4 4

corrected

eff 2 2 4 24 2 4 4 systematic

b e

( ) (3.3) df 49.2

1 ( ) 1 (3.6) (2.8) (0.4) ˆ ( ) ˆ

df df df 4 5 84 19

u y

s ub u e

n

 

        

  

,

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

 

2 eff corrected corrected 1 ,df

0.975,49

( )

61.3 J 3.3 J

61.3 J 2.010 3.3 J

61.3 J 6.6 J.

y t uy

t

  a

 

 



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 

systematic ˆ (y ) ( ) ˆ

(57.6 J 6.6 J) ( 3.7 J 0 J)

(51.0 J, 64.2 J) 3.7 J.

  U be

  



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 fl exible 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 verifi cation limits with respect to results obtained in a 
20

Computing UnCertainty for Charpy Impact Machine Test Results

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 verifi cation specimen or 

the variability in the verifi cation 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 

specifi c measurement result, while the verifi cation limits describe the acceptable variation 

among means for a test method. These two items are not necessarily related. 

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, 1993 (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 Confi gurations 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/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/

div898/handbook/, July 18, 2006.
21

Appendix A 

Appendix A. Uncertainty Details

The following information is provided for completeness and to document the justifi cation 

for the recommended uncertainty procedures.

A.1 Test Material

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

,inhomogeneity ,repeatability i, other random systematic y b e e e e i PY  Y  i  i  

for i n  1, 2, ,  measurements. Terms on the right side of the equation having the “i” 

subscript denote random errors that change from measurement to measurement.

 represents the true mean breaking energy of the test material if the material could 

have been tested on the three NIST reference machines.

Y b 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.

i,inhomogeneity e represents the material inhomogeneity.

i, repeatability e represents the machine repeatability.

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

systematic e 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 

Y Y inhomogeneity repeatability other random systematic y be e e e     P ,

 

and the true variance of y is

 i YYi i ,inhomogeneity , repeatability i, other random systematic y be e e e     P 
22

Computing UnCertainty for Charpy Impact Machine Test Results

 2 2 2

inhomogeneity repeatability other random var( ) Y n nn

V V V

  ,

which is estimated by 2 S

n , with df 1  n degrees of freedom. The three random 

errors cannot be estimated separately. The corrected value is 

corrected systematic ˆ ˆ Y y yb e   .

The indirect verifi cation results will be used to estimate ˆ

Y b and its uncertainty, and 

we will assume that systematic eˆ is zero. There is uncertainty associated with each of the 

estimated systematic errors. The combined standard uncertainty of corrected value is 

2

2 2

corrected systematic ˆ ( ) () ( ) ˆ Y

s uy u b u e n   .

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

 

4

corrected

eff 2 4 4 2 systematic

b e

( ) df ˆ ( ) ˆ 1 ( )

df df df

Y

u y

s u b u e

n



     

 

 . 

The expanded uncertainty associated with corrected y is 

2 eff 1 ,df corrected U t uy a ( )    .

The corrected value reported by the Charpy laboratory has the form corrected y 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

systematic systematic ˆ ˆ ( ) or (y ) ( ) ˆ ˆ Y Y yb e U U b e      .

A.2 Indirect Verifi cation Test

The Charpy laboratory’s indirect verifi cation test will be used to estimate machine bias 

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

indirect verifi cation test is defi ned as
23

References 

 V b i ZV i i P ,inhomogeneity , repeatability i, other random systematic     G GG G ,

where 1, 2, , V i n   measurements ( Vn is usually fi ve). The “i” subscript denotes errors 

that change from measurement to measurement.

PY represents the true mean breaking energy of the reference material if the material 

could have been tested on the NIST reference machines.

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

machine differences that are constant for the duration of the set of Vn measurements.

i,inhomogeneity e represents the reference material inhomogeneity.

i, repeatability e represents the machine repeatability.

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

systematic e 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 Vn

measurements.

The mean of Vn measurements is

 V b     PG G G G Z V inhomogeneity repeatability other random systematic ,

and the variance of V ,

2 2 2

inhomogeneity repeatability other random var( )

V VV

V n nn

V V V

 

 ,

is estimated by 

2

V

V

S

n , with V df 1 V   n degrees of freedom. The three random 

errors cannot be estimated separately.

A.3 NIST Reference Value

The NIST reference value will be used to estimate machine bias in conjunction with the 
24

Computing UnCertainty for Charpy Impact Machine Test Results

customer’s associated verifi cation test.

According to ASTM E 23-06, the reference value of Charpy indirect verifi cation 

specimens is established using three master machines maintained by NIST.

In the NIST Charpy verifi cation program, the reference value and its associated 

uncertainty are based on two sets of measurements. The fi rst set of measurements 

involves breaking 75 verifi cation specimens (25 on each master machine) from a 

“pilot” lot to determine if the material meets the rigid specifi cations of the verifi cation 

program. If the material is acceptable, the remaining verifi cation 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 signifi cantly from the original pilot lot, the material is sold to the public in 

sets of fi ve specimens as a Standard Reference Material. The reference value R is 

established using the 75 verifi cation lot and 75 production lot specimens.

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

uncertainty. 

1. The reference value is defi ned 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 t-test for means and an F-test for variances.) In 

the event that the verifi cation lot and production lot have signifi cantly different 

means and/or variances, the reference value will be based solely on the production 

lot data.

A.3.1 Reference Machine

We defi ne a single measurement taken on a NIST reference machine as 

Z1 1 , inhomogeneity , repeatability , other random systematic kk k k     PJ J J J 

,

where 1 k n  1, 2, ,  measurements ( 1 n is usually 50). The “k” subscripts on the right 

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

P1 represents the true mean breaking energy of the reference material as measured by 

the NIST reference machine.

k ,inhomogeneity J represents the reference material inhomogeneity.

k , repeatability J represents the machine repeatability.
25

Appendix A: Uncertainty Details 

k ,other random J represents all other sources of errors due to random effects.

systematic J represents the errors due to all systematic effects. Systematic errors 

remain constant for the duration of the set of 1 n measurements. Although we

 assume systematic J is zero, it does have some uncertainty.

The mean of 1 n measurements taken on a NIST reference machine is

 Z1 1 inhomogeneity repeatability other random systematic     PJ J J J , 

and the variance associated with the mean, 

 2 2 2

inhomogeneity repeatability other random

1

1 11

var( ) Z n nn

V V V

  ,

is estimated by 

2

1

1

S

n , with 1 1 df 1   n degrees of freedom. The three random errors

cannot be estimated separately.

The corrected value for the NIST reference machine is

 1, corrected 1 systematic Z Z Jˆ . 

The combined standard uncertainty of the corrected value is

 2

1 2

1, corrected systematic

1

( ) () ˆ S uZ u n  J ,

which has effective degrees of freedom 

 1

4

1, corrected

Z 2 2 4

1 systematic

1 1 Ȗ

( ) df

1 ( ) ˆ

df df

u Z

S u

n

J 

§ ·  ̈  ̧ 

© 1

 ,

based on the Welch-Satterthwaite approximation.

The procedure for computing Z1, corrected , 1, corrected u Z( ) , and Z1 df for one reference machine 

also applies to the remaining two NIST reference machines so that we obtain 
26

Computing UnCertainty for Charpy Impact Machine Test Results

Z2, corrected , 2, corrected u Z( ) , and Z2 df for the second reference machine, and Z3, corrected , 

3, corrected u Z( ) , and Z3 df 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 

defi ned as

 

123

3 Z

P P P P  

 

,

where 12 3 P , , and P P denote the respective true mean breaking energies for each of the 

three reference machines. The NIST reference value defi nes the true breaking energy of 

the material.

We estimate the NIST reference value using

1,corrected 2, corrected 3, corrected

3

ZZ Z

R

   ,

which has combined standard uncertainty

22 2

1, corrected 2, corrected 3,corrected

1 () ( ) ( ) ( ) 9

uR u Z u Z u Z       ,

and effective degrees of freedom

 

 

123

4

R 11 1 44 44 44

33 3 1,corrected 2, corrected 3,corrected

ZZ Z

( ) df

() ( ) () ( ) () ( )

df df df

u R

uZ uZ uZ 

 

,

based on the Welch-Satterthwaite approximation. The reference value expanded 

uncertainty is

 2 R 1 , df ( ) U t uR R  D .

A.4 Estimating Machine Bias 

Assume the machine bias is the same for the new material ( Y b ) and the reference material 
27

Appendix A. Uncertainty Details 

( Vb ), so that Y V bbb   . The best estimate of the machine bias b is

 systematic ˆ ˆ bV R   G ,

which has combined standard uncertainty

 

2

2 2

systematic ˆ ˆ () ( ) ( ) V

V

S ub u u R n   G

 

.

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

4

b 2 2 4 4 systematic

V į R

ˆ ( ) df ˆ 1 () ( )

df df df

V

V

u b

S u u R

n

G 

§ ·  ̈  ̧  

© 1

.

Appendix B. Direct Verifi cation Components of Uncertainty

B.1 Anvils and Supports, A

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

uncertainty associated with anvil confi gurations. 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.1 lists the uncertainty estimates for 

low, high, and super-high energies.

Table B.1. 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 A df   , 

which implies that we know u A( ) exactly. 

B.2 Height of Pendulum Fall, h

The height of the pendulum fall is

h S  (1 cos ) E ,
28

Computing UnCertainty for Charpy Impact Machine Test Results

where S is the measured length of the pendulum, and E is the measured fall angle. 

Letting S and 'E denote the manufacturer’s stated error bounds, respectively, and 

assuming a rectangular distribution bounded by S and r'E , the uncertainties for S

and E are

( ) 3

S u S   and ( ) 3 u E E ' .

Then the uncertainty of h is

2 2

222

22 22

( ) ( ) ( ) 2 ( , ),

( ) ( ) 2 ( , ), S S

h h hh u h u S u uS

S S

c u S c u cc uS E E

E E E E

E E

§· §· w w ww §· §·

    ̈ ̧  ̈ ̧  ̈ ̧  ̈ ̧ ©1 ©1 w w ww ©1 ©1

 

where

1 cos S

h

c

S E w  w and (sin ). h c S E E E

w  w

If ( S, E ) are independent, then only the fi rst two terms are needed to determine the 

uncertainty. We can assume that S df   and ȕ df f , which implies that we know u S( )

and u( ) E exactly. (See the ISO-GUM, G.4.3 [2] for details.) The effective degrees of 

freedom associated with u h( ) are

4

h 4 4 4 4

S ȕ

( ) df

( ) ( )

df df

S

u h

cu S c u E E 



,

based on the Welch-Satterthwaite approximation.

B.3 Potential Energy, E

The potential energy is

E hF   ,
29

Appendix B 

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

position, and h is the height of the pendulum fall defi ned in Section B.2. Letting F

denote the manufacturer’s stated error bound of the measurement instrument, and 

assuming a rectangular distribution bounded by F , the uncertainty of F is

( ) 3

F u F   .

Then the uncertainty of E is

2 2

2 22

22 2 2

( ) () ( ) 2 (, )

( ) ( ) 2 ( , ), h F hF

E E EE u E u h u F uhF

h F hF

c u h c u F ccuhF

         

                  

 

where

h

E c F

h

   

 and F

E c h

F 

   .

The uncertainty associated with h is defi ned in Section B.2. If ( h F, ) are independent, 

then only the fi rst two terms are needed for the uncertainty. We can assume F df   , 

which implies that we know u F( ) exactly. The effective degrees of freedom associated 

with u E( ) are

4

E 4 4 4 4

h F

( ) df

( ) ( )

df df

h F

u E

cu h cu F 



,

based on the Welch-Satterthwaite approximation.

B.4 Impact Velocity, v

The impact velocity is v

Q  2 g h ,

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

defi ned in Section B.2. Letting g denote the manufacturer’s stated error bound of 

the measurement instrument (0.001 m/s2

 according to ASTM E 23), and assuming a 
30

Computing UnCertainty for Charpy Impact Machine Test Results

rectangular distribution bounded by g , the uncertainty for g is

( ) 3

g u g  .

The uncertainty of v is

2 2

222

22 22

() ( ) ()

( ) ( ), g h

v v uv ug uh g h

cu g cu h

               

 

where

2

2 g

v h

c

g g h

     and 2 . 2 hv g c

h g h

    

The uncertainty associated with h is defi ned in Section B.2. We can assume g df  

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

with u v( ) are

4

v 4 4 4 4

g h

( ) df

( ) ( )

df df

g h

u v

cu g cu h 



,

based on the Welch-Satterthwaite approximation.

B.5 Center of Percussion, L

The center of percussion is

 

2

2 4

g p L

S

 ,

where g is the local acceleration of gravity defi ned 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 p s , so the uncertainty of the mean period 

is
31

 Appendix B 

( ) 3

p s

u p 

,

with p df 3 1 2   degrees of freedom. 

The uncertainty of L is

2 2

2 22

22 22

( ) ( ) ( ),

( ) ( ), g p

L L uL ug up

g p

cu g cu p

   

       

 

where 

2

2 4 g

L p c

g S

w w and 2 . 2 p

L gp c

p S

w  w

The uncertainty associated with g is defi ned in Section B.4. From the WelchSatterthwaite approximation, the effective degrees of freedom associated with u L( ) are

 

4

L 44 44

g p

( ) df

() ()

df df

g p

u L

cu g cu p 



 

.

 

B.6 Friction Loss, D

The friction loss is

DE E  0 1 ,

where E0 is the potential energy due to the combined indicator and pendulum, and E1 is 

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

22 2

0 1 01 u D u E u E uE E ( ) ( ) ( ) 2( , )   .

Assuming perfect correlation between E0 and E1 , a conservative estimate of the 

covariance 0 1 uE E (,) is 

2 2

01 0 1 uE E u E u E (,) () ()   .
32

Computing UnCertainty for Charpy Impact Machine Test Results

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

0 1

4

D 4 4

0 1

E E

( ) df

( ) ( )

df df

u D

u E u E 



,

based on the Welch-Satterthwaite approximation.

B.7 Scale Accuracy, r

Let r represent the bias in the scale mechanism and ± r be the specifi ed error bounds of 

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

( ) 3

r u r  .

We will assume r df   , which implies that we know u r( ) exactly.
33

Appendix C. t-Table

The following t-table values were taken from NIST/SEMATECH e-Handbook of 

Statistical Methods [5].

Table C.1 Upper critical values of Student’s t distribution with degrees of freedom, df.

df 0.900 0.950 0.975 0.990 0.995 0.999

1 3.078 6.314 12.706 31.821 63.657 318.313

2 1.886 2.920 4.303 6.965 9.925 22.327

3 1.638 2.353 3.182 4.541 5.841 10.215

4 1.533 2.132 2.776 3.747 4.604 7.173

5 1.476 2.015 2.571 3.365 4.032 5.893

6 1.440 1.943 2.447 3.143 3.707 5.208

7 1.415 1.895 2.365 2.998 3.499 4.782

8 1.397 1.860 2.306 2.896 3.355 4.499

9 1.383 1.833 2.262 2.821 3.250 4.296

10 1.372 1.812 2.228 2.764 3.169 4.143

11 1.363 1.796 2.201 2.718 3.106 4.024

12 1.356 1.782 2.179 2.681 3.055 3.929

13 1.350 1.771 2.160 2.650 3.012 3.852

14 1.345 1.761 2.145 2.624 2.977 3.787

15 1.341 1.753 2.131 2.602 2.947 3.733

16 1.337 1.746 2.120 2.583 2.921 3.686

17 1.333 1.740 2.110 2.567 2.898 3.646

18 1.330 1.734 2.101 2.552 2.878 3.610

19 1.328 1.729 2.093 2.539 2.861 3.579

20 1.325 1.725 2.086 2.528 2.845 3.552

21 1.323 1.721 2.080 2.518 2.831 3.527

22 1.321 1.717 2.074 2.508 2.819 3.505

23 1.319 1.714 2.069 2.500 2.807 3.485

24 1.318 1.711 2.064 2.492 2.797 3.467

25 1.316 1.708 2.060 2.485 2.787 3.450

26 1.315 1.706 2.056 2.479 2.779 3.435

27 1.314 1.703 2.052 2.473 2.771 3.421

28 1.313 1.701 2.048 2.467 2.763 3.408

29 1.311 1.699 2.045 2.462 2.756 3.396

30 1.310 1.697 2.042 2.457 2.750 3.385

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

∞ 1.282 1.645 1.960 2.326 2.576 3.090

Appendix B 
34

Computing UnCertainty for Charpy Impact Machine Test Results

Appendix D. Glossary of Terms 

n Number of test material samples measured 

y Mean absorbed energy of test material samples 

s Standard deviation of test material samples 

df Degrees of freedom for test material standard deviation 

Vn Number of indirect verification samples measured 

V Mean absorbed energy of verification samples 

V S Standard deviation of verification samples 

dfV Degrees of freedom for verification material standard deviation 

R NIST reference value 

u(R) Standard uncertainty of NIST reference value 

dfR Degrees of freedom for reference value standard uncertainty 

systematic eˆ Systematic error estimate associated with test material 

(ˆ ) systematic u e Standard uncertainty of test material systematic error 

dfe Degrees of freedom for standard uncertainty of test material systematic error 

systematic

ˆ Systematic error estimate associated with verification material 

) ˆ ( systematic u  Standard uncertainty of verification material systematic error 

dfδ Degrees of freedom for standard uncertainty of verification material systematic 

error 

corrected y Corrected test result 

( ) corrected u y Combined standard uncertainty of corrected test result 

dfeff Degrees of freedom for combined standard uncertainty of corrected test result 

U Expanded uncertainty of corrected test result