NIST Special Publication 260-181r1

The ABCs of Using

Standard Reference Materials

in the Analysis of Foods and Dietary Supplements:

A Practical Guide

Katherine E. Sharpless

Katrice A. Lippa

David L. Duewer

Andrew L. Rukhin

This publication is available free of charge from:

http://dx.doi.org/10.6028/NIST.SP.260-181r1

100% Fat

0% Protein

0% Carbohydrate

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+

0% Carbohydrate

100% Protein

0% Fat

100% Carbohydrate

0% Fat

0% Protein

1

2 3 4

5 6 7 8 9

NIST Special Publication 260-181r1

The ABCs of Using

Standard Reference Materials

in the Analysis of Foods and Dietary Supplements:

A Practical Guide

Katherine E. Sharpless

Katrice A. Lippa

David L. Duewer

Chemical Sciences Division

Materials Measurement Laboratory

Andrew L. Rukhin

Statistical Engineering Division

Information Technology Laboratory

This publication is available free of charge from:

http://dx.doi.org/10.6028/NIST.SP.260-181r1

September 2015

U.S. Department of Commerce

Penny Pritzker, Secretary

National Institute of Standards and Technology

Willie May, Under Secretary of Commerce for Standards and Technology and Director
ii

National Institute of Standards and Technology Special Publication 260-181rev1

Natl. Inst. Stand. Technol. Spec. Publ. 260-181r1, 41 pages (Revised, September 2015)

CODEN: NSPUE2

This publication is available free of charge from:

http://dx.doi.org/10.6028/NIST.SP.260-181r1
iii

Abstract

Although ‘nutraceuticals’ and ‘functional foods’ seem to be ill-defined terms, both are often used 

to indicate a food that contains compounds providing benefits beyond basic nutrition, often with 

the expectation that these compounds are protective against chronic disease. The National 

Institute of Standards and Technology (NIST) has been producing food-matrix Standard 

Reference Materials (SRMs) since the mid-1970s. Early materials were characterized solely for 

elements. Values were assigned for organic constituents of food-matrix SRMs beginning in 

1996 and in dietary supplements beginning in 2006. Although none of the NIST food or dietary 

supplement SRMs were categorized as functional foods per se, many have values assigned for 

components that put the ‘functional’ in functional foods – e.g., antioxidants, phytonutrients, 

minerals, vitamins, etc. Recommendations for use of these and other natural-matrix SRMs as 

quality assurance tools are discussed in this paper: from selecting an appropriate material to 

validating analytical methods, characterizing in-house quality control materials, and establishing 

traceability.

Keywords

Food, Dietary Supplement

Certified Reference Material (CRM), Standard Reference Material (SRM)

Quality Control (QC), Metrological Traceability

Disclaimer: Certain commercial products are identified in this paper in order to specify the 

experimental procedure 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 materials or equipment identified are necessarily the best available for the purpose.
iv

Table of Contents

Abstract..........................................................................................................................................iii

Keywords.......................................................................................................................................iii

Table of Contents........................................................................................................................... iv

Tables............................................................................................................................................. iv

Figures............................................................................................................................................ iv

Introduction..................................................................................................................................... 1

Step A. Selection of SRMs ............................................................................................................ 3

Step B. Use of SRMs for Establishing Metrological Traceability................................................. 8

Step C. Use of SRMs for Precision Assessment............................................................................ 9

Step D. Use of SRMs to Establish Trueness of Results............................................................... 12

D.1 Comparing your results to an SRM.................................................................................... 13

D.2 What n is needed? (Estimate from literature or your own historical data)....................... 15

D.3 What n is needed? (Estimation from SRM data)................................................................ 16

D.4 Quantitative estimation of trueness (bias) ......................................................................... 18

D.5 The perils of checking trueness with two or more validation materials............................ 18

Step E. Use of an SRM for Characterization of an In-House Quality Control Material ............. 19

Step F. Use of an SRM in Value Assigning a Secondary Reference Material ............................ 21

Conclusion .................................................................................................................................... 22

References..................................................................................................................................... 23

Appendix I: Acronyms and Symbols........................................................................................... 25

Appendix II: Statistics and Examples.......................................................................................... 27

Appendix III: Installation of R and metRology ........................................................................... 34

Appendix IV: Data for the Examples........................................................................................... 38

Tables

Table A.1. Organic measurands of interest in SRMs in sector 6 of the AOAC food triangle....... 5

Table A.2. Elements of interest in SRMs in sectors 2, 5, 6, and 7 of the AOAC food triangle. ... 6

Table A.3. SRMs with values assigned for catechins.................................................................... 7

Table C.1. Catechin monomer validation “data.” ........................................................................ 10

Table C.2. One-way ANOVA for catechin monomer “data.” ..................................................... 10

Figures

Figure 1. Uses of an SRM.............................................................................................................. 2

Figure A.1. Location of SRMs in AOAC Food Triangle .............................................................. 4

Figure D.1. Graphical Evaluation of Method Trueness............................................................... 13

Figure D.2. Number of results needed to confidently assess trueness......................................... 17

Figure E.1. Comparison of Measurement Results for the SRM and Candidate QC Material ..... 19
1

Introduction

Various labeling requirements are imposed on food and dietary supplement manufacturers [1,2], 

and regulatory agencies and consumers expect product labels to be accurate. In addition to 

providing accurate product labels, dietary supplement manufacturers are required to follow 

current good manufacturing practices (cGMPs) [3], including setting product specifications and 

ensuring that analytical methods are appropriate for assessing whether or not these specifications 

are met. If a particular material is studied in a clinical trial, investigators must demonstrate that 

constituents in the test material are the same among batches. Certified reference materials 

(CRMs) can be used to address these requirements.

An analytical chemist’s primary concern in the laboratory is whether or not his/her results are 

consistently correct. One way to determine whether an analytical method is working properly 

and generating accurate results is through use of suitable CRMs. CRMs come in two flavors:

those intended for use in calibrating measurement systems and those primarily intended to help 

validate the performance of measurement systems. Figure 1 shows how these two types of 

materials fit within the complete measurement process, connecting your measurements to the 

International System of Units (SI).

A calibration CRM is typically either a “neat” material of established high purity or a simple

solution gravimetrically prepared from a material of established high purity. Such materials are 

used to establish the relationship between the quantity of a measurand and the results from a 

measurement process. (A “measurand” is the “quantity intended to be measured” [4]; for our 

purposes this refers to the quantity of a given analyte in a particular sample matrix.)

Validation/verification materials are typically preparations of naturally occurring materials of a 

comparable matrix as the unknown samples that a) contain desired levels of the measurand(s) of 

interest, b) have been processed to have uniform composition, and c) may have been modified 

for improved stability. When developing an analytical method, these materials are used to 

evaluate whether the calibration relationship established for a measurement process is valid when 

analyzing a particular sample matrix. When implementing a method in a given situation 

(particular equipment, reagents, analysts, etc.), these materials are used to verify that the 

developed measurement process indeed provides the expected results.

What your measuring system actually measures may differ from what you intend to measure. 

However, if the result of your measurement on a suitable natural-matrix CRM agrees with its

certified value, it is likely that your measurement process has been suitably characterized and is 

capable of producing correct results for unknown samples and, ultimately, that the composition 

of a product can be specified and labeled accurately.

Standard Reference Materials® (SRMs®) are CRMs characterized by the U.S. National Institute 

of Standards and Technology (NIST). In this paper, we describe the steps that you would take to 

effectively use SRMs: selecting appropriate materials, establishing traceability, validating and 

verifying analytical methods, and characterizing in-house quality control materials. While our 

focus is on the use of our SRMs, the steps are similar for all CRMs regardless of the source [5,6]. 

Certificates of Analysis on NIST’s website [7] (and the websites of other CRM manufacturers 
2

and distributors) as well as other publications provide the information that you will need to 

evaluate and select an appropriate material [8,9,10].

Figure 1. Uses of an SRM

The boxes to the left and right in Figure 1 represent materials, those in the center represent 

activities, and the arrows indicate the flow of the complete measurement process from the SI 

reference system to the desired measurement value. The boxes and lines in black denote the 

materials and activities provided by NIST and other CRM providers. The boxes and lines in blue 

denote the materials and activities that you must provide.
3

Step A. Selection of SRMs

Choose an SRM with a matrix that is similar to that of your test samples: any matrix effects or 

sample preparation issues encountered in one sample should be encountered in the other. For 

example, if your unknown samples contain a high level of fat, a high-fat SRM would likely be 

more appropriate for quality assurance than would a high-protein SRM. However, along with a 

matrix match, measurand quantities should also be matched. Interferences and blank issues may 

arise if the quantities in the different materials are quite different. Also, results for test samples 

and the SRM should lie within the same general range on the calibration curve. Often the 

Official Methods of Analysis of AOAC International (and standard test methods developed by 

other standards organizations) will specify an analyte range when defining the scope of the 

method, and this range is also an important consideration.

Next, check that the uncertainty of the SRM's assigned value(s) is fit for your intended purpose 

[11]. The assigned uncertainty should be small relative to the total uncertainty targeted for your 

test samples. As a rule of thumb, the SRM's assigned uncertainties should be about one-third or

less of the target uncertainty to ensure that uncertainty in the certified value will have negligible 

influence on the results of your measurements.

To determine the appropriate SRM for analysis with food samples, the AOAC International 

triangle can be used (Figure A.1). This triangle was developed by AOAC in an effort to promote 

analytical methods to address nutrition labeling requirements. If a method provides accurate 

results for one or two foods within a sector, the method is expected to provide accurate results 

for other foods in that sector. NIST extended this model, expecting that one or two SRMs in a 

sector should be useful as quality assurance tools when testing other foods within that sector. If 

the fat, protein, and carbohydrate content of the test sample are not known, approximations can 

be obtained from the U.S. Department of Agriculture (USDA) nutrient database [12] and used to 

estimate the test sample’s position in the triangle. Once an appropriate natural-matrix SRM is 

identified, the expected measurand content should be compared to its certified value.

For dietary supplements, where there is not a fat, protein, and carbohydrate composition to 

consider, the analyst might look at food-matrix SRMs and also at botanical and agricultural 

materials, environmental matrices (e.g., for measurement of contaminants), and geological 

materials (e.g., for measurement of elements in solid oral dosage forms) to see whether one of 

those matrices might have values assigned for the measurand of interest. Again, the optimal 

SRM will be of a “similar” matrix with a “similar” measurand quantity, but the user must decide 

what is similar enough based on available knowledge about the materials and on his/her own 

analytical methods.

As an example, we will consider the analysis of an imaginary product, a 50 gram dark chocolate 

protein bar. This protein bar contains mass fractions of 15 % protein, 15 % fat, and 50 % 

carbohydrate, which puts it in sector 6 of the AOAC triangle. An additional consideration is that 

our product is fortified with fat- and water-soluble vitamins and several minerals. Fortification 

often results in a single form of the vitamin that largely swamps out any naturally occurring 

forms. Therefore, it would also be best to have the same forms in the SRM as in our material. 

Again, the compositional information needed to evaluate the suitability of an SRM is available 
4

on the material’s Certificate of Analysis on the NIST website [7]; other CRM providers make 

available similar information in their own websites and catalogs.

Figure A.1. Location of SRMs in AOAC Food Triangle

The measurands currently required on nutrition labels for processed foods sold in the U.S. 

include the organic constituents cholesterol and vitamins A (retinol) and C (ascorbic acid), as 

well as fat, protein, carbohydrate, dietary fiber, and sugar [13]. Since measurement processes for 

some organics are relatively sensitive to the sample matrix, we look first at the SRMs that are in 

sector 6 (Table A.1). SRM 1849a Infant/Adult Nutritional Formula appears to be a good choice 

for use as a quality control material for vitamins and proximates because it 1) has assigned 

values for the organic measurands of greatest interest and (2) is fortified with the same forms of 

vitamins as those found in our imaginary protein bar.

SRM 1548a

SRM 1549a 

SRM 1845a SRM 2384

SRM 2387

SRM 1849a 

SRM 1946

SRM 1947

SRM 3252 

SRM 2383a

SRM 1566b 

SRM 2385 

SRM 3287 

SRM 1546 

SRM 3234 

100 % fat

0 % protein

0 % carbohydrates

0 % fat

100 % protein

0 % carbohydrate

0 % fat

0 % protein

100 % carbohydrate

SRM 3274 

SRM 3290

SRM 1544 

SRM 3233 

0 % fat

100 % protein

0 % carbohydrate

0 % fat

0 % protein

100 % carbohydrate

SRM 3274 

SRM 3290

SRM 1974c

1

5 6 7 8 9

2 3 4

IMAGINARY PRODUCT
5

Table A.1. Organic measurands of interest in SRMs in sector 6 of the AOAC food triangle.

Material

Cholesterol,mg/kgRetinol,mg/kgAscorbic Acid,mg/kgFat, %Protein,%Carbohydrates,%Sugar,%

Anticipated mass fraction in the

imaginary 50 g protein bar low ≈3 ≈300 15 15 50 ?

SRM 1544 Fatty Acids and Cholesterol 

in a Frozen Diet Composite

148.3

± 9.4

not 

assigned

not 

assigned

3.7

± 0.6

5.3

± 0.3

16.9

± 1.5

not 

assigned

SRM 1548a Typical Diet not 

assigned

not 

assigned

not 

assigned

19.41

± 1.45

18.08

± 0.42

58.36

± 1.53

not 

assigned

SRM 1549a Whole Milk Powder 981

± 71

not 

assigned

41.9

±2.5

26.98

± 0.66

25.64

± 0.31

38.43

± 0.95

not 

assigned

SRM 1849a Infant/Adult Nutritional 

Formula

137.4

± 2.9

7.68

± 0.32

784

±65

30.4

± 0.95

13.72

± 0.92

51.6

± 1.3

47.6

± 5.5

U.S. nutrition regulations also require labeling the contents of sodium, calcium, and iron [13]. 

You expect the protein bar to contain sodium, calcium, and iron mass fractions of no more than 

2500 mg/kg, 300 mg/kg, and 15 mg/kg, respectively. You also want to ensure that your product

is not contaminated with lead. While we first look at the SRMs in sector 6, measurement 

processes for the total quantity of elements can be somewhat less sensitive to the sample matrix

than those for organic measurands, so we also look at materials in neighboring sectors 

(Table A.2).

While it’s best to have a close match between analyte levels in the validation SRMs and the 

target levels in the protein bar, SRMs with levels that are within an order-of-magnitude (between 

10-fold smaller to 10-fold larger) of the targets are likely be acceptable as long as they are within 

the scope of your analytical method. Levels that are fairly close to the levels that you anticipate 

in your protein bar are highlighted in bold red; levels that are within an order of magnitude are 

identified by footnote “a”. The material you identified as the organic control material, SRM 

1849a, is a reasonable match for the desired sodium content, although SRM 1549a Whole Milk 

Powder and SRM 1566b Oyster Tissue are better. However, SRM 1849a has calcium and iron 

contents that are much greater than the targets and does not have an assigned value for lead. 

While you can dilute your digests of SRM 1849a to measure calcium and iron, this would also 

dilute sodium and other matrix elements that might affect the determination of calcium and iron. 

You decide instead that your quality assurance program will need to use several different 

materials.
6

None of the sector 6 materials is a good match for calcium. The best matches are SRM 2383a 

Baby Food Composite, SRM 3287 Blueberry (Fruit), and possibly SRM 2385 Slurried Spinach. 

While the sector 6 SRM 1548a Typical Diet would be a reasonable choice for iron (and maybe 

also sodium, depending on the scope of the method and the range of your calibration curve), both 

SRM 3287 and SRM 2385 are better matches. To control costs, you wish to minimize the 

number of SRMs needed. Exploring other information available through the website, you find 

that SRM 2383a requires refrigeration while SRM 3287 delivers both calcium and iron, can be 

stored at room temperature (five packets of ≈ 5 g each). While SRMs 2385 is an acceptable 

choice for calcium and is perhaps the best choice for iron, (four jars of ≈ 70 g each) and requires 

refrigeration. On balance, you decide that SRM 3287 is the material of choice for calcium and 

iron, but would not be appropriate for sodium or lead.

Table A.2. Elements of interest in SRMs in sectors 2, 5, 6, and 7 of the AOAC food triangle.

Sector Material

Sodium,

mg/kg

Calcium,

mg/kg

Iron,

mg/kg

Lead,

mg/kg

6

Anticipated mass fraction in the

imaginary 50 g protein bar <2500 300 15 none

6

SRM 1544 Fatty Acids and Cholesterol in 

a Frozen Diet Composite

not

assigned

not

assigned

not

assigned

not

assigned

6 SRM 1548a Typical Diet 8132a

± 942

1967a

± 113

35.3

± 3.77

0.044

± 0.009

6 SRM 1549a Whole Milk Powder 3176

± 58

8810

± 240

1.8a

± 0.7

not

assigned

6

SRM 1849a Infant/Adult Nutritional 

Formula

4265

± 83

5253

± 51

175.6

± 2.9

not

assigned

2 SRM 2384 Baking Chocolate 40

±2

840a

±74

132a

±11

not

assigned

5 SRM 2383a Baby Food Composite 195

±29

324.6

±5.0

4.42a

±0.51

not

assigned

5 SRM 3233 Fortified Breakfast Cereal 6830a

±120

36910

±920

766

±36

not

assigned

5 SRM 3287 Blueberry (Fruit) 16.39

±0.74

323

±16

12.20

±0.74

not

assigned

7 SRM 1566b Oyster Tissue 3297

±53

838a

±20

205.8

±6.8

0.308

±0.009

7 SRM 2385 Slurried Spinach 47

±1

624

±40

17.1

±1.9

not

assigned

7 SRM 3234 Soy Flour 2.52

±0.45

3191

±56

80.3a

±2.7

not

assigned

a Values within an order of magnitude of the target analyte levels.
7

SRM 1548a is the best available match to the low lead content you require for your product, and 

may be good enough for sodium. SRM 1566b has a higher lead level than you’d want to find in 

your protein bar, but you could dilute it validate your calibration at several different levels 

including the very low level you anticipate. Given that the oyster tissue has a reasonable amount 

of sodium in it, you decide to buy SRM 1566b for your sodium and lead control.

And finally, you want to measure catechins present in the chocolate in your product. You’ve 

developed a new method and want to make sure that it works well. You search the NIST SRM 

website by the keyword "catechin" and discover five SRMs: a baking chocolate; a threecomponent set of catechin calibration solutions; and green tea as leaves, extract, and solid oral 

dosage forms (Table A.3). Because you want to make sure that your method extracts the 

catechins adequately from a matrix, you eliminate the calibration solution from consideration as 

a control material; but you do decide to select SRM 3257 to calibrate your method. (Note that if 

you did decide to use SRM 3257 to make sure your method was working properly, you could not 

also calibrate with it.) Since you’re interested specifically in catechins from the chocolate in 

your protein bar, the baking chocolate SRM is the logical choice, but it doesn’t hurt to look at the 

green tea materials as well. The levels in the green tea are much higher than those that you 

would expect in your product, where chocolate is merely an ingredient and the catechins in it 

will be diluted by other ingredients. So you choose SRM 2384 Baking Chocolate for method 

validation.

Table A.3. SRMs with values assigned for catechins.

Material

Catechin,

mg/kg

Epicatechin

mg/kg

Catechin 

monomers

mg/kg

SRM 2384 Baking Chocolate 245 ±51 1220 ±240 1490 ±220

SRM 3254 Camellia sinensis (Green Tea) Leaves 1010 ±410 9000 ±1600 10000±1700a

SRM 3255 Camellia sinensis (Green Tea) Extract 9170 ±930 47300 ±6700 56500±6800a

SRM 3256 Green Tea-Containing

Solid Oral Dosage Form 2630 ±180 12000 ±2600 14600±2600a

SRM 3257 Catechin Calibration Solutions 23.54 ±0.53 93.9 ±2.1 117.4±2.2a

a Calculated from the certified values
8

Step B. Use of SRMs for Establishing Metrological Traceability

Metrological traceability is the “property of a measurement result whereby the result can be 

related to a reference through a documented unbroken chain of calibrations, each contributing to 

the measurement uncertainty” [4]. Metrological traceability is different from chain-of-custody 

traceability: metrological traceability follows a series of measurement comparisons rather than a 

material passing through a series of possessions or a supply chain. In the case of your protein 

bar measurements, the traceability chain extends from the SI through the SRMs used to calibrate 

your measurement systems to the measurements you’ve made on the protein bar. Since every 

link in the measurement chain contributes uncertainty, your confidence in your protein bar 

measurement results will be maximized (that is, the measurement uncertainty will be minimized) 

by keeping the length of the chain as short as possible.

The most direct method of establishing metrological traceability to the SI is through the 

calibration of your measurement system with a certified calibration material, such as SRM 3128 

Lead (Pb) Standard Solution or SRM 3257 Catechin Calibration Solutions, followed by 

validation of your measurement process with the appropriate natural-matrix SRM. The 

calibration solutions have been prepared from materials of established purity using appropriately 

calibrated balances and instruments under controlled environmental conditions. Neat materials 

of certified purity, such as SRM 911c Cholesterol, may be available in addition to or instead of 

pre-made solutions. In either case, assuming that you a) use suitably calibrated equipment under 

suitably controlled conditions to prepare whatever working standards are needed to calibrate 

your measurement systems, b) suitably document your use of those procedures, and c) correctly 

calculate the uncertainties, the measurements made with those systems can be made traceable to 

the certified value of the calibrant and through it to the SI units of mass (kilogram) or mole and, 

if required, volume (cubic meter).

When neither certified calibration solutions nor neat materials are available, traceability can be 

established by calibrating with a natural-matrix material. However, by definition, the traceability 

chain for natural-matrix materials has at least one more link than that of a calibration material 

and so will not provide the smallest possible measurement uncertainty. In the case of catechin, 

the relative uncertainty in the certified value in the baking chocolate (≈ 21 %) is almost an order 

of magnitude greater than that in the calibration solution (≈ 2.3 %). Further, calibrating with a 

natural-matrix material may bias your results unless your measurement process is very similar to 

the processes used to certify the SRM. Should you calibrate to SRM 2384 Baking Chocolate you 

are assuming that extracting catechins from the protein bar and from the baking chocolate have 

similar efficiencies and that the chromatography of the extracts has similar selectivity.

You can establish direct traceability to SI units by evaluating the purity of a non-certified neat 

material yourself. This can be an expensive and time-consuming process [14], but it is routinely 

accomplished by sophisticated analytical laboratories. Traceability would then be linked to the 

SI through the procedures you used to establish purity in addition to those used to prepare the 

working solutions.

For additional discussion of traceability, see: [15,16].
9

Step C. Use of SRMs for Precision Assessment

Depending on how you design your measurement program, repeated measurements of an analyte 

(call it measurand X) can be used to characterize different sources of variability (imprecision) in 

your analytical method for X. These measurements do not have to be made on SRMs but rather 

on any homogenous, stable material available in sufficient quantity having an appropriate matrix 

and analyte content.

Multiple analyses of a single preparation of a sample reflect the imprecision of the post-sample 

preparation aspects of your method. The post-preparation standard deviation, spost, is readily 

estimated from the generic formula 

s(x) = √∑

(xi−x̅)2

n−1

n

i=1

x̅= ∑

xi

n

n

i=1

where s(x) is the estimated standard deviation of the measurements of X, each xi is a measured 

value, n is the number of such values, and 

x

is the mean (also known as the arithmetic average)

of the values. Note: Appendix I lists and defines all of the symbols used in this document.

When the xi are merely repeated measurements of a single preparation of the sample made by 

one analyst over a short period of time, spost = s(x). Since the precision uncertainty components 

are combined in quadrature (i.e., the square root of the sum of the squares), then as long as spost is 

less than about 0.3 times the total target imprecision (i.e., standard deviation), your postpreparation measurement process should contribute less than about 10 % of your targeted

uncertainty. If the factor is more than about 0.3, you should try to improve the instrumental 

analysis aspects of your method – including confirming that the method’s detection limit is 

adequate for your purpose.

If you analyze independently prepared subsamples of a material over a short period of time, you 

can estimate your method’s repeatability imprecision. The repeatability standard deviation, sr,

can be estimated as above when the xi represent single measurements of n independently 

prepared subsamples. As above, if sr is less than about 0.5 times the targeted imprecision, the 

combined sample preparation and post-preparation processes should contribute less than about 

30 % of your targeted uncertainty. If the factor is more than about 0.5, you should try to improve 

the sample preparation aspects of your method.

If you and several other analysts analyze independently prepared subsamples or you analyze 

your independently prepared subsamples over a sufficiently long period of time, you can 

estimate some form of what is termed “intermediate” imprecision, sI. If the variability 

introduced by combining data from multiple analysts and/or collecting data over a long period of 

time is not within the target uncertainty, the method must be made more robust to small changes, 

the contributing influence factors that give rise to the variability must be identified and 

controlled, and/or the analysts must be provided with additional training. However, it’s best to 

estimate sI using one-factor analysis of variance (ANOVA) – described in most statistics texts 

and guides to laboratory statistics: see, for example, [17,18,19,20]. While the calculations are 
10

fairly simple, they are much more efficiently (and reliably) performed using appropriate 

software.

Table C.1 lists (dry-lab) “results” for total catechin monomers in SRM 2384 of measurements 

performed by five analysts, along with basic summary statistics for each analyst. Let’s say that 

all five groups of data were obtained using nominally the same analytical process, equipment, 

and supplies over roughly the same time period and that each value is the result for an 

independently prepared subsample. Table C.2 shows typical summary results from a one-factor 

ANOVA of these data. For our current purpose, only the number of measurements per group, nj, 

and the within- and between-groups mean-squares, MSwth and MSbtw, are of interest.

Table C.1. Catechin monomer validation “data.”

Mass fractions in mg/g.

Analyst

Repeat KS DD BN KL AR

1 1.362 1.494 1.389 1.460 1.357

2 1.388 1.504 1.415 1.530 1.453

3 1.392 1.647 1.424 1.560 1.510

4 1.412 1.650 1.441 1.580 1.573

5 1.415 1.692 1.455

6 1.419 1.698 1.472

7 1.426 1.741 1.482

8 1.458 1.770 1.483

9 1.467 1.519

10 1.511 1.588

n 10 8 10 4 4

Mean 1.425 1.649 1.467 1.533 1.473

s 0.044 0.102 0.057 0.053 0.092

Table C.2. One-way ANOVA for catechin monomer “data.”

Mass fractions in mg/g

One-Factor ANOVA

Group nj Sum Mean Variance Component SumSq df MS F P-value

KS 10 14.252 1.425 0.001896 Between 0.2522 4 0.0631 12.85 2.8E-06

DD 8 13.195 1.649 0.010329 Within 0.1522 31 0.0049

BN 10 14.668 1.467 0.003258 Total 0.4044 35

KL 4 6.130 1.533 0.002758

AR 4 5.893 1.473 0.008408

The repeatability standard deviation is just the square-root of MSwth

sr = √MSwth = √0.0049 = 0.070 mg/g.
11

The between-analyst and/or between-chunk-of-chocolate intermediate imprecision is

sI = √

MSbtw−MSwth

n′

n

′ =

N2−∑ nj

m 2

j

(m−1)N

N = ∑ nj

m

j

where n' is the effective number of independent measurements per group, m is the number of 

groups, and N is the total number of measurements. If all analysts had made the same number of 

measurements (a “balanced design”) then n' would just equal that number. In this example (an 

“unbalanced design” more typical of routinely collected data)

n

′ =

362−296

(5−1)36

≅ 6.94

and so

sI ≅ √

0.0631−0.0049

6.94

= 0.092 mg/g.

With the data at hand, you can’t tell whether differences among the analysts or heterogeneity in 

the SRM unit is the major source of sI imprecision, but this could be addressed by having 

analysts KS and DD analyze subsamples of the same chunk of the sample material. (You know 

that KS and DD are the analysts who should make additional measurements because of the five 

analysts the mean of KS's measurements is the smallest while the mean of DD's measurements is 

the largest.) Regardless, the expected combined standard deviation for the measurement process 

under the stated conditions of measurement combines the repeatability and intermediate 

components

s(x) = √sr

2 + sI

2 = √0.0702 + 0.0922 ≅ 0.12 mg/g.

If the within-group mean-square is greater than the between-group, MSwth > MSbtw, then there is 

no significant intermediate imprecision and s(x) can be taken to be sr.

Note: you can characterize more than one source of additional variability in a single data set 

given enough (good) data (e.g., several analysts each reporting data for the same control 

materials using a number of different extraction protocols and/or instruments multiple times a 

week for many weeks) and appropriate data analysis tools. However, the design and evaluation

of such studies is well beyond the scope of this document. Consult with your favorite statistical 

consultant for further information.
12

Step D. Use of SRMs to Establish Trueness of Results

This section contains a lot of detailed explanation on interpreting your results relative to the 

certified value because these are the types of questions that we are most frequently asked when a 

customer’s results don’t seem to “agree” with the assigned value.

Different CRM suppliers express certified values and their uncertainties somewhat differently. 

For NIST natural-matrix SRMs characterized for chemical composition, the certified value of 

measurand X in the sample matrix is typically stated as xNIST ± U95(xNIST), where U95(xNIST) is 

an expanded uncertainty such that the interval from xNIST − U95(xNIST) to xNIST + U95(xNIST) is 

expected to contain the true value of X in all units of the SRM with about a 95 % level of 

confidence. Note that this statement does not guarantee that the true value of X in your unit of 

the SRM is xNIST, only that its true value is expected with high confidence to be somewhere 

within the certified interval. Conversely, there is a small probability (about 5 %) that the true 

value for your unit is somewhere outside the range.

While NIST’s natural-matrix SRMs are intended to be homogeneous and relatively stable, the 

vast majority of these SRMs are batch-certified, and some degree of within-unit and betweenunit heterogeneity is inevitable. The Certificate of Analysis for these SRMs will indicate the 

minimum sample mass that you should use for your analyses. This information is generally 

under the “Instructions for Use” section of the Certificate and is stated either as: a) the minimum 

sample mass for the certified value to be valid or b) the sample size at which homogeneity was 

assessed. The certified intervals take into account measurand heterogeneity at this sample size as 

well as the imprecision and between-method bias (systematic differences or “lack of trueness”) 

of the measurement processes used to assign the certified value.

If you could make very many (say, 30 or more) measurements of independently prepared 

subsamples of a particular unit of a given SRM over a long period of time using an unbiased 

analytical method, then the mean of all your measurements, x̅lab, should be within the xNIST ±

U95(xNIST) interval. However, it is not practical for any SRM user to make such a large number 

of measurements: laboratory resources and the amount of material in each unit of the SRM are 

both limited. You are more likely to make a few measurements (say, three to five) and base your 

assessment of your method’s bias on those. But there is a tension between the two possible 

conclusions: you don't want to conclude that a method is unbiased when it is in truth biased nor 

conclude that it is biased when in truth it is unbiased. Unfortunately, for any given number of 

measurements there is no way to minimize both risks at the same time; it is up to you to decide 

the relative importance of the two risks [19].

And always keep in mind that natural-matrix CRMs generally provide only “yellow light” 

(cautionary) validation of the fitness-for-purpose of your measurement procedure for real 

samples. While failure to agree with the certified value is a “red light” signal that your 

measurement procedure may not be fit for any similar sample, agreement is not a “green light” 

that all is well for every sample that your laboratory may need to analyze but only that it is fit for

samples having analyte content and matrix similar to the SRM. For such materials, proceed, but 

with due caution.
13

D.1 Comparing your results to an SRM.

The comparison of your results to a certified value is generally known as “compliance 

assessment” but is more formally the determination of the “metrological compatibility of 

measurement results” [4]. The various aspects of assessing the compliance of chemical 

measurement procedures as well as practical assessment tools are discussed in the 

Eurachem/CITAC Guide "Use of uncertainty information in compliance assessment" [21] and 

the fundamental statistical limitations of these approaches have recently been discussed [22]. 

Appendix II outlines the philosophy for formal statistical assessment of compliance and provides 

detailed instructions for using sophisticated free computer programs for evaluating the examples, 

and Appendix III describes how to install and use the freeware software required to run the 

programs.

However, it is often possible to assess whether a measurement procedure is "true enough" using 

very simple data analysis methods. In our protein bar example, say you’re measuring total 

catechin monomers (the sum of catechin and epicatechin). If your analytical method gives you 

the “right” answer for catechins in SRM 2384 Baking Chocolate, you will have confidence in 

your ability to reliably measure catechins in your product samples. Figure D.1 displays (dry lab) 

“results” for seven measurement processes, each set of results consisting of three independent 

measurements of total catechin monomers in single units of SRM 2384. The individual data and 

the summary statistics for the seven methods are provided in Appendix IV. The results are 

labeled “A” through “G.” Your results are likely to be similar to one of these examples.

Figure D.1. Graphical Evaluation of Method Trueness

A B C D E F G

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Catechin Monomers, mg/g

Method
14

The horizontal dashed lines in Figure D.1 bound the NIST-certified xNIST ± U95(xNIST) interval 

of 1.49 mg/g ± 0.22 mg/g. The open circles represent individual independent measurement 

results, xlab,i. The solid circles represent the arithmetic mean of your measurements,

x̅lab = (∑ xlab,i

n

i=1

)⁄n,

where n is the number of your measurements. The thick vertical lines through those circles 

represent approximate 95 % level of confidence intervals about your mean

U95(x̅lab) = k ∙ s(x̅lab)

s(x̅lab) = s(x)⁄√n

s(x) = √∑

(xlab,i −̅xlab)

2

(n−1)

n

i=1

where s(x̅lab) is the standard deviation of the mean (also known as the standard error of the 

mean), s(x) is the standard deviation of your measurements, and k is an appropriate expansion

factor. When the number of independent measurements is not large, say less than 10, then k

should be estimated using Student’s t with the appropriate number of degrees of freedom. When 

the number of measurements is large or you have confidence in the estimate based on other 

evidence, it is typically asserted that you have “large” effective degrees of freedom and the value 

for k is 2 [23]. For convenience, Figure D.1 uses k = 2. The thin red vertical lines to the 

immediate right of the solid circles for methods E through G represent the approximate 95 % 

level of confidence intervals after a number of additional measurements have been made.

Without any further analysis, it can be safely concluded that method A is unbiased and that 

method B is biased. The independent measurement results within each of these methods are very 

self-consistent, with a U95(x̅lab) of 0.087 mg/g that is much less than the 0.22 mg/g of 

U95(xNIST). For method A, the entire x̅lab ± U95(x̅lab) interval is well inside the 

xNIST ± U95(xNIST) certified interval, indicating that results from this method are reliably 

unbiased. For method B, the entire x̅you ± U95(x̅lab) interval is well outside the certified 

interval, indicating that results from this method, while self-consistent, are consistently biased. 

Additional measurements would be unlikely to change these conclusions.

Methods C and D also produce reliably consistent results and have U95(x̅lab) of 0.087 mg/g. The 

x̅lab of the method C results is just at the upper edge of the certified interval; assuming the results 

are roughly normally distributed, there is at least a 50 % probability that the “very many ... long 

term” mean is actually within the certified interval and the method can be considered unbiased. 

For method D, it is the lower limit, x̅lab − U95(x̅lab) that is aligned with the upper edge of the 

certified interval and thus there is only about a 2.5 % chance that more measurements would 

yield an x̅lab that is within the certified interval. Therefore it is rather likely that method D 

provides results that are somewhat biased (although whether that bias is significant depends on 

your needs.)

But by NIST's definition of the certified interval, the true value of X in about 2.5 % of the SRM 

units may be equal to or a bit greater than the upper dotted line. If in this situation, a rigorous 

bias evaluation is required and more measurements are to be made, they would best be made on a 

different bar of the chocolate SRM to better identify whether the slight bias reflects the intrinsic 
15

characteristics of the measurement processes or the true value of X in the particular bar that was 

analyzed. (Many food and dietary supplement SRMs contain several packets or bottles per SRM 

unit; analyzing another packet does not necessarily mean that you must buy another unit of the 

SRM.)

In contrast to methods A through D, the relatively large intervals for methods E, F, and G are 

attributable to widely dispersed individual results. Of course, a pragmatic first step would be to 

improve the measurement process to reduce the overall variability. However, for this illustrative 

case, more measurements on the same SRM unit will improve the confidence in the trueness 

assessment for methods E, F, and G. The x̅lab ± U95(x̅lab) intervals for all three methods 

include all of the xNIST ± U95(xNIST) interval. Thus the available evidence is that the methods 

could be unbiased; however, the intervals are too large to assert that conclusion with much 

confidence. As identified in [22], a meaningful bias assessment requires that the 

x̅lab ± U95(x̅lab) interval shouldn't be much wider than the xNIST ± U95(xNIST) interval. How 

can that be achieved?

Assuming that the results for a measurement procedure are more or less symmetrically 

distributed around the true value of the quantity of X in your packet of the SRM, as indicated by 

the thin red lines in Figure D.1, your knowledge of this true value will improve as additional 

independent measurements are made. The rate of improvement is proportional to the square root 

of the number of measurements, √n. The next sections discuss how to estimate the minimum 

number of measurements needed.

D.2 What n is needed? (Estimate from literature or your own historical data).

If you have a reliable estimate of the expected measurement standard deviation, s(x), for given 

amounts of analyte, then estimating the n needed to get an appropriately small U95(x̅lab) is fairly 

straightforward. You can estimate generic values for s(x) from historical data on replicate 

analyses of unknowns and/or quality control materials [24]. If your measurement method is the 

same or very similar to one characterized in the literature, then a usable estimate of s(x) may be 

available from robustness evaluations or interlaboratory reproducibility studies [25]. Standards 

Development Organizations such as AOAC International generate such estimates in the process 

of evaluating their Official Methods of Analysis. These estimates are often given in the form of 

relative standard deviations expressed as percent, CV = 100 ∙ s(x)⁄x̅, and would then need to be 

converted back into the standard deviations for each particular x: s(x) = x̅∙ CV/100 .

Using the method C example, assume that about half of the x̅lab ± U95(x̅lab) interval must fit

within xNIST ± U95(xNIST); that is U95(x̅lab)⁄2 ≤ U95(xNIST). Since we are assuming here that 

U95(x̅lab) has been estimated from a confident estimate of s(x̅lab),

s(x̅lab) = U95(x̅lab)⁄2

and so

s(x̅lab) ≤ U95(xNIST).

Given s(x̅lab) = s(x)⁄√n, then

√n = s(x) s(x̅lab ⁄ ) ≈ s(x) U95(xNIST ⁄ )

and thus

n ≈ (s(x) U95(xNIST ⁄ ))

2

.
16

The s(x) for methods E and F is 1.22 mg/g; assuming that this value is actually known with high 

confidence, then the n required to confidently assess bias is at least (1.22/0.22)2 = 5.62 ≈ 31. The 

thin red vertical line to the right of the x̅lab for the three methods in Figure 2 represents the 

length of the resulting ±U95(x̅lab) intervals from a total of 31 measurements. With the 

additional measurements, method E is seen to be unbiased and method F to be biased.

For most situations, this high number of independent measurements will not be feasible. It is 

thus questionable whether assessing the bias of very imprecise methods is a useful exercise. 

However, the extremely high result for method F (3.58) could well reflect some one-off 

analytical glitch in the procedure rather than intrinsic variability of the method. If the root cause 

of the extreme value could be determined (and the procedure corrected), this result could be 

excluded from the estimate of x̅lab and the method assessed as unbiased – although another 

measurement or two would be needed to make that conclusion convincing. Thus, it is quite 

possible that relatively few additional measurements would be required to confidently assess 

trueness for method F.

Although the x̅lab ± U95(x̅lab) for method G spans the entire certified interval, there is less than 

a 50 % chance that the x̅lab is within xNIST ± U95(xNIST). If the method’s s(x) of 0.61 mg/g is 

known with high confidence then n = (0.61/0.22)2 = 2.82 ≈ 8 measurements are needed to 

confidently establish whether x̅lab is more likely than not to be within the certified interval.

D.3 What n is needed? (Estimation from SRM data)

If, for any reason, s(x) must be estimated only from measurements on the SRM itself, the process 

is a bit more complicated. Rather than using a constant expansion factor of 2 to estimate the 

uncertainty from the standard deviation from just a few measurements, you should use the 

Student's t distribution at the 0.95 confidence level for n-1 independent analyses, t95,n−1. Under 

the same assumption that half of your interval must fit within the certified uncertainty, 

t95,n−1 ∙ s(x̅)⁄2 ≤ U95(xNIST), then

n ≈ (t95,n−1⁄2)

2

(s(x) U95(xNIST ⁄ ))

2

.

Since the value of t95,n−1 depends on n, you can estimate n most easily with statistical software 

(see Appendices 3 and 4) or from a plot of n as a function of the ratio s(x) U95(xNIST ⁄ ). Figure 

D.2 displays ratios corresponding to n from 2 to 32 when s(x) is well known (open diamonds) 

and when it is estimated from the same results used to estimate the mean (solid diamonds). The 

solid vertical lines represent the ratio values for the 1.22 mg/g s(x) of methods E and F and the 

0.61 mg/g s(x) of method G.

The estimates are quite similar for ratios greater than about 5 (and less than about 0.5, because a 

minimum of two values is required to estimate a standard deviation). But for ratios between 

these values, you must make a few more measurements to confidently assess trueness than is 

indicated using a fixed expansion factor of 2. The Student’s t expansion suggests that 11 rather 

than 8 results are needed when the ratio is 2.8 (method G). Both estimates suggest that more 

than 30 results would be required when the ratio is 5.6 (methods E and F).
17

Remember that these values are appropriate when you require that there be at least a 50 % 

overlap of the 95 % level of confidence interval about your estimated mean with the certified 

interval. The extent of overlap required for a particular purpose reflects value judgments about 

the relative costs associated with false positive and false negative risks. The smaller the overlap, 

the greater the risk of accepting a biased method as unbiased. The greater the overlap, the 

greater the number of independent results that will be needed. It’s up to you to decide what risks 

and costs are fit for your purpose, and thus the degree of overlap that is acceptable [26,27].

Figure D.2. Number of results needed to confidently assess trueness

Open diamonds (◊) represent the function when s(x) is well known.

Closed diamonds (♦) represent the function when x̅and s(x) are estimated from the same data.

5

10

15

20

25

30

0.2 0.3 0.4 0.6 0.8 1 2 3 4 5 6

n

s(x) / U95(xNIST) 
18

D.4 Quantitative estimation of trueness (bias)

The graphical analysis described above is often sufficient to establish whether or not your 

measurement process is capable of providing fit-for-purpose results. Assuming that s(x̅) is at 

least as small as U95(xNIST), the difference between the assigned value for the SRM and your 

mean value is a good quantitative summary of your method's bias:

Bias = x̅lab − xNIST .

The combined standard uncertainty (i.e., the standard deviation) associated with Bias is:

u(Bias) = √s

2(x̅lab) + u2(xNIST) ; u(xNIST) = U95(xNIST)⁄2 .

The 95 % level of confidence expanded uncertainty estimate on the Bias is again

U95(Bias) = k ∙ u(Bias) .

Since u(xNIST) is known with high confidence, the number of degrees of freedom associated 

with u(Bias) will, in general, be the number associated with u(x̅lab). However, if s(x̅lab) is

estimated from a small number (n) of independent measurements and much smaller than

u(xNIST), the effective number of degrees of freedom for k, k = t95,veff, can be calculated using

the Welch-Satterthwaite formula [23]:

veff =

(s

2(x̅lab)+u2(xNIST))

2

s

4(x̅lab)⁄(n−1) + u4(xNIST)⁄60

.

The “60” is an arbitrary stand-in for the “large” number of degrees of freedom associated with an 

SRM’s certified value, but it is convenient in that t95,60 = 2.000 and so t95,veff will not be less 

than 2.

If zero is contained within the interval [Bias − U95(Bias) to Bias + U95(Bias)], then with 

about a 95 % level of confidence your method is not significantly biased. If zero is not contained 

within the interval, you should investigate why your method appears biased and take corrective 

action.

D.5 The perils of checking trueness with two or more validation materials

When two or more appropriate validation materials are available, it is possible to get “mixed 

signals”: the Bias determined relative to one material may be significant while the Bias for 

another is insignificant. Reference [28] reviews various mathematical ways that have been 

proposed to deal with such ambiguous situations by including the uncorrected bias (and its 

estimated uncertainty) into the estimated uncertainty. This approach should be avoided since (in 

addition to increasing the uncertainty beyond what should be attainable) it provides a 

disincentive to discovering why your method appears biased with some samples and not with 

others.

It is also possible that one or another of the natural-matrix SRMs has degraded or even been 

incorrectly value assigned; if you have strong evidence that this may be the case, you can e-mail

the SRM’s Technical Contact. This person’s name is listed on the SRM’s page on the NIST 

website, www.nist.gov/srm; enter the SRM number and you’ll be taken to a second page, where 

you can select the link to the SRM in which you are interested.
19

Step E. Use of an SRM for Characterization of an In-House Quality Control Material

SRMs are expensive and are prepared in limited quantities. While you need to have quality 

control (QC) materials that you analyze on a regular basis to document that your measurement 

process is stable (and to discover instabilities quickly if it isn’t), using an SRM just for QC

wastes scarce resources. You should consider preparing one or more in-house QC materials.

A QC material must be stable and homogeneous and have a composition similar to that of the

samples you will be analyzing. For the protein bar example, you might want to grind and 

package a sufficient quantity of a single batch of the protein bar itself. Grinding will help to 

ensure homogeneity. Packaging in single-use portions (under argon or nitrogen if possible) and 

cold storage (frozen, preferably at -80 °C) will enhance long-term stability.

You will need to characterize your in-house QC material(s) by analyzing them and the SRM(s)

together at least 10 times over a relatively long period of time. You must prepare all samples 

independently from start to finish: you cannot merely analyze a single preparation multiple 

times. The results for the SRM are used to establish that your measurement process is 

acceptably stable. Assuming that the results for the candidate QC material(s) are acceptably

precise, the mean and standard deviation of these results can be confidently used to establish 

action limits [17].

Figure E.1 displays a comparison of (dry-lab) results for total catechin monomers in SRM 2384 

(solid circles) and your candidate in-house QC material (open circles). These “results” represent 

one or two measurements of the candidate control material per working day over a period of one 

month and a few measurements of the SRM per week (data and calculations are in 

Appendix IV.)

Figure E.1. Comparison of Measurement Results for the SRM and Candidate QC Material

0.0

0.5

1.0

1.5

2/29 3/7 3/14 3/21 3/28

Catechin Monomers, mg/kg

Date
20

The dashed line in the middle of the SRM results in Figure E.1 represents the mean value of your 

12 results for the SRM, x̅lab,SRM = 1.43 mg/kg, with the solid bracketing lines representing an

approximate 95 % level of confidence interval on the measurements,

U95(x̅lab,SRM) = 2 ∙ s(x̅lab,SRM) = 0.26 mg/g.

Note that this interval is based only on the measurements you made. Likewise for the QC

material, the dashed and bracketing lines represent the mean and its 95 % level of confidence 

interval, x̅lab,QC ± U95(x̅lab,QC) = 0.62 mg/g ± 0.13 mg/g, for the 34 measurements that you 

made on the QC material.

The lack of any apparent trend in either set of results suggests that the measurement process and 

the QC material are adequately stable. The percent relative standard deviations of the two 

materials, CVlab,SRM =

100×0.26/2

1.43

= 9.1 % and CVlab,QC =

100×0.13/2

0.62

= 10.5 %, are similar

enough to conclude that the performance of the measurement process is about the same for the 

two materials and that the QC material is adequately homogenous. Rather than using the SRM 

to monitor your measurement process, you can now confidently – and much less expensively –

use your own material.

Should future measurement results for the QC material start to diverge from the trend or become 

significantly less precise, you can identify whether the material or the measurement process itself 

is at fault by again evaluating the SRM. If you conclude that the QC material has degraded, you 

will need to prepare a new material and characterize it against the SRM as with the original 

material.
21

Step F. Use of an SRM in Value Assigning a Secondary Reference Material

Monitoring the stability of a measurement process requires QC materials that are stable and 

homogenous, but you don’t have to know the true value of X in the material in order to monitor 

the imprecision of the method (see Step E). However, if the measurement process is traceably 

calibrated (see Step B), the sources of imprecision identified and characterized (Step C), and the 

trueness of the calibration confirmed by analysis of matrix-matched natural-matrix SRMs (Step 

D), then the QC material can become a secondary reference material (RM). If sufficiently well 

characterized, such RMs can be used in place of SRMs for (nearly) all purposes, and with proper 

documentation [29,30] can be used and even sold as CRMs. To minimize the accumulation of 

uncertainty and propagation of unrecognized biases, secondary RMs should not be used to value 

assign other secondary RMs if CRMs of “higher metrological order” are available.

While the control chart analysis presented in Step E is not fully adequate for value assigning a 

commercial CRM, it may be suitable for within-organization tasks where the assessment of 

accuracy rather than just imprecision is important. A data and calculation example of value 

assignment of a QC material as a secondary reference material is provided in Appendix IV.

Recall that the 12 independent measurements of the SRM yielded x̅lab,SRM = 1.43 mg/kg with 

U95(x̅lab,SRM) = 2 ∙ s(x̅lab,SRM) = 0.26 mg/g and that 34 measurements on the QC material 

yielded x̅lab,QC = 0.62 mg/g with U95(x̅lab,QC) = 2 ∙ s(x̅lab,QC) = 0.13 mg/g. The standard 

deviations of the mean values are then s(x̅lab,SRM) = 0.13⁄√12 = 0.038 mg/g and 

s(x̅lab,QC) = 0.065⁄√34 = 0.011 mg/g.

The certified value for total catechin monomers in SRM 2384 is xNIST ± U95(xNIST) = (1.49 ±

0.22) mg/g so Bias = 1.43 − 1.49 = −0.06 mg/g, U95(Bias) = 2√0.0112 + (0.22⁄2)

2 =

0.22 mg/g. Since 0.06 mg/g is considerably less than 0.22 mg/g, Bias can be considered 

insignificant and, with yellow-light caution, the measurement procedure considered to be 

unbiased. The appropriate assigned value for the QC material is then x̅lab,QC ± U95(x̅lab,QC) =

(0.62 ± 0.13) mg/g.
22

Conclusion

So there you have it – selection and use of SRMs for quality assurance and establishing 

traceability in six (perhaps-not-so) easy steps. As a greater variety of materials become 

available, it becomes easier to find a match to your samples, although there will always be cases 

in which no good match exists. The analyses themselves require a lot of work, but that work is 

necessary to assure the quality of all of your results. Fortunately, the symbols in the equations 

seem more cumbersome than the mathematical ideas and operations themselves, and so once 

you’re to that point, you’re pretty much home free. Good luck and happy measuring!

Acknowledgements

We thank our NIST colleague Michael Nelson for verifying the instructions and calculations in 

Appendices II and III. We thank Will Cone (Cone Bioproducts, Seguin, TX) for calling our 

attention to an error in the veff value in Appendix IV, Step F.
23

References

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http://www.nist.gov/traceability
24

16 Ellison SRL, King B, Rösslein M, Salit M, Williams A (eds) (2003) Traceability in chemical 

measurement: a guide to achieving comparable results in chemical measurement. 

http://www.eurachem.org/index.php/publications/guides/trc

17 Hibbert DB, Gooding JJ (2006) Data Analysis for Chemistry: An Introductory Guide for 

Students and Laboratory Scientists, Oxford University Press, New York

18 Prichard E, Barwick V (2007) Quality Assurance in Analytical Chemistry, Wiley, Chichester 

England.

19 Ellison SLR, Barwick VJ, Farrant TJD (2009) Practical Statistics for the Analytical Scientist: 

A Bench Guide, 2

nd Ed, Royal Society of Chemistry

20 Kallner A (2014) Laboratory Statistics: Handbook of Formulas and Terms, Elsevier, 

Waltham MA, USA

21 Eurachem/CITAC Guide (2007) Use of uncertainty information in compliance assessment 

http://www.eurachem.org/index.php/publications/guides/uncertcompliance

22 Rukhin AL (2013) Assessing compatibility of two laboratories: formulations as a statistical 

hypothesis testing problem. Metrologia 50:49-59

http://iopscience.iop.org/0026-1394/50/1/49/

23 JCGM 100:2008. Evaluation of measurement data – Guide to the expression of uncertainty in 

measurement (GUM). JCGM.

http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf

24 Thompson M, Wood R (2006) Using uncertainty functions to predict and specify the 

performance of analytical methods. Accredit Qual Assur 10:471-478

25 Appendix D, Official Methods of Analysis of AOAC International, 17th Edition, AOAC 

International, Gaithersburg, MD (2000)

26 Rukhin AL (2014) Compatibility verification of certified reference materials and user 

measurements. Metrologia 51:11-17. http://iopscience.iop.org/0026-1394/51/1/11/

27 AOAC Guidelines for Single Laboratory Validation of Chemical Methods for Dietary 

Supplements and Botanicals,

http://www.aoac.org/dietsupp6/Dietary-Supplement-web-site/DSHomePage2.html

28 Magnusson B, Ellison SLR (2008) Treatment of uncorrected measurement bias in uncertainty 

estimation for chemical measurements. Anal Bioanal Chem 290, 201-213.

29 ISO Guide 34:2009 General requirements for the competence of reference material 

producers. ISO

30 ISO Guide 35:2006 Reference materials – general and statistical principles for certification. 

ISO
25

Appendix I: Acronyms and Symbols

Bias The difference between your mean value for X in your unit of the SRM and 

NIST’s certified value

CRM Certified Reference Material, an RM accompanied by a certificate issued by an 

authoritative body

CV Relative standard deviation expressed as percent (aka “coefficient of variation”)

CVlab,SRM CV of your measurements of X on the SRM

CVlab,QC CV of your measurements of X on your QC material

i Index over measurement values

j Index over subsets (groups) of a set of measurements of X

k coverage factor for transforming standard uncertainties (u – think standard 

deviations, where ±u spans about 68 % of the possible values) into 95 % level of 

confidence expanded uncertainties (U95, where ±U95 is expected to span about 

95 % of the possible values). When u is estimated with great confidence, k = 2; 

when u is estimated from a limited number of measurements, k = t95,n−1 .

MSwth Within-subset mean squares (see [17,19] for further information)

MSbtw Between-subset mean squares (see [17,19] for further information)

m Number of subsets (groupings) within a given set of measurements of X

NIST National Institute of Standards and Technology

n Number of data values, x, in a given set of measurements of X

nj Number of data values, x, in a given subset (group) of the entire set of 

measurements of X

n' Effective number of data values per subset (group) when the nj are not the same

QC Quality Control

RM Reference Material, “material sufficiently homogeneous and stable with reference 

to specified properties, which has been established to be fit for its intended use in 

measurement or in examination of nominal properties” (VIM 5.13,[4])

SRM Standard Reference Material, a CRM provided by NIST

s(x) Standard deviation of your measurements of X

s(x̅) Standard deviation of the mean of your measurements of X , aka “standard error 

of the mean”

s(x̅lab) Standard deviation of the mean of your measurements of X on a material

s(x̅lab,QC) Standard deviation of the mean of your measurements of X on independently 

prepared subsamples of your QC material

s(x̅lab,SRM) Standard deviation of the mean of your measurements of X on independently 

prepared subsamples of the SRM

sI Intermediate imprecision, the standard deviation of measurements made on 

independently prepared subsamples of a material made by multiple analysts 
26

within a single laboratory or one analyst of that laboratory over a long period of 

time.

spost Post sample preparation imprecision, the standard deviation of a series of

measurements made on a single preparation of a sample made by one analyst over 

a very short period of time

sr Repeatability imprecision, the standard deviation of measurements on 

independently prepared subsamples of a material performed over a short period of 

time by the same analyst

t95,n−1 Student’s t value for expanding u to U95 given that u has been estimated from n

independent measurements. See “k”

veff Number of effective degrees of freedom used to estimate k when u is estimated as 

a combination of two or more uncertainty components

u(Bias) Combined standard uncertainty (i.e., standard deviation) of the Bias estimate

u(xNIST) One-half of NIST’s U95(xNIST): xNIST ± 2 ∙ u(xNIST) provides the same 

approximate 95 % level of confidence coverage as xNIST ± U95(xNIST).

U95(Bias) NIST certified approximate 95 % level of confidence interval about the certified 

value.

U95(xNIST) NIST certified approximate 95 % level of confidence interval about the certified 

value.

X the measurand of interest, where “measurand” is a particular analyte in a defined 

sample matrix

x a measured value of X

xlab,i the i

th measurement of X that you make under defined experimental conditions

xNIST NIST certified value for the level of X in a given SRM. With high confidence, 

xNIST ± U95(xNIST) is expected to include the true value of X in your unit of the 

SRM

x̅ Arithmetic mean of your measurements of X

x̅lab Arithmetic mean of your measurements of X

x̅lab,QC Arithmetic mean of your measurements of X made on your own candidate quality 

control material

x̅lab,SRM Arithmetic mean of your measurements of X made on the SRM
27

Appendix II: Statistics and Examples

The specifications indicated in a NIST reference material certificate provide the estimated 

measurand xNIST, i.e., the certificate value, and the expanded uncertainty U95(xNIST). Thus, 

xNIST ± U95(xNIST ) is the uncertainty interval for the measurand x. Commonly the interval is 

written with an expansion factor 2, U95(xNIST) = 2σNIST.

The user’s replicated measurements, say, x1, ..., xn are summarized by the value x̅, which is the 

estimated measurand (typically the sample mean) and s, which estimates the repeatability 

standard deviation. Sometimes s/√n has the meaning of the uncertainty u(x̅) that may include 

components estimated from information other than the statistical analysis of series of 

observations in place of or in addition to the statistical analysis of observations. In the following 

we suppose that s is the sample standard deviation which does not depend on x̅. In many 

situations this independence can be approximately achieved by a suitable transformation of x’s.

In the simplest setting accepted here, xi is a realization of a Gaussian random variable with some 

mean xNIST + ∆ and some unknown standard deviation τ. The “no bias” (trueness or 

compatibility) hypothesis H0 means that ∆ = 0. The error τ represents the repeatability of the 

lab’s measurements.

The commonly used statistical procedure, the classical t-test, rejects compatibility or 

“compliance” of results to the certified value when

|x̅− xNIST| ≥

t(

α

2

,n−1) s

√n

.

When ∆ = 0, the probability of false positives (Type 1 error) by using t-test is exactly α. If ∆ ≠ 0, 

the distribution of the ratio √n(x̅− xNIST)/s is known as a noncentral t-distribution with the 

same degrees of freedom v and the noncentrality parameter √n∆/τ. Besides controlling for the 

Type 1 error (say, α = 0.05), one would like to have the probability (say, β) of false negatives 

(Type 2 error) as small as possible.

For that purpose you must specify the metrologically important critical bias, ∆c ≠ 0, whose value 

is large enough to worry about. For the bias to be deemed significant, this value cannot be 

smaller than U95(xNIST), but realistically ∆c should not be taken too large. We recommend to 

limit the choice of ∆c to the range U95(xNIST) ≤ ∆c ≤ 2U95(xNIST).

The larger ∆c, the smaller is the sample size n needed to attain a given Type 2 error, the false 

negatives probability β, at ∆c. This balance can be achieved only if there is some information 

about the unknown τ. Indeed the probability of the Type 2 error is a function of the noncentrality 

parameter √n∆/τ. If τ were known, you could solve for n in the equation, Type 2 error = β, to 

get the needed sample size n, n ≈ (z(α/2) + z(β))

2

τ

2

/∆c

2

, where z(α/2) and z(β) are the critical 

values of the standardized normal distribution at the specified probabilities. This value of n is 

necessary to have the test of power 1 - β.

Since τ is unknown, you must estimate it. For this purpose it may be helpful to compare τ to 

σNIST which by itself is not employed in the t-test. Unless U95(xNIST) includes substantially large 
28

components that were not estimated by statistical analysis of a series of measurements, τ must be 

larger than U95(xNIST). One can take τ = BσNIST , where the corresponding factor B, say, 

1 ≤ B ≤ 5, is determined from your preparatory work.

This factor B also can be described via the measurement capability index which here is the ratio 

of (expected) widths of two coverage intervals, Cm = 2√nσNIST/ (t(

α

2

, n − 1) τ). Indeed, for 

the desired measurement capability index Cm, B = 2√nCm

−1

/ t(

α

2

, n − 1) . See [i] for the 

discussion of other capability characteristics in quality control problems. The recommended 

value Cm should exceed 1.5 [ii]. According to the rule of thumb, B is about 3 [iii].

Returning to the issue of controlling the Type 2 error, by taking τ = BσNIST, one gets the 

estimated value of the noncentrality parameter, √n∆c/(BxNIST), so that the numerical evaluation 

of the smallest n such that Type 2 error ≤ β, becomes feasible. Modern statistical software, in 

particular the R language (free software for statistical computing and graphics; http://www.rproject.org/) available in the public domain, offers you several routines to determine numerically 

the needed sample size for any given values of α, β and ∆c/τ. Additionally, the software package 

metRology (http://www.nist.gov/itl/sed/gsg/metrology.cfm) provides specialized functions for 

statistical metrology as a package for the R software environment, with an option of a convenient 

user interface using Microsoft Excel (http://www.nist.gov/itl/sed/gsg/metrology-for-microsoftexcel.cfm).

Example 1. Compute power of t test

We present here exact calculations for the example ‘dry lab’ catechin monomer data for seven 

different methods in Figure 3 in the main text.

Assume that the critical value is ∆c = 2U95(xNIST) = 0.44 and τ1 = 0.087/2 (for methods A, B, C, 

and D); τ2 = 1.22/2 (methods E and F); and τ3 = 0.88/2 (method G). With di = ∆c/τi, you

determine d1(A-D) = 10.115, d2(E,F) = 0.721, and d3(G) = 1 that represents the three grouped 

methods that have comparable uncertainty.

First, in R (refer to Appendix III for the installation of R and metRology), set your working 

directory through the File – Change dir menu dropdown list (this will remain the working 

directory as long as you maintain the R session). Now, install the Basic power calculations ‘pwr’ 

package through the Packages – Install package(s) menu dropdown list. At the R Console 

command prompt ( >), load the ‘pwr’ package within the R environment:

(Note: In this document red typeface denotes commands to be entered by the user into the R 

console and blue typeface is of appropriate outputs.)

> library(pwr)

Now compute a series of power tests for the three grouped scenarios (d1, d2 and d3) and store the 

results in the associated data structures (P1, P2 and P3, respectively):
29

> P1=pwr.t.test(d=10.115,power=0.8,sig.level=0.05,type="one.sample",alternative="two.sided")

> P2=pwr.t.test(d=0.721,power=0.8,sig.level=0.05,type="one.sample",alternative="two.sided")

> P3=pwr.t.test(d=1,power=0.8,sig.level=0.05,type="one.sample",alternative="two.sided")

Typing P1, P2 or P3 at the command prompt provides a summary of the results for each of the 

individual power tests:

> P1

One−sample t test power calculation

n = 2.054403

d = 10.115

sig.level = 0.05

power = 0.8

alternative = two.sided

> P2

One-sample t test power calculation

n = 17.11833

d = 0.721

sig.level = 0.05

power = 0.8

alternative = two.sided

> P3

One−sample t test power calculation

n = 9.93785

d = 1

sig.level = 0.05

power = 0.8

alternative = two.sided

According to these calculations, about 2 observations are required to have the type 2 error of ttest about 0.2 when ∆c/τ = 10.115 (P1), while for ∆c/τ = 1.0 one needs nearly 10 measurements 

(P3).
30

Example 2. Analysis of Variance (ANOVA) for the Catechin Data

We present here an ANOVA for the example catechin monomer validation data that was 

acquired by five separate analysts in Table 4 in the main text.

For the ANOVA calculations with the following R code, the raw catechin data should be 

organized into two separate columns, one labeled ‘analyst’ with values 1 to 5 representing the 

five different analysts and the second labeled ‘mass’ with the corresponding catechin mass 

fraction readings. You can generate the data in Excel and then save as either a .csv or .txt file. 

Import or ‘read’ the data (as either a .csv file or a .txt file) and convert it to a data frame in R:

> cate = read.csv(file="catechin.csv", head=TRUE) or

> cate = read.table(file="catechin.txt", head=TRUE)

Print the data frame and verify that is has the comparable structure:

> cate

analyst mass

1 1 1.362

2 1 1.388

3 1 1.392

.....

34 5 1.453

35 5 1.510

36 5 1.573

Load the Linear and Nonlinear Mixed Effects Models ‘nlme’ package through the Packages –

Load package menu dropdown list and then add it to the R environment library:

> library(nlme)

Construct a new ‘cate’ data object with the data grouped:

> cate=groupedData(mass ~ 1| analyst, data=read.table("catechin.txt",header=TRUE))

Generate a summary to present the basic data information within the ‘cate’ data structure:

> summary(cate)

analyst mass 

1:10 Min. :1.357 

5: 4 1st Qu.:1.423 

4: 4 Median :1.477 

3:10 Mean :1.504 

2: 8 3rd Qu.:1.563 

 Max. :1.770

Here, the number of measurements for each analyst is displayed as well as the descriptive 

characteristics of their mass fraction readings.
31

Calculate the linear mixed-effects model fit of the ‘cate’ data and print a summary:

> cate_lme=lme(mass ~ 1, data=cate, random = ~ 1|analyst)

> summary(cate_lme)

Linear mixed-effects model fit by REML

Data: cate

AIC BIC logLik

-67.83977 -63.17373 36.91989

Random effects:

Formula: ~1 | analyst

(Intercept) Residual

StdDev: 0.08472037 0.06996101

Fixed effects: mass ~ 1

Value Std.Error DF t-value p-value

(Intercept) 1.50919 0.03992659 31 37.79912 0

Standardized Within-Group Residuals:

Min Q1 Med Q3 Max

-2.06514039 -0.56280795 0.02422057 0.56949594 1.87991441

Number of Observations: 36

Number of Groups: 5

Determine the analysis of variance table and a one-way analysis of means for the ‘cate’ data:

> anova(lm(mass ~ analyst, data=cate))

Analysis of Variance Table

Response: mass

Df Sum Sq Mean Sq F value Pr(>F)

analyst 4 0.25264 0.06316 12.865 2.772e−06 ***

Residuals 31 0.15220 0.00491

> oneway.test(mass ~ analyst, data=cate)

One−way analysis of means (not assuming equal variances)

data: mass and analyst

F=8.6657, num df = 4.000, denom df=10.406, p−value=0.002438

The analysis of variance table unequivocally rejects the equality of means hypothesis if 

homogeneity of variances holds. The p-value 2.772 × 10-6is quite small. The same hypothesis 

is also rejected with the p-value 0.002 when variances are not assumed to be equal.

Install the companion to applied regression ‘car’ package through the Packages – Install 

Package(s) menu dropdown list. Load the ‘car’ package within the R environment:

> library(car)
32

Compute the Levene’s test for homogeneity of variance across the individual groups (i.e., 

analysts):

> leveneTest(mass ~ analyst, data=cate)

Levene’s Test for Homogeneity of Variance (center=median)

Df F value Pr(>F)

group 4 1.4716 0.2347

31

This provides the estimates of the parameters in a random effects model which assumes a 

random bias presence. Namely, the measurements xij of the analyst i = 1, ..., 5, 1 ≤ j ≤ ni, 

n1 = 10, n2 = 8, n3 = 10, n4 = 4, n5 = 4, are supposed to have the form

xij = μ + ∆i + εij.

Here μ is the common true mean, (estimated as 1.506 with uncertainty of 0.038), εij are 

independent normal disturbances with zero mean and unknown variance σi

2

depending on the 

analyst i. The between-analyst effect values, ∆i, which can be viewed as realizations of a random 

bias supposed to have zero mean and some unknown standard deviation which is determined to 

be 0.080 (Intercept Stdev).

Individual σi can be found from parameter estimates which give the ratios, σi/σ1 with 

σ1 = 0.043 (Residual StDev). The ∆i predicted values can be found as x̅i − μ, e.g. 

∆1 = 1.425 − 1.506 = −0.081, ∆2 = 1.649 − 1.506 = 0.143, ∆3 = 1.467 − 1.506 = −0.041, 

∆4 = 1.533 − 1.506 = −0.026, and ∆5 = 1.473−1.506 = −0.033. Thus in this example there was 

just one analyst (2) with a high bias, and all others were negatively biased. However, Levene’s 

test of variances equality does not reject the homogeneity of variance hypothesis at the 95 % 

confidence level as the p-value is 0.234.

As additional information, Excel users may want to use metRology for Microsoft Excel, an addin which allows to call R functions as worksheet functions [iv]. There are several web sites, 

intended mainly for applications in biostatistics, that also provide the needed sample size 

calculations [v]. Some additional R codes useful in metrology applications are provided in 

metRology. Support for metrological applications which can be accessed at 

http://metrology.sourceforge.net.
33

References in Appendix II

i NIST (2010) NIST/SEMATECH e-Handbook of Statistical Methods.

http://www.NIST.gov/itl/sed/gsg/handbook project.cfm

ii ASME B89.7.3.1-2001 (2002) Guidelines for decision rules: considering measurement 

uncertainty in determining conformance to specifications. ASME, New York

iii Instone I (1996) Simplified method for assessing uncertainties in commercial, production 

environment. http://metrology forum.tm.agilent.com/easy.shtml

iv Heiberger, R. M., Neuwirth, E. (2009) A Spreadsheet Interface for Statistics, Data Analysis, 

and Graphics Springer NY

v E.g.: http://hedwig.mgh.harvard.edu/sample size,

http://homepage.stat.uiowa.edu/  ̃rlenth/Power, 

http://calculators.stat.ucla.edu/powercalc.
34

Appendix III: Installation of R and metRology

1. Install the newest version of R on your computer (Windows, MacOS or UNIX)

Download the R package from the R project website (http://www.r-project.org/) by either 

selecting the download R or Comprehensive R Archive Network (CRAN) mirror (CRAN 

mirror) link under the ‘Getting Started’ window:

After selecting your CRAN mirror most suitable for your physical location (geographically 

proximal), select the Download and Install R link for your particular operating system (e.g., 

Download R for Windows). Install the base subdirectory, which are the binaries for base 

distribution and are needed to install R for the first time (the links are CRAN mirrorspecific):
35

Click on the Download R 3.2.1 for Windows (or the current version. R 3.2.1 was current on 

9-July-2015), but link to download the installation binary file (e.g., R-3.2.1-win.exe) to your 

computer. Save and then run the R executable file and select language preferences, directory 

locations, desktop icon shortcut, etc. during the installation. R is now installed and ready to 

start. Double-click the (desktop) icon and the initial R console window should appear:

Set your working directory for file storage under the File – Change dir... menu dropdown 

list.
36

2. Install the metRology data analysis package within R

Using the Packages Menu, select ‘Install package(s)...’ You will then need to select a 

CRAN mirror. Select the ‘metRology’ package from the dropdown menu that is 

automatically displayed and it should install. You may be asked whether or not to use a 

personal library; if so, select ‘Personal’. (Note: the details of the loading process may 

change with time as new versions replace the old.) You can type ‘library()’ at the command 

prompt ( > ) to see the list of packages stored in your library:

You now can load (or attach) the metRology package either through the 

Packages – Load package... menu dropdown list or by typing ‘library(metRology)’ at the 

command prompt ( > ): 

You are now ready to use the functions within the metRology package.
37

More details regarding the metRology software and its ongoing development are described 

on the NIST website: http://www.nist.gov/itl/sed/gsg/metrology.cfm and at: http://cran.at.rproject.org/web/packages/metRology/.

3. Install the metRology add-in for Microsoft Excel

Detailed instructions on the installation of the metRology add-in for Microsoft Excel are 

provided on the NIST website: http://www.nist.gov/itl/sed/gsg/upload/metRology-forMicrosoft-Excel-installation-instructions.pdf . You are encouraged to open these instructions 

and use them as a guide for the installation and testing of the add-in.

In brief, the installation file (metRology-for-Microsoft-Excel-v1-03-setup.exe) for the 

metRology add-in for Microsoft Excel can be downloaded from the NIST website: 

http://www.nist.gov/itl/sed/gsg/metrology-for-microsoft-excel.cfm. Save and then run the R 

executable file (metRology-for-Microsoft-Excel-v1-03-setup.exe). Prior to installation, make 

sure that it is to be installed in the following directory: C:\Program Files\metRology

for Microsoft Excel. If you have a 64-bit operating system, you may need to modify the 

destination location to C:\Program Files\ by deleting “(x86)”. It is also important the 

Rscript.exe directory is added to the path:

Lastly, open the appropriate test file (e.g., http://www.nist.gov/itl/sed/gsg/upload/testmetRology-for-Microsoft-Excel.xlsm) and ensure that clicking on the ‘Compute Uncertainty’ 

button generates a result.
38

Appendix IV: Data for the Examples

Step D. Use of SRMs to Establish Trueness of Results

Data for Figure D.1: Catechin Monomers, mg/g:

Catechin Monomers, mg/g Summary Statistics, mg/g

Method Rep1 Rep2 Rep3 n x̅ s(x) s(x̅) 2 · s(x̅)

A 1.362 1.419 1.511 3 1.431 0.075 0.043 0.087

B 0.562 0.619 0.711 3 0.631 0.075 0.043 0.087

C 1.641 1.698 1.790 3 1.710 0.075 0.043 0.087

D 1.722 1.779 1.871 3 1.791 0.075 0.043 0.087

E 2.710 1.490 0.270 3 1.490 1.220 0.704 1.409

F 3.582 1.419 1.511 3 2.171 1.223 0.706 1.412

G 2.010 2.110 1.010 3 1.710 0.608 0.351 0.702
39

Data for Figure D.2: Number of results needed to confidently assess trueness

s(x) U95(xNIST ⁄ ) = 2√n⁄k

n t95,n−1 k = 2 k = t95,n−1

2 12.706 1.414 0.223

3 4.303 1.732 0.805

4 3.182 2.000 1.257

5 2.776 2.236 1.611

6 2.571 2.449 1.906

7 2.447 2.646 2.163

8 2.365 2.828a2.392

9 2.306 3.000 2.602

10 2.262 3.162 2.796

11 2.228 3.317 2.977a

12 2.201 3.464 3.148

13 2.179 3.606 3.310

14 2.160 3.742 3.464

15 2.145 3.873 3.612

16 2.131 4.000 3.753

17 2.120 4.123 3.890

18 2.110 4.243 4.022

19 2.101 4.359 4.150

20 2.093 4.472 4.273

21 2.086 4.583 4.394

22 2.080 4.690 4.511

23 2.074 4.796 4.625

24 2.069 4.899 4.736

25 2.064 5.000 4.845

26 2.060 5.099 4.952

27 2.056 5.196 5.056

28 2.052 5.292 5.158

29 2.048 5.385 5.258

30 2.045 5.477 5.356

31 2.042 5.568b5.453

32 2.040 5.657 5.547

33 2.037 5.745 5.640b

34 2.035 5.831 5.732

35 2.032 5.916 5.822

36 2.030 6.000 5.911

37 2.028 6.083 5.999

38 2.026 6.164 6.085

39 2.024 6.245 6.170

40 2.023 6.325 6.254

a First ratio greater than 2.8

b First ratio greater than 5.6
40

Step E. Use of an SRM for Characterization of an In-House Quality Control Material

Data for Figure E.1: Comparison of Measurement Results for the SRM and the QC Material

Total Catechin Monomers, 

mg/g

Date and Time QC Material SRM 2384 Statistic QC Material SRM 2384

3/1/2012 9:15 0.615 n 34 12

3/1/2012 15:09 0.647 1.471 x̅, mg/g 0.616 1.426

3/2/2012 13:10 0.619 s(x), mg/g 0.063 0.131

3/5/2012 14:40 0.722 2 ∙ s(x), mg/g 0.126 0.262

3/5/2012 15:34 0.702 1.333 CV, % 10.2 9.2

3/6/2012 11:26 0.490

3/6/2012 15:18 0.629 1.399

3/7/2012 11:48 0.616

3/8/2012 10:40 0.685

3/8/2012 13:53 0.597 1.258

3/9/2012 8:36 0.558

3/12/2012 12:18 0.565

3/13/2012 13:53 0.697

3/14/2012 13:25 0.566

3/14/2012 15:42 0.660 1.639

3/15/2012 8:03 0.616

3/15/2012 14:36 0.652 1.587

3/16/2012 11:20 0.662

3/19/2012 13:08 0.669 1.354

3/19/2012 14:26 0.666

3/20/2012 12:14 0.643

3/20/2012 14:55 0.519 1.446

3/21/2012 13:13 0.464

3/22/2012 14:35 0.593

3/23/2012 9:54 0.618

3/23/2012 14:49 0.594 1.439

3/26/2012 8:17 0.559 1.519

3/26/2012 11:55 0.636

3/27/2012 8:05 0.667 1.180

3/27/2012 11:04 0.490

3/28/2012 9:10 0.646

3/29/2012 8:57 0.585

3/29/2012 12:21 0.691 1.481

3/30/2012 15:12 0.610
41

Step F. Use of an SRM in Value Assigning a Secondary Reference Material

Validation of absence of significant bias in SRM measurement values, mg/g

Your Measurements NIST Certificate Comparison

Statistic Value Statistic Value Statistic Value

n 12 n 60 veff 70

x̅lab,SRM 1.426 xNIST 1.49 Bias -0.06

s(x) 0.131

k = t95,11 2.20 k 2 k 2

u(x̅lab,SRM) 0.038 u(xNIST) 0.11 u(Bias) 0.12

U95(x̅lab,SRM) 0.083 U95(xNIST) 0.22 U95(Bias) 0.23

QC material Value Assignment, mg/g

Statistic Value

n 34

x̅lab,QC 0.616

s(x) 0.063

k = t95,33 2.03

u(x̅lab,QC) 0.011

U95(x̅lab,QC) 0.02