NISTIR 6919

Recommended Guide for

Determining and Reporting

Uncertainties for Balances and

Scales

Val Miller

State Laboratory Program

Weights and Measures Division

National Institute of Standards and Technology

Technology Administration

U.S. Department of Commerce

 
 
NISTIR 6919

Recommended Guide for

Determining and Reporting

Uncertainties for Balances

and Scales

Val Miller

State Laboratory Program

Weights and Measures Division

National Institute of Standards and Technology

Technology Administration

U.S. Department of Commerce

January 2002

U.S. DEPARTMENT OF COMMERCE

Donald L. Evans, Secretary

TECHNOLOGY ADMINISTRATION

Phillip J. Bond, Under Secretary of Commerce for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

Arden L. Bement, Jr., Director

 
 
Preface 

The Recommended Guide to Determining and Reporting Uncertainties for Balances and 

Scales was developed to meet the need expressed by members of the scale service 

community who were attempting to comply with quality system requirements contained in 

QS9000, and ISO/IEC 17025, for an interpretive document applying the concepts contained 

in the Guide to the Expression of Uncertainty in Measurement (GUM) to scale and balance 

calibrations. 

Acknowledgement: 

Special thanks regarding this Recommended Guide are given to Georgia Harris, NIST

Weights and Measures Division, who originally responded to inquiries and suggested 

possible solutions and provided technical and editorial guidance, and to Jeffery Horlick, 

NIST Standards Services Division, who provided technical and editorial comment 

throughout the process of generating this document. Thanks is also given to those members 

of the scale service community and NIST WMD office staff who contributed to the creation 

of the Recommended Guide by providing feedback as to the workability and 

understandability of the text from the field service perspective and by answering the author's

many questions. 

These include: 

Charles Francis, Mettler Toledo, Inc., 

Mark Ruefenacht, Heusser Neweigh, 

Bernd Damm, Deka Scale, Inc., 

Terry Benjamin, Grand Rapids Scale Company, Inc., 

Henry Meyers, Antibus Scales and Systems, Inc., 

Aaron Aldinger, Aldinger Company, 

Julie Beffrey, Cech Corporation, 

Richard Suiter, NIST Weights and Measures Division, 

Tina Butcher, NIST Weights and Measures Division, and 

Kenneth Butcher, NIST Weights and Measures Division. 

Additionally, thanks must go to the writers of the many reference documents for their varied 

approaches to the calculation of measurement uncertainty. Those works are listed in the 

bibliography contained in Appendix B. 

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Recommended Guide for Determining and Reporting 

Uncertainties for Balances and Scales 

Preface....................................................................................................... i 

Chapter 1: Purpose & Background ........................................................................................... 1 

Introduction........................................................................................................................... 2 

Scope..................................................................................................................................... 2 

Background........................................................................................................................... 3 

History and development ...................................................................................................... 3 

Chapter 2: General concepts..................................................................................................... 7

Introduction........................................................................................................................... 8 

General Description of Method ............................................................................................ 9 

Application.......................................................................................................................... 10 

Chapter 3: Balance Calibrations in a Laboratory Environment.............................................. 13 

Introduction......................................................................................................................... 14 

1. Specify the Process and Equation............................................................................... 14 

2, 3 & 4. Uncertainty Identification, Characterization and Quantification........................ 14 

Process standard deviation (sp) ....................................................................... 18 

Uncertainty of the standards (us)..................................................................... 20 

5. Calculating the combined uncertainty ........................................................................ 21 

6. Calculating the expanded uncertainty......................................................................... 21 

7. Evaluating the expanded uncertainty.......................................................................... 21 

8. Reporting Uncertainty................................................................................................. 22 

Quick Guide for Laboratory Balance Calibration Uncertainties ........................................ 25 

Uncertainty Worksheet ....................................................................................................... 26 

Chapter 4: Scale Calibrations Performed in an Uncontrolled Environment.......................... 27 

Introduction......................................................................................................................... 28 

Scales in an uncontrolled environment............................................................................... 28 

1. Specify the Process and Equation............................................................................... 28 

2, 3 & 4. Uncertainty Identification, Characterization and Quantification........................ 28 

Process standard deviation (sp) ....................................................................... 31 

Uncertainty due to display resolution (udr)...................................................... 33 

Uncertainty of the standards (us)..................................................................... 33 

Wind (uw)........................................................................................................ 34 

Vibration ......................................................................................................... 35 

Rain................................................................................................................. 35 

Substitution Loads .......................................................................................... 36 

Strain Load Tests............................................................................................ 36 

5. Calculation of the combined uncertainty .................................................................... 36 

6. Calculating the expanded uncertainty......................................................................... 36 

7. Evaluating the expanded uncertainty.......................................................................... 37 

8. Reporting Uncertainty................................................................................................. 37 

Quick Guide for Scale Calibration Uncertainties ............................................................... 39 

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Uncertainty Worksheet ....................................................................................................... 40 

Chapter 5: Scale Calibrations in a Semi-Controlled Environment ......................................... 41 

Introduction......................................................................................................................... 42 

1. Specify the Process and Equation............................................................................... 42 

2, 3 & 4. Uncertainty Identification, Characterization and Quantification........................ 42 

Process standard deviation (sp) ....................................................................... 45 

Uncertainty due to display resolution (udr) ..................................................... 47 

Uncertainty of the standards (us)..................................................................... 47 

5. Calculation of the combined uncertainty .................................................................... 48 

6. Calculating the expanded uncertainty......................................................................... 48 

7. Evaluating the expanded uncertainty.......................................................................... 49 

8. Reporting Uncertainty................................................................................................. 49 

Quick Guide for Scale Calibration Uncertainties ............................................................... 51 

Uncertainty Worksheet ....................................................................................................... 52 

Chapter 6: Sample Calculations and Recommendations for Reducing Uncertainty .............. 53 

Introduction......................................................................................................................... 54 

Sample Uncertainty Calculation #1 (Relates to Chapter 3)................................................ 54 

Sample Uncertainty Calculation #2: (Relates to Chapter 4)............................................... 59 

Changes Needed in Current Calibration Procedures to Address Uncertainty .................... 66 

Appendix A Common Terminology and Definitions ............................................................. 67 

Appendix B Bibliography....................................................................................................... 73

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Chapter 1 

Chapter 1: 

Purpose & Background

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Chapter 1 

Introduction

The goal of this guide is to provide service personnel with a method for identifying and 

quantifying the uncertainty components for scale and balance calibrations in both the field 

and laboratory environments. This guide includes specific information regarding uncertainty 

in on-site calibration of laboratory balances and large capacity scales. Information compiled 

from the many reference documents has enabled the writing of this guide; a comprehensive 

bibliography provides supplementary reading. 

Scope 

The Recommended Guide to Determining and Reporting Uncertainties for Balances and 

Scales provides a useful methodology to enable service personnel to identify, quantify, 

evaluate, combine and report the uncertainty components most likely to be encountered 

during the calibration of a scale or balance. The guide cannot be all-inclusive, since each 

calibration is unique, but it identifies the most common uncertainty contributors and attempts 

to educate the user so that less significant or less frequently encountered uncertainty 

contributors may also be identified and included in an expanded uncertainty statement. 

This Guide does not address scale and balance calibration methods. Procedural and 

specification documents for the testing of scales and balances have been in place for quite 

some time in the form of: 

- OIML R 76-1, “Nonautomatic weighing instruments, Part1: Metrological and technical 

requirements-Tests”, 

- ASTM E 319-85, “Standard Practice for the Evaluation of Single-Pan Mechanical 

Balances”, 

- ASTM E 898-88, “Standard Method of Testing Top-Loading, Direct-Reading Laboratory 

Scales and Balances”, 

- ASTM E 1270-88, “Standard Test Method for Equal-Arm Balances”; 

- NIST Handbook 44, “Specifications, Tolerances, and Other Technical Requirements for 

Weighing and Measuring Devices”, and 

- NIST Handbook 112, "Examination Procedure Outlines for Commercial Weighing and 

Measuring Devices." 

However, these procedural documents do not provide adequate instructions to complete a 

valid uncertainty analysis, hence should be updated by the standards writing community to 

include specific information regarding calibration uncertainty analyses. 

Scale and balance service personnel who perform calibrations are required to properly 

identify a weighing device and to follow a calibration procedure similar to those listed above. 

This guide will assist balance and scale service personnel in developing an uncertainty 

statement that will accompany any reported values resulting from the calibration procedure. 

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Chapter 1 

Background 

ISO/IEC 17025, "General Requirements of the Competence of Testing and Calibration 

Laboratories", states that a calibration or testing laboratory performing calibrations shall 

have, and shall apply, a procedure to estimate the uncertainty of measurement for all 

calibrations. It also requires that the customer be provided calibration reports that contain the 

measurement results and a statement regarding the measurement uncertainty. According to 

the definition of traceability, measurement results must be traceable to a national standard 

through an unbroken chain of calibrations or comparisons, each having a stated uncertainty. 

Each value indicated by a scale or balance is an estimate of the true value of the material 

weighed. Each value also has a portion about which there is uncertainty, doubt, skepticism,

suspicion or mistrust. This guide provides a practical method for calculating the uncertainty 

of a scale or balance calibration, in a manner useable by scale and balance service personnel 

who are not trained statisticians. 

History and development 

Over the years the NIST, Office of Weights and Measures (OWM), now designated as NIST 

Weights and Measures Division (WMD), has received many inquiries regarding the proper 

methods for calculating uncertainties of laboratory balances. More recently, due to industry 

requirements for uncertainty statements, NIST WMD has been receiving similar questions 

regarding uncertainties for scale calibrations. This comprehensive interpretive guide was 

developed to provide general guidance for the proper calculation of uncertainties for both 

scale and balance calibrations. 

Complaints have been heard about how this "new" uncertainty requirement is being forced on 

the balance and scale service industry. However, the need for an understanding of 

uncertainty associated with measurements is not at all new. Research reveals that as early as 

1960, U.S. Government requirements stated that the “collective uncertainty of the 

measurement standards shall not exceed 25 percent of the acceptable tolerance for each

characteristic being tested”. To ensure that this requirement was met, a rudimentary 

uncertainty analysis was required. Unfortunately, this was normally accomplished by

evaluating only the allowable tolerances or uncertainties of the standards that were used in 

the calibration process, and did not include process-related uncertainty components such as 

repeatability and operator input. This typically resulted in an underestimation of the 

measurement uncertainty value. 

Starting in the mid 1980's quality management system standards, in the form of the ISO/IEC 

9000 series, were implemented as manufacturers tried to identify the causes of quality

problems in their products and as competition on the international market intensified. The 

ISO/IEC 9000 documents provided a standardized methodology for dealing with all facets of 

a quality system through documentation of processes. Traceability of measurement results 

was one of the requirements set forth, with the quality of a measurement result directly 

relating to the traceability. In turn, traceability of measurement results requires documented 

evidence of the chain of preceding measurement results, and their associated uncertainties, to 

national standards. During that time, the impact of the process uncertainty on measurement 

results also became a topic of discussion, culminating in the 1993 publication of the ISO 

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Chapter 1 

Guide to the Expression of Uncertainty in Measurement (GUM). The GUM addresses the 

issue of uncertainty by looking at the complete measurement process. This includes the effect 

of the uncertainty of the calibration standards, as well as the impact of the processes involved 

in making the measurement and an engineering evaluation of data. This was not a totally new 

concept, for some organizations had been including variability of measurement in their 

uncertainty statements. But with the adoption of the GUM, the inclusion of process 

variability in uncertainty calculations became a widespread practice. 

In 1995, the U.S. Government requirements expressed in MIL-STD-45662A were 

superseded by adoption of ISO/IEC Guide 25, General Requirements for the Competence of 

Calibration and Testing Laboratories, as ANSI/NCSL Z 540-1-1994, Part 1, which became 

the U.S. industry guideline. Laboratories fully compliant with ISO/IEC Guide 25 could be 

confident that their measurement results would be accepted by ISO/IEC 9000 compliant 

organizations because Guide 25 was written to address the requirements of ISO/IEC 9000 as 

they applied to calibration and testing laboratories. ISO/IEC Guide 25 required that 

measurement results be traceable to national standards, with measured values and reported 

uncertainties. At approximately the same time, ISO/IEC 9000: 1994 was issued and became 

the accepted quality management criteria of much of the manufacturing marketplace. 

ISO/IEC 9000 criteria, dealing primarily with manufacturing process and customer service 

issues, did not address the technical needs of the laboratories, but contained wording that 

required that calibration support be provided by ISO/IEC Guide 25 compliant laboratories. 

In 1999, ISO/IEC 17025 replaced ISO/IEC Guide 25 as the requirements document for 

calibration and testing laboratories, and since has been adopted as the new U.S. industry 

standard. The traceability requirements remain virtually unaltered. Measurement results 

must be traceable to national standards through an unbroken chain of measurement results, 

with documented uncertainties. ANSI/NCSL Z 540-1-1994 was also reaffirmed in 2001 and 

is still in service. However, ISO/IEC 17025 takes precedence over ANSI/NCSL Z 540-1-

2001 for accreditation bodies. 

Inside the front cover of the GUM is this statement: “This Guide establishes general rules for 

evaluating and expressing uncertainty in measurement that are intended to be applicable to a 

broad spectrum of measurements.” The GUM describes the general process that should be 

followed to calculate and document the uncertainty of measurement results. The writers of

the GUM used partial derivatives and other mathematical processes requiring an extensive 

statistical background for proper implementation. In general, scale service personnel are 

unable to easily interpret and understand the GUM. (This is evidenced by the numerous 

inquiries received by WMD.) 

As a result, numerous interpretive documents have been generated to address the calculation 

of uncertainties associated with specific calibration processes. Yet none have addressed one 

of the most fundamental and common measurement processes, the calibration of scales and 

balances. Scales and balances are used extensively in commerce and in analytical processes. 

Virtually everything bought or sold has a scale or balance somewhere in the process. Too 

often, the stated uncertainty of a scale or balance calibration has reflected only the tolerance 

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Chapter 1 

or uncertainty of the calibrating mass standards, which is an extremely minor component of 

the overall measurement uncertainty. 

As stated above, compliance with ISO/IEC 17025 requires that an uncertainty estimation 

process, based on the GUM, be part of the daily operations of service organizations. Since 

few service organizations employ statisticians, NIST WMD has received a steadily 

increasing number of calls requesting assistance in establishing uncertainty statements for 

scale and balance system calibrations. 

This Recommended Guide for Determining and Reporting Uncertainties for Balances and 

Scales has been developed to provide scale and balance service organizations with a practical 

interpretation of the GUM as it relates to the processes and measurement influences routinely 

encountered by their service personnel. 

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Chapter 1 

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Chapter 2 

Chapter 2: 

General concepts 

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Chapter 2 

Introduction

The calibration of a scale or balance consists of applying known loads to the device in a 

specified order, expecting the indicator to correctly indicate the weight of the applied load. 

The purpose of a balance or scale calibration is to measure the ability of the scale or balance 

to perfectly indicate the weight of the applied load under all conditions. The purpose is not

to measure the weight of the applied standard mass artifacts. But no matter how perfectly the 

balance or scale indications appear to be, there will always be an uncertainty, or unknown 

quantity, associated with the calibration. The technician must assign a value to that unknown 

quantity for each measurement. 

To be able to assign that value, the technician must have a good understanding of the 

equipment and the measurement process being used. This will allow him to take into 

consideration all of the factors that contribute to the uncertainty of the calibration. For 

instance, a scale indicator that has the zero tracking function turned ‘on’ may cause an 

indication error if a small load is applied to the weighing pan in a very slow manner. The 

indicator may interpret some initial portion of the applied weight as scale drift and zero the 

indication. This causes an erroneous shift in the zero reference and translates into an equal 

shift at all subsequently applied loads. If a technician is not completely familiar with the 

scale indicator, he may interpret that shift as an indication error when it is actually improper 

use of the weighing device. The impact of the zero tracking function on measurement results 

depends on a number of parameters that the manufacturer deems significant. The technician 

must be aware of potential error sources and the proper method for eliminating as many as 

possible. 

There are many possible sources of uncertainty in a balance or scale calibration. Among the 

most common uncertainty contributors are: 

• The uncertainty or tolerance of the applied load, 

• repeatability of the weighing system, 

• readability, 

• reproducibility of the weighing system, and 

• the effects of: 

- temperature changes, 

- drafts or wind, 

- off center loading, 

- indicator drift, 

- electrical noise and variation 

- vibration, and 

Please note: This list is not all inclusive.

The balance or scale calibration technician must be capable of identifying those measurement 

influences that affect the measurement result and be able to estimate how each influence 

affects the balance or scale indication. These estimated quantities are then combined 

according to a documented procedure and reported as the uncertainty of the balance or scale 

calibration process. Randomly deciding to include or not include measurement influences in 

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Chapter 2 

uncertainty calculations does not comply with requirements in ISO/IEC 17025. Service 

providers must have a documented procedure that includes guidance on how decisions are to 

be made as to what is significant and must be included as part of the uncertainty. This may 

even include a requirement to describe in writing how the decisions are made at the time of 

each calibration. 

The Guide to the Expression of Uncertainty in Measurement (GUM) describes a method for 

determining the uncertainty of a calibration. 

General Description of Method 

There are eight basic steps in the process of determining the uncertainty of a calibration. 

They are: 

1. Specify the Process and equation: 

Write down a clear, concise statement of what is being measured and the relationship 

between it and the parameters on which it depends. It must be remembered that the 

weighing device calibration process measures the ability of the weighing device to 

properly represent the applied calibration load. It is not measuring the mass of the 

applied load. A possible equation statement would be: y = (mx + b)±U 

where y is the balance indication, 

m is the sensitivity of the weighing device, 

x is the applied load, 

b is the zero offset, and 

U is the assigned measurement uncertainty. 

2. Identify and characterize the uncertainty sources 

Use a Cause and Effect diagram or uncertainty budget to help identify uncertainty 

contributors. The diagram or budget provides a systematic approach for listing all of the 

measurement influence factors that can cause an error in the balance or scale indication. 

3. Quantify the resulting uncertainty components 

Looking at the list of error contributors, assign a value to each, remembering that not all 

of them will be measured in mass units. For example, a change in temperature is 

measured in degrees Celsius or Fahrenheit and must be converted to mass units in the 

next step. 

4. Convert the influences of the uncertainty components on the measurement to 

standard deviation equivalents 

Using the example of temperature identified in step 3, convert that change in 

temperature to mass units. It may be necessary to consult the scale manufacturer’s 

specifications to make conversions. All of the final values must be in terms of mass 

units. 

Another factor that may require estimation is the standard deviation of the weighing 

system (the repeatability of the device). If the process of making repeated 

measurements to calculate a standard deviation of the scale indication is not practical, 

cost-effective or feasible, as may be the case for a large capacity scale, the standard 

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Chapter 2 

deviation may be approximated by using a portion of either the vendor’s specification or 

the device readability, whichever is greater. This process is covered in detail in chapters 

3, 4 and 5. 

5. Calculate the combined standard uncertainty (uc) 

Use the root-sum-squared (RSS) method to combine the standard (one standard 

deviation) uncertainty components into a combined uncertainty value. 

2 2

5

2

4

2

3

2

2

2

1

2 ...... c p i u = s + u + u + u + u + u + + u

where sp is the process standard deviation and the terms containing 'u' are other standard 

uncertainties. 

6. Calculate the expanded uncertainty (U) 

Multiply the combined standard uncertainty by the appropriate coverage factor for the

confidence interval desired for the expanded uncertainty. Normally, k=2 for a 95 % 

confidence interval, will be used as the coverage factor. The use of other coverage 

factors requires that the selection and use be documented and justified when reported. 

7. Evaluate U for appropriateness 

Ask the following questions: 

Does the expanded uncertainty make sense? 

Is the expanded uncertainty at least two (or ‘k’, if some other coverage factor was used) 

times the largest standard uncertainty component? 

Is the expanded uncertainty large enough to encompass the normal indication errors that 

experience tells you are possible? If so, continue to step 8. If not, investigate the 

calculations for mathematical errors or go back to step 2 and re-evaluate the calibration

process looking for other causes of uncertainty that must be included. 

8. Report the uncertainty

Report the expanded uncertainty value including the ‘k’ factor used. NIST has adopted 

k=2 as the standard value to be used in reporting the uncertainty of measurement results 

reported in the United States. If another k value is used, such as k = 3 for a 99.73 % 

confidence interval, the use of the nonstandard k factor must be justified. 

Additionally, a statement of traceability that includes a description of how the 

measurement and its associated uncertainty are traceable to a national metrological 

institute (NMI), such as NIST, should be included. 

Each of these steps is expanded and explained as appropriate for the process being discussed 

in the examples in chapters 3, 4 and 5. Examples of each common uncertainty contributor 

are provided along with discussion that will enable the user to categorize each uncertainty 

contributor. 

Application

This guide provides the necessary tools to evaluate the calibration process being used, 

identify uncertainty contributors for the measurements made, quantify the impact of the 

uncertainty contributors on the measurement results, combine the uncertainty contributions in 

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Chapter 2 

a standardized manner, obtain and evaluate an expanded uncertainty, and report the 

measurement results with a properly computed, properly documented, uncertainty statement. 

Chapters 3, 4 and 5 address specific processes. Each chapter addresses some special issues 

that are typically encountered in those calibration processes. Instructions are provided for the 

most appropriate method of calculating a reasonable uncertainty for a weighing device in 

each situation. 

Chapter 6 contains sample calculations and includes the rationale that might be used in 

calculating and evaluating the resulting uncertainties. 

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Chapter 2 

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

Chapter 3: 

Balance Calibrations in a Laboratory 

Environment

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

Introduction

Laboratory balances are typically located in a controlled environment that minimizes many 

of the measurement influence factors that contribute to measurement uncertainty. 

Additionally, it is possible to perform repeated measurements due to the relatively small 

capacity of these weighing devices and the availability of suitable standards for all applied 

loads. The controlled environment and the ability to make repeated measurements simplify 

the process of calculating the calibration uncertainty. Therefore, service personnel must 

make fewer subjective decisions. 

The following steps outline the process recommended for use by balance service personnel to 

calculate the uncertainty associated with the balance calibration process. 

1. Specify the Process and Equation 

Write down a clear concise statement of what is being measured and the relationship between 

it and the parameters on which it depends. Remember, the balance indication is being tested, 

not the mass of the standard mass artifacts. 

Example: y = (mx + b)±U where; 

y is the balance indication, 

m is the sensitivity of the weighing device, 

x is the applied load, 

b is the zero offset, and 

U is the assigned measurement uncertainty. 

Ideally, b = 0 if the balance indication was properly zeroed, and m = 1 because the balance 

indicates one mass unit for each mass unit applied. 

2, 3 & 4. Uncertainty Identification, Characterization and Quantification 

The uncertainty associated with the calibration of a laboratory balance is comprised of many 

influence factors. A cause and effect diagram is used here to identify the measurement 

influence factors and to show their relationship to other factors. 

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

Sample Cause and Effect Diagram

Uncertainty

Design Installation Staff & Procedures

Standards Facility

Environment/Location

Method of Use

A Capacity ccuracy

Sensitivity Readability

Draft Shield

Off center

loading sensitivity

Design

Experience

of Installer

Repeatability Supports

Experience

Attitude

Procedure selection

Frequency

of test Training

Auto zero tracking

Reported

Uncertainty/

Tolerance

Substitution

load

Handling/

condition

Storage

Internal

standards

Use of error

weights

Temperature

stability Product

contamination

Drafts

RFI/EMI Vibration

 Static

Electricity

Range of use

Type product

weighed

Gross/net weight

Auto zero-tracking

on/off

Repair

Shock

loading

Dynamic vs static

weighing

Factors

affecting

Uncertainty

Drift

Levelness

* Not all inclusive

Draft

Shield

In addition to the influences shown on the diagram, other sources of uncertainty may include 

magnetism of weights, standard mass errors, thermal equilibration of standards, balance 

warm-up, power-line noise, etc. 

Evaluate each measurement influence for its significance. Consider an uncertainty 

component significant if it can cause a change in the value of the second most significant 

digit, leading zeros excluded, when included in the uncertainty calculations. For example, if 

the expanded uncertainty value is currently 0.052 g and including the value of an uncertainty 

contributor causes the new value to be 0.053 g, that contributor is considered significant. If 

no change results in the second significant digit, the ‘2’, the contributor is not considered 

significant for that particular calibration. When an uncertainty contributor is considered 

significant, a value for that uncertainty component must be included in the uncertainty 

calculations for that calibration process. Whether included in the uncertainty calculations or 

not, the GUM requires that the process of performing the evaluation, and the results of the 

evaluation, be documented for each known uncertainty contributor. 

Many of the influence factors contained in the Sample Cause and Effect Diagram are 

addressed in the following paragraphs. The individual(s) performing the test must decide 

how to deal with each influence, whether by avoidance, correction or inclusion as part of the 

uncertainty. 

The service staff must evaluate the environment and use of the balance and eliminate or 

minimize as many of the measurement influence factors as possible. Engineering changes 

put in place to minimize a measurement influence must remain in place at all times for the 

calibration uncertainty to remain valid.

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

Levelness of the balance should be checked and corrected if necessary. 

Drafts should be eliminated by the installation of draft shields, air diffusers or by 

redirecting air vents, or closing doors. These draft elimination measures must be in 

place at all times (including during use) for the calibration uncertainty statement to 

remain valid during daily use. 

Off-center loading errors should be evaluated and corrected before recording final 

calibration data. 

Thermal equilibrium of the balance should be ensured by maintaining the balance in an 

energized state (i.e., turned-on) for sufficient time to ensure that all circuitry and 

hardware have reached a stable temperature. 

Vibration sources must be identified and eliminated or minimized to limit the vibration 

levels to which the weighing device is exposed during daily use, as well as during 

calibration. 

Zero-tracking features should be disabled to avoid undetectable zero indication errors. 

Zero-tracking is designed to eliminate minor changes in the balance indication due to 

drift of the weighing system and may be turned 'on' to maintain the 'zero' indication of 

the weighing device. Due to the correction limits included in the system programming 

regarding quantity and motion detection, detrimental effects on measurement results are 

typically not evident. However, service personnel may encounter times while attempting 

to load the pan that the zero tracking circuitry will cause an offset in the indication, 

particularly if placing weights in a very slow, gentle manner. While this would be an 

unusual occurrence, service personnel must be aware of the possibility of such errors and 

take steps to prevent them from occurring. 

Mass Standards must be brought into close proximity to the balance, even installed 

inside the balance chamber where possible, to ensure thermal equilibrium with the 

balance and surrounding environment prior to calibration. This becomes more critical 

for balances and weights of higher or improved accuracy. Table B.2 of the 2002 Draft 

of OIML R 111, the OIML specification for weights, recommends that weights be 

permitted to undergo thermal stabilization for as long as 79 hours, depending on the 

desired use, nominal mass, weight accuracy class and temperature differential. 

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

Excerpt from Table B.2 * 

Thermal stabilization in hours 

∆T* Nominal Value Class E1 Class E2 Class F1 Class F2

1 000, 2 000, 5 000 kg - - 79 5 

100, 200, 500 kg - 70 33 4 

10, 20, 50 kg 45 27 12 3 

1, 2, 5 kg 18 12 6 2 

100, 200, 500 g 8 5 3 1 

10, 20, 50 g 2 2 1 0.5 

± 20°C 

< 10 g 1 1 1 0.5 

* OIML R 111-2002 Draft 

Failure to allow the weights to sufficiently stabilize prior to beginning balance 

calibrations can cause significant errors in the measurement of the applied mass. This is 

because thermal air currents are generated around the balance weighing pan due to the 

temperature variations within the measuring system. Only adequate thermal stabilization 

times will prevent this phenomenon from occurring. The service personnel should apply 

stabilization considerations similar to those in OIML R 111 when calibrating balances 

requiring the use of OIML Class E2, F1, F2, and ASTM Class 1, 2 or 3 weights. 

Standard mass errors can be minimized by using the mass values reported during 

calibration of the mass artifacts. 

Balance indication drift errors should be corrected to the maximum extent possible. 

Though slight drift in the indicated value is somewhat normal, significant indication drift 

may be a sign of thermal instability of the weighing system. Allowing a longer 

temperature stabilization period for the weighing device and standard mass artifacts may 

minimize drift due to thermal instability. Exercising the weighing system may also act 

to minimize drift. Repeatedly applying and removing loads within the range of the 

device causes the electronic and mechanical components of the system to acquire a 

thermal equilibrium of use. If indication drift is excessive and cannot be eliminated by 

longer warm-up times or exercising the weighing system, other corrective action must be 

taken. 

Magnetic fields surrounding weights or weight handling devices can cause severe, yet 

difficult to detect, errors in the indication generated by an applied mass load. Whenever 

possible, mass artifacts and handling devices made of non-magnetic materials should be 

used. When influences due to magnetic fields are suspected, measurements using 

different weight artifacts should be performed. Magnetic effects may be detected by 

making a series of mass measurements of the weight, half of the measurements with the 

weight in an upright position and half of the measurements with the weight inverted. 

Magnetism is probable if a detectable difference, greater than the repeatability of the 

balance, is noted between the average of the upright indications and the average of the 

inverted indications. If magnetism is suspected, weights should be tested more 

extensively for magnetic fields. A further description of the test for magnetism is 

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

beyond the scope of this document. Use weights known to be non-magnetic for 

calibration of balances using the force balance principle. 

RFI/EMI susceptibility tests should be performed. Changes in indication due to 

random RFI/EMI influences can be difficult to detect during normal use, and may be 

interpreted as repeatability errors, drift or erratic readings. Corrective action should be 

taken if RFI/EMI susceptibility is detected. 

Note: RFI/EMI sources include two-way radios, cell phones and other electronic devices. Laboratories

are available specializing in detecting and measuring RFI/EMI. 

Power-line noise or variations can cause random display indications to occur. The 

specific cause of these random indications may be difficult to determine, but they will 

affect the repeatability of the weighing system. Where possible, it is best that balances 

be powered by a dedicated power circuit or by an AC line conditioner to prevent these 

measurement influences. 

Operator errors result when individuals are inadequately trained. All operators of 

weighing devices should have proper training and be knowledgeable about the weighing 

instrument and the process in which it is used. Inadequately trained personnel may 

record data with significant errors, improperly influencing critical process decisions. 

Weighing system operators must be equipped with correct and complete work 

instructions to minimize the likelihood of operator error. 

Careful planning or suitable corrective action will minimize or eliminate many balance 

measurement uncertainty sources. However, there are several uncertainty influences that 

cannot be eliminated. The results of these influences must be included in the calibration 

uncertainty calculations. Each are discussed in the following paragraphs. 

Process standard deviation (sp) 

The source of a value for sp can be based on balance repeatability, balance reproducibility or 

the size of a balance division. Each method is discussed below. 

sp from measured repeatability 

Repeatability is a measure of a balance’s ability to produce the same indication every 

time the same weight is placed on the sensing device. Repeatability is presented as a 

standard deviation. Obtaining repeated measurement results is possible in the laboratory 

due to the small size and ready availability of the weights used. Repeatability tests 

should be conducted with weights approximating the typically measured load. If a wide 

range of load values are normally measured, evaluation of repeatability at several loads, 

e.g., 50 % capacity and 100 % capacity, may be desirable. If the repeatability value at 

one load is more than two times the value at a second load, it may be desirable to report 

the repeatability results at specific load ranges of the balance. If only a single 

repeatability value is to be reported, the largest measured value must be used. 

Seven or more measurement results of a weight are required to calculate the repeatability

or standard deviation of the balance at a given load. Increasing the number of 

measurement results provides greater confidence in the value. The calculated standard 

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

deviation obtained in this manner is a measure of the weighing system’s ability to repeat 

measurement results over a short time interval and does not represent the long-term

reproducibility. 

Due to balance resolution, it is possible to make seven or more measurements with every 

measurement resulting in the same value. A true balance standard deviation of zero is 

not statistically possible, though the standard deviation may be less than one display 

increment (d). In this situation, assuming that a control chart of a check standard is not 

available, the standard deviation of the balance can be estimated as d d

p 0.577

3

s = =

(0.577 times the value of ‘d’) as described in the GUM. This method of calculating the 

sp value is recommended only if the result of a series of repeatability weighings is zero. 

If it is possible to perform a discrimination test as described in NIST Handbook 44, 

N.1.5 and T.N.7.1 or OIML R 76-1, A.4.8, another equation may be used. The equation 

d ds p 0.29

2 3

1 = = (0.29 times the value of ‘d’) can be used as it has been proven that 

the true value is statistically most probable to lie near the center of the scale display 

interval. This estimation equation should only be used when the calculated standard 

deviation is zero and the balance has passed a discrimination test as described in NIST 

Handbook 44, N.1.5 and T.N.7.1 or OIML E 76-1, A.4.8. 

sp from measured reproducibility 

When the uncertainty of the balance during routine weighing operations is desired, a 

check standard and control chart should be used to determine the long-term repeatability 

of the balance under varying operational conditions. This is called reproducibility and is 

also evaluated as a standard deviation. Reproducibility is a measure of long-term

repeatability and may be used in place of the short-term repeatability when calculating

measurement uncertainty. The standard deviation value of the control chart will be 

included in the combined uncertainty as sp. The load chosen for use as the check 

standard should represent the items typically weighed on the balance. 

When using values from control charts to determine the sp value, attention must be given 

to the range of measurement results over which that value has been determined. When 

readings from a wide range of measurement results are included in the calculation of the 

sp value and the resulting standard deviation is acceptable, an uncertainty calculated 

from that standard deviation is valid for the range over which the measurement results 

were obtained. However, it is possible that the calculated uncertainty is greater than 

desired or greater than is acceptable. Repair of the weighing device may not correct the 

problem. For those situations, it may be desirable to divide the data into measuring 

ranges and to calculate the standard deviation of each of several narrower measuring 

ranges. These standard deviations are applicable to the measuring range over which the 

data was obtained. Especially in a calibration facility, a balance may have multiple sp

values each covering a different range of the balance capacity. The existence of multiple 

sp values requires the calculation of expanded uncertainty values for each assigned 

measuring range. 

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Chapter 3: 

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sp from manufacturer’s specification 

The value for sp is best determined from repeated readings of a weight, whether shortterm or long-term. The manufacturer’s specification sheet is not a recommended source 

for determining the uncertainty contribution due to repeatability. Manufacturers'

specifications are established for a specific set of conditions that may not be 

representative of the actual environment in which the scale is tested and used. 

Manufacturers' specifications are an excellent tool when comparing the expected 

performance of one weighing device to another similar device, but they are established 

to indicate the expected performance of a family of weighing systems, not a specific 

weighing system. As the calibration uncertainty is being estimated for a specific

weighing system, it is best to establish the calibration uncertainty contribution due to 

repeatability from measured data, not from an expected performance parameter. 

Manufacturers' specifications should not be used to estimate calibration uncertainty. 

Uncertainty of the standards (us) 

The uncertainty associated with the standard mass artifacts, us, may come from one of several 

possible sources. us may be based on the calibration uncertainty assigned to the calibrated 

value of the standard mass artifact as taken from the calibration report, or it may be based on 

the tolerance to which the standard mass has been verified. A calibration load may include 

one mass artifact or multiple mass artifacts. Each of these situations requires a different

method for determining the proper way to calculate the us value.

us from mass artifact calibration uncertainty 

When using the reported calibration value of the standard mass artifact in the calibration 

of the balance, calculate the uncertainty of the mass standard (us) from the uncertainty 

associated with the reported values of the individual mass artifact. Normally, standard 

mass uncertainties are reported as expanded uncertainties with a stated coverage factor

(k). Divide the expanded uncertainty of the standard mass artifacts by the reported 

coverage factor (k) to obtain the standard uncertainty for the mass artifact. 

us from mass artifact calibration tolerance 

When basing the standard mass uncertainty (us) on the tolerance of the standard mass 

artifact, treat the tolerance as having a uniform probability distribution. The standard 

mass uncertainty calculated from the standard mass calibration tolerance is 

0.577( ). 3tolerance toleranceus = = 

Dealing with Multiple standards: Additional Guidance 

When using multiple weights to generate a single load the most conservative value for the 

uncertainty of the mass calibrations (us) is the linear sum of the standard uncertainty 

associated with each weight. For this situation us = usm + usm + usm + + usmi ..... 1 2 3 . While this 

method provides the most conservative estimate of standards uncertainty, it also typically 

provides the largest estimate for the standards uncertainty and may result in an uncertainty 

value that is greater than is allowed by the customer's weighing process. 

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Chapter 3: 

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A less conservative method, the root-sum-squared (RSS) method, results in a smaller 

estimated standard mass uncertainty value. The equation for the RSS method is 

2 2

3

2

2

2

1 ...... s sm sm sm smi u = u + u + u + + u . This method of combining the standard uncertainties of 

the mass standards requires that you prove independence between the reported value of each 

mass standard. Values are independent when no portion of the value of one weight affects 

the value of another. Obtain information about mass value independence from the calibrating 

organization. When the user is unsure of the independence of values, he must assume that the 

values are dependent and use the sum of the individual uncertainties for us. 

5. Calculating the combined uncertainty 

The combined uncertainty, ‘uc’, of the weighing system calibration is calculated as the rootsum-squared of the influence factors. 

2 2

3

2

2

2

1

2 2 .... c s p u u u ui u = u + s + + + + + where ui are any other uncertainty 

components that the scale technician includes in the uncertainty calculations. 

Remember: 

- All of the uncertainty components must be in like terms of the mass units. Differing 

units, such as °C and mg, cannot be combined. Convert the impact of the non-mass units 

to appropriate mass units to calculate the uncertainty. 

- All uncertainty components must be in terms of standard (one standard deviation) 

uncertainties. 

6. Calculating the expanded uncertainty 

The expanded uncertainty, ‘U’, is calculated by multiplying the value obtained for uc by the 

coverage factor, ‘k’, for the confidence interval to be used. By convention, as defined in 

NIST Technical Note 1297, “Guidelines for Evaluating and Expressing the Uncertainty of 

NIST Measurement Results”, a coverage factor of k = 2 is used for a confidence interval of 

approximately 95 %. The use of k values other than k = 2 requires an explanation of the 

reason for the deviation from convention. 

7. Evaluating the expanded uncertainty 

There are several things to consider when evaluating the uncertainty. First, does the final 

expanded uncertainty make sense? The expanded uncertainty must be at least ‘k’ times

greater than the largest uncertainty component included. 

Second, does the calculated expanded uncertainty seem appropriate for the tested device? An 

uncertainty value seems unreasonable and should be investigated if the expanded uncertainty 

is calculated to be significantly less than one balance division. Likewise, if the calculated 

uncertainty is excessively large, that value should be investigated. Investigation should 

include verifying the use of proper evaluation techniques. Ensure that arithmetical errors 

were not the cause of the questionable uncertainty value. 

Third, does the calculated uncertainty meet the requirements for weighing with that weighing 

device? Different quality systems have different requirements concerning the allowable 

measurement uncertainty; requirements may state that the uncertainty of a measuring device 

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

may not exceed 33 %, 25 %, or even as little as 10 %, of the tolerance of the object being 

tested. Will the calculated uncertainty meet the requirements that are in place? If not,

reexamine the entire calibration process for uncertainty contributors that can be reduced. 

Reduction may be accomplished by selection of more accurate standards, repair of the 

weighing device to obtain a smaller standard deviation, or perhaps making multiple 

measurements to determine the true repeatability of a device rather than using an estimated 

repeatability. Evaluate each uncertainty contributor, beginning with the most significant, to 

determine how it can legitimately be reduced until the required uncertainty level is obtained 

or until the decision is made that the weighing device cannot meet the quality requirements 

and must be replaced or that the weighing system must be moved to a more hospitable 

environment. 

Remember, the reported calibration uncertainty is valid only for the environment in which it 

is determined. Temporarily improving the environment for the calibration, then returning it 

to the prior conditions for normal weighing operations invalidates the uncertainty statement. 

Engineering changes made to reduce calibration uncertainty must remain in place during 

normal weighing operations for the calibration uncertainty to remain valid. 

8. Reporting Uncertainty

Reporting the expanded uncertainty value is no longer a matter of simply stating that a 

measurement result is ‘x ± y’ where ‘x’ is the reported value and ‘y’ is the expanded 

uncertainty. The Guide to the Expression of Uncertainty in Measurement requires that you 

identify the various components of the uncertainty. You must also explain why that 

component was included and how it was evaluated. Specifically, the GUM provides a test of 

the stated uncertainty statement: “Has enough information been provided in a sufficiently 

clear manner that the result can be updated in the future if new information or data became 

available?” Another test is to ask: “Would another individual, not associated with the 

measurement process, be able to understand how the stated uncertainty was calculated and 

what was included, and then properly apply it to his/her own uncertainty calculation?” 

It must be understood that the measurement uncertainty must be calculated for each balance 

test load. The needs of the customer will determine whether a table format is used to report 

the uncertainty at each load, or the maximum uncertainty value of all test loads is reported as 

a single value covering the entire range of the weighing system. In either case, the 

uncertainty at each test load must be calculated and documented. 

It is very important that uncertainty calculations be understandable so that the user can 

properly include the uncertainty components in their process uncertainty calculations. Some

components of the uncertainty need to be addressed differently than other components, and 

sufficient information must be available to make that possible. For example, when 

calculating the uncertainty of his production process the customer would likely not wish to 

include reproducibility if the reproducibility value from a control chart of the measuring 

process was used to calculate the uncertainty of the balance calibration. To include the 

reproducibility value a second time would needlessly increase the expanded uncertainty of 

the production measurement process. Sufficient information must be provided to the user so 

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

that it is clear that the reproducibility should be included only once, in the originally reported 

balance uncertainty. 

The importance of a clear, understandable and defensible uncertainty statement, based on a 

properly documented set of instructions, cannot be emphasized too strongly. The uncertainty 

statement is a critical portion of the calibration process. Write the uncertainty statement so 

that a user can make informed decisions concerning proper use of the uncertainty 

components provided. 

The following tables provide a quick reference guide to the proper equation to use for 

calculating the calibration uncertainty in most common situations. They also provide an 

example of a form that might be used to document the calibration uncertainty. 

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

Quick Guide for Laboratory Balance Calibration Uncertainties 

Reference information: 

Process (sp) Source (in order of desirability): Distribution 

Type 

Value to 

use: 

1 Calculated standard deviation at the test load from an upto-date control chart Normal As calculated

2 Calculated standard deviation from 10 or more readings of 

the same load over a short period of time

Normal As calculated

Standards (us) Sources 

(in order of desirability): 

Distribution 

Type 

Value to 

use: 

1 Reported k=1 uncertainty from Report of Test Normal As calculated

2 Tolerance of weight(s) used Uniform

0.577 x 

tolerance 

Additional equations when using multiple standards 

A Multiple standard masses (if independence is proven) 2 2

3

2

2

2

1 ...... s sm sm sm smi u = u + u + u + + u

B Multiple standard masses (if independence is unknown) ..... . us = usm1 + usm2 + usm3 + + usmi

Additionally, there may be other known measurement uncertainty contributors. Evaluate the 

calibration process carefully to ensure that all significant contributors are properly included 

in the uncertainty calculations. Consult the text of this Guide for additional guidance.

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Chapter 3: 

Balance Calibrations in a Laboratory Environment 

Equipment Identification

Manufacturer Model Serial ECN Other

Uncertainty Worksheet 

at _____________________ Test Load 

Uncertainty Influence Description Identifier Estimated 

value 

Distribution 

type 

(Normal/Uniform)

Estimated 

Standard 

Uncertainty

Uncertainty of the standards used us

Standard deviation of the process 

(Determined at ___________ test load) sp

Uncertainty due to u1

Uncertainty due to u2

Uncertainty due to u3

Uncertainty due to u4

Uncertainty due to u5

Uncertainty due to u6

Uncertainty due to u6

Uncertainty due to u7

Combined standard uncertainty 

2 2

5

2

4

2

3

2

2

2

1

2 2 .... c s p u u u u u u i u = u + s + + + + + + + uc = 

Expanded uncertainty

U = k(uc), k = ________ k = 2 is recommended U = 

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Chapter 4: 

Scale Calibrations Performed in an Uncontrolled Environment 

Chapter 4: 

Scale Calibrations Performed in an 

Uncontrolled Environment 

(E.g., a truck scale in a quarry or steel mill) 

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Chapter 4: 

Scale Calibrations Performed in an Uncontrolled Environment 

Introduction

This chapter identifies many of the measurement influences affecting scale calibrations in an 

uncontrolled environment and provides instructions on how to include the resulting 

uncertainty contributors in a calculated expanded uncertainty for the calibration. It is not 

possible to discuss all possible measurement influences, but the examples will provide 

adequate guidance to enable the service personnel to make informed decisions when 

calculating calibration uncertainty. 

Scales in an uncontrolled environment 

Large capacity scales, such as truck scales or floor scales, are often installed out-of-doors in 

an uncontrolled environment. Thus, many influences affect the calibration uncertainty. 

Many of these influences are beyond the ability of service personnel to control, so they must 

be identified and evaluated for the impact they have on the calibration process. 

The following steps outline the process recommended for use by scale service personnel to 

calculate the uncertainty associated with the scale calibration process. 

1. Specify the Process and Equation 

Write down a clear concise statement of what is being measured and the relationship between 

it and the parameters on which it depends. Remember, the scale indication is being tested, not 

the mass of the standard mass artifacts. 

Example: y = (mx + b)±U where; 

y is the scale indication, 

m is the sensitivity of the weighing device, 

x is the applied load, 

b is the zero offset, and 

U is the assigned measurement uncertainty. 

Ideally, b = 0 if the scale indication was properly zeroed, and m = 1 because the scale 

indicates one mass unit for each mass unit applied. 

2, 3 & 4. Uncertainty Identification, Characterization and Quantification 

The uncertainty associated with the calibration of a laboratory balance is comprised of many 

influence factors. A cause and effect diagram is used here to identify the measurement 

influence factors and to show their relationship to other factors. 

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Chapter 4: 

Scale Calibrations Performed in an Uncontrolled Environment 

* Not all inclusive

Sample Cause and Effect Diagram

Uncertainty

Design Installation Staff & Procedures

Standards Facility

Environment/Location Method of Use

A Capacity ccuracy

Sensitivity Readability

Draft Shield Off center

 loading sensitivity

Design

Experience

of installer

Repeatability

Foundation/

 Supports

Experience

Attitude

Procedure selection

Frequency

of test Training

Auto zero tracking CLC

Reported

Uncertainty/

Tolerance

Substitution

load

Handling/Condition

Storage

Internal

Standards

Use of error

weights

Temperature

stability Product

contamination Snow

Wind

Drafts

Vibration

RFI/EMI

Rain

Range of use

Type product

weighed

Gross/net weight

Auto zero-tracking

on/off

Repair

Shock

loading

Dynamic vs static

weighing

Factors

affecting

Uncertainty

Draft

Shield

In addition to the influences shown on the diagram, other possible causes of uncertainty are 

standard mass errors, thermal equilibration of standards, scale warm-up, power-line noise, 

etc. 

Evaluate each uncertainty influence for its significance. Consider an uncertainty component 

significant if it can cause a change in the value of the second most significant digit, leading 

zeros excluded, when included in the uncertainty calculations. For example, if the expanded 

uncertainty value is currently 0.52 kg and including the value of an uncertainty contributor 

causes the new value to be 0.53 kg, that contributor is considered significant. If no change 

results in the second significant digit, the ‘2’, the contributor is not considered significant for 

that particular calibration. When an uncertainty contributor is considered significant, a value 

for that uncertainty component must be included in the uncertainty calculations for that 

calibration process. Whether included in the uncertainty calculations or not, the GUM 

requires that the process of performing the evaluation, and the results of the evaluation, be 

documented for each known uncertainty contributor. 

Many of the influence factors contained in the Sample Cause and Effect Diagram are 

addressed in the following paragraphs. The individual(s) performing the test must decide 

how to deal with each influence, whether by avoidance, correction or inclusion as part of the 

uncertainty. 

The service staff must evaluate the environment and use of the scale and eliminate or 

minimize as many of the measurement influence factors as possible. Engineering changes 

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Chapter 4: 

Scale Calibrations Performed in an Uncontrolled Environment 

put in place to minimize a measurement influence must remain in place at all times for the 

calibration uncertainty to remain valid.

Levelness of the scale should be checked and corrected if necessary. 

Wind influences are typically very difficult and expensive to eliminate for large scales

in an uncontrolled environment. However, any controls in place for the calibration must 

remain in place for all weighing operations. Guidance for including an uncertainty 

component for wind influences is provided later in this chapter. 

Off-center loading errors should be evaluated and corrected before recording final 

calibration data. 

Thermal equilibrium of the scale should be ensured by keeping the scale in an 

energized state (i.e., turned-on) for sufficient time to ensure that all circuitry and 

hardware have reached a stable temperature. 

Vibration sources must be identified and eliminated or minimized to limit the vibration 

levels to which the weighing device is exposed during daily use, as well as during 

calibration. 

Zero-tracking features should be disabled to avoid undetectable zero indication errors. 

Zero-tracking is designed to eliminate minor changes in the scale indication due to drift 

of the weighing system and may be turned 'on' to maintain the 'zero' indication of the 

weighing device. Due to the correction limits included in the system programming 

regarding quantity and motion detection, detrimental effects on measurement results are 

typically not evident. However, service personnel attempting to load the load receiver 

may at times find that the zero tracking circuitry will cause an offset in the indication, 

particularly if placing weights in a very slow, gentle manner or when using a weight cart 

to load the scale. While this would be an unusual occurrence, service personnel must be 

aware of the possibility of such errors and take steps to prevent them from occurring. 

Mass Standards must be brought into close proximity to the scale to ensure thermal 

equilibrium with the scale and surrounding environment. Thermal stabilization times for 

weights typically used for a scale in this environment are not specified, but sufficient 

time must be allowed to ensure that condensation is not present on the weights. 

Mass standard error effects are minimized by using the mass values determined during 

calibration of the mass artifacts. For most large capacity scales this will not be 

necessary as the tolerances of the scales and standard masses have been set to minimize

the effects of the standard mass errors. The concept is presented here as an option 

should correction be desirable. The standard mass uncertainty for large capacity scales 

is normally based on the tolerance of the mass standards. 

However, if the mass corrections of the standard masses are used, the reported 

calibration uncertainty of the mass calibration must be used as us. The complete 

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Chapter 4: 

Scale Calibrations Performed in an Uncontrolled Environment 

calculation of us is covered in a later section of this chapter. Additionally, the length of

re-calibration intervals and care and handling of weights directly affects the validity of

the mass corrections determined at calibration. Mass corrections for weights that are 

abused or stored in harsh environments, such as on an open truck bed without benefit of 

cover, may no longer be valid. Thus, mass corrections must be used with discretion. 

 

Scale indication drift errors should be corrected to the maximum extent possible. 

Though slight drift in the indicated value is somewhat normal, significant indication drift 

may be an indication of thermal instability of the weighing system. Allowing a longer 

temperature stabilization period for the weighing device and standard mass artifacts can 

minimize drift due to thermal instability. Exercising the weighing system may also 

minimize drift. Repeatedly applying and removing loads within the range of the device 

causes the electronic and mechanical components of the system to acquire a thermal 

equilibrium of use. If indication drift is excessive and cannot be eliminated by the 

means described, other corrective action must be taken. 

RFI/EMI susceptibility tests should be performed. Changes in indication due to 

random RFI/EMI influences can be difficult to detect during normal use and may be 

interpreted as repeatability errors, drift or erratic readings. Corrective action should be 

taken if RFI/EMI susceptibility is detected. 

Note: RFI/EMI sources include two-way radios, cell phones and other electronic devices. Laboratories

are available specializing in detecting and measuring RFI/EMI. 

Power-line noise or variations can cause random display indications to occur. The 

specific cause of these random indications may be difficult to determine, but they will 

affect the repeatability of the weighing system. Where possible, it is best that scales be 

powered by a dedicated power circuit or by an AC line conditioner to prevent these 

measurement influences. 

Operator errors result when operators are inadequately trained. All operators of

weighing devices should have proper training and be knowledgeable about the weighing 

instrument and the process in which it is used. Inadequately trained personnel may 

record data with significant errors, improperly influencing critical process decisions. 

Weighing system operators must be equipped with correct and complete work 

instructions to minimize the likelihood of operator error. 

Careful planning or suitable corrective action will minimize or eliminate many scale 

measurement uncertainty sources. However, there are several uncertainty influences that 

cannot be eliminated. The results of these influences must be included in the calibration 

uncertainty calculations. Each are discussed in the following paragraphs. 

Process standard deviation (sp) 

The source of a value for sp can be based on scale repeatability, scale reproducibility, the size 

of a scale division or a manufacturer’s specification. Each are discussed below. 

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sp from measured repeatability 

Repeatability is a measure of a scale’s ability to produce the same indication every time

an identical load, under the same conditions, is placed on the sensing device. 

Repeatability is presented as a standard deviation. The most desirable method for 

determining repeatability is to make a number of measurements at a load that is typical 

of routine weighing operations. At least seven measurement results of a repeated 

specific scale load are required to calculate the standard deviation of a scale. Typically, 

due to time, energy or cost constraints, service personnel do not perform repeatability 

testing of large capacity scales. Instead, a method for estimating the repeatability is 

required. 

It is possible that when performing repeatability testing, due to scale resolution, the 

same value is obtained for every measurement resulting in a standard deviation of zero. 

A true scale standard deviation of zero is not statistically possible, though the standard 

deviation may be significantly less than one display increment (d). In this situation, if a 

control chart of a check standard is not available, the standard deviation of the scale can 

be estimated from the scale display resolution as described below. 

sp from display resolution 

The repeatability may be estimated from display resolution as d d

s p 0.577

3 = = (0.577 

times the value of ‘d’) as described in the GUM. This equation assumes that no 

discrimination test, as described in NIST Handbook 44, N.1.5 and T.N.7.1 or OIML R 

76-1, A.4.8, has been performed to evaluate the scale response to scale load changes 

smaller than one scale division. 

If it is possible to perform a discrimination test, a slightly different equation may be 

used. That equation is: d d

s p 0.29

2 3

1 = = (0.29 times the value of ‘d’) since passing a 

discrimination test indicates that the true value most probably lies near the center of the 

scale display interval. This estimation equation should only be used when the calculated 

standard deviation is zero, or the sp is being estimated from display resolution, and the 

scale has passed the discrimination test. 

sp from measured reproducibility 

When desiring to ascertain the uncertainty of the scale during routine weighing 

operations, a check standard and control chart should be used to determine the long-term 

repeatability of the scale under varying operational conditions. This is called 

reproducibility, also evaluated as a standard deviation. Reproducibility is a measure of 

long-term repeatability and may be used in place of the short-term repeatability when 

calculating measurement uncertainty. The standard deviation of the control chart values

will be used in the combined uncertainty as sp. Multiple check standard loads may be 

required when the scale is used to measure loads that vary widely within the range of the 

scale. The check standard loads should be selected to represent the loads typically 

weighed on the scale. 

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When using values from control charts to determine the sp value, attention must be given 

to the range of measurement results over which that value has been determined. When 

readings from a wide range of measurement results are included in the calculation of the 

sp value, an uncertainty calculated from that standard deviation is valid for the range 

over which the measurement results were obtained. However, the calculated uncertainty 

is sometimes greater than desired, or greater than is acceptable, and repair of the 

weighing device may not correct the problem. For those situations, it may be desirable 

to divide the data into measuring ranges and to calculate the standard deviation for the 

narrower measuring ranges. These standard deviations are applicable to the measuring 

range over which the data was obtained. The existence of multiple sp values requires the 

calculation of expanded uncertainty values for each affected measuring range. 

sp from manufacturer’s specification 

The value for sp is best determined from repeated readings of a weight, whether shortterm or long-term. The manufacturer’s specification sheet is not a recommended source 

for determining the uncertainty contribution due to repeatability. Manufacturers'

specifications are established for a specific set of conditions that may not be 

representative of the actual environment in which the scale is tested and used. 

Manufacturers' specifications are an excellent tool when comparing the expected 

performance of one weighing device to another similar device, but they are established 

to indicate the expected performance of a family of weighing systems, not a specific 

weighing system. As the calibration uncertainty is being estimated for a specific

weighing system, it is best to establish the calibration uncertainty contribution due to 

repeatability from measured data, not from an expected performance parameter. 

Manufacturers' specifications should not be used to estimate calibration uncertainty. 

Uncertainty due to display resolution (udr)

When estimating the value of sp display resolution, an additional uncertainty component for 

indicator resolution must be included. This value (udr) will also be equal to 

d dudr 0.29

2 3

1 = = (0.29 times ‘d’). This uncertainty component will not be included in the 

uncertainty calculations if repeated measurement results were used to determine a value for 

sp. The effect of the display resolution uncertainty component will already have been part of 

the measured values and an additional uncertainty component is not required. 

Uncertainty of the standards (us) 

The uncertainty contribution of the standard masses used to perform the scale calibration 

may be estimated from one of two sources, the tolerance to which the masses have been 

tested, or the uncertainty associated with the calibration of the masses. Specific guidance 

follows. 

us from standard weight tolerances 

When calibrating large capacity scales, the nominal values of the standard weights are 

typically used. In this situation, a portion of the tolerance to which the weights were 

tested is used as the standard uncertainty of the standards value, us, when calculating the 

calibration uncertainty. Only a portion of the tolerance is used as the tolerance is 

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considered to follow a uniform probability distribution. The standard uncertainty of the 

standards is calculated as 

3

tolerance

us = or (0.577 times the tolerance of the applied 

weights). When using multiple weights, the tolerance used in the us calculation will be 

the sum of the tolerances for all of the weights used. 

us from calibration uncertainties 

Standard mass uncertainties normally are reported as expanded uncertainties with 

reported coverage factor (k). This expanded uncertainty must be divided by the stated 

coverage factor to obtain the standard uncertainty for the mass artifacts (us). 

us with multiple standard weights 

When multiple weights are used, the most conservative value for the uncertainty of the 

mass calibrations is the linear sum of the uncertainties associated with each of the 

weights used simultaneously. For this situation us = usm + usm + usm + + usmi ..... 1 2 3 . While 

this method provides the most conservative estimate of standards uncertainty, it also 

typically provides the largest estimate for the standards uncertainty and may result in an 

uncertainty value that is greater than allowed by the customer's weighing process. 

When a smaller uncertainty value is required and sufficient information about the weight 

calibrations is available, the root-sum-squared (RSS) value of the individual weight 

uncertainties may be used, but only if independence of the mass values is proven. 

Values are independent when no portion of the value of one weight affects the value of 

another. (If the same standard mass was used to calibrate each of the weights being 

used, the calibrations are not independent since the uncertainty and error of the 

standard’s value contribute to the error and uncertainty for each of the weights. This is 

typically the case for weights on a large scale service truck as they are usually all 

calibrated at the same time.) Information concerning independence must be obtained 

from the calibrating organization. If the user is unsure of the independence of values, it 

should be assumed that the values are not independent. Thus, the sum of the individual 

uncertainties must be used. The RSS value of us is to be used only if the actual 

measured values of recently calibrated mass artifacts were being used in the calibration 

of the scale. The equation for the RSS method is 2 2

3

2

2

2

1 ...... s sm sm sm smi u = u + u + u + + u . 

Wind (uw) 

The influence of the wind on large capacity scale systems is a very common contributor to 

uncertainty since most large capacity scales are installed in an outdoor environment. 

Depending on wind speed, direction, and the effect of nearby structures, it is possible to 

experience scale indication fluctuations many times the scale tolerance. In these situations, 

the service technician has to make a judgement call as to whether or not calibration is 

possible. This situation requires balancing the cost of re-scheduling the test against the 

ability to make a reasonable estimate of the true scale reading. It is recommended that the 

scale test be rescheduled for a time when the wind is less of an influence, thus minimizing 

the scale indicator fluctuations. 

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uw for scales with a recording system available 

If a technician decides to proceed with a scale calibration that is being affected by wind 

influences, an estimate of the additional uncertainty involved in the measurement must 

be included in the uncertainty calculations. Unless some means of recording a large 

number of scale indications over a period of time is available, the uncertainty due to 

wind influences is not a value that can be derived from statistical methods. When an 

indication recording system is available, an average and standard deviation of thirty or 

more readings should be calculated for each applied load value. The average of the 

measurement results will be reported as the scale indication and the standard deviation 

of the values will be included as part of the calibration uncertainty (uw). 

uw for scales without a recording system 

In situations where a recording device is not available, a scale technician will typically 

observe the weighing system indicator for a time and mentally calculate an average 

value based on his observations. The scale technician will report this estimated average 

as the scale indication for that applied load. The technician must also mentally note the 

maximum and minimum values observed. The reported scale indication will have an

uncertainty equal to some portion of the observed fluctuations. As the difference 

between the estimated average and each observed scale indication is a fixed number of 

scale divisions, this uncertainty influence has a uniform probability distribution. The 

uncertainty will be estimated as w

w w 0.577

3

u = = , (0.577 times w) where w is the 

maximum difference between the reported scale indication and any observed scale 

indication. 

Vibration 

In the absence of wind effects that mask many uncertainty contributors, other influences, 

such as vibration, become more of a factor. Potential vibration sources include blasting, rock 

crushers, heavy equipment, railroad tracks, highways with heavy vehicular traffic, etc. An 

approach similar to that described for wind effects should be applied to suspected vibration 

effects and an uncertainty (uv) included in the calibration uncertainty calculations. 

Rain 

The effect of rain on a scale calibration is quite difficult to quantify, thus calibrations where 

rain is a factor are to be avoided. Each weight artifact used in a calibration will have its mass 

changed by an amount related to the quantity of water accumulated on its surface. In the 

calibration laboratory, large errors have been measured for mass artifacts that were covered 

with a layer of water. Additionally, the zero indication of the scale itself will vary as the 

quantity of water pooled on, or absorbed into, the scale deck varies during the test. This can 

also be a factor as a scale deck dries after the rain stops. Because this error is quite variable, 

depending on the amount of water accumulated, it is not possible to develop a correction that 

can be applied to a scale indication. For the same reason, developing an uncertainty factor 

for the effect of accumulated water is not possible. Calibration of scales where water 

accumulation is a factor is to be avoided. 

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Substitution loads 

Substitution loading involves applying a known load to a weighing system, removing that 

known load and then loading the weighing system with uncalibrated material to obtain a 

reading identical to that of the known load. The known load is then reapplied, along with the 

substitution load, effectively doubling the known load. According to current procedures, this 

process can be repeated two more times for a total of three substitutions, effectively 

multiplying the calibrated load by a factor of four. The substitution process results in an 

increased uncertainty as the substitution loads are being weighed by the system being 

calibrated. The uncertainty attributed to each substitution is equal to the repeatability factor 

(sp) calculated previously. Thus, the uncertainty contribution due to substitution tests (usub) 

will be calculated as the number of substitutions times the calculated repeatability value. The 

result is included in the uncertainty as usub. 

Strain Load Tests 

Strain load testing involves loading the scale with an unknown load, setting the scale 

indication with the load as the reference, followed by applying additional known load to the 

scale deck. The additional uncertainty associated with a properly performed strain load test 

is negligible, because the possible measurement effects have been included as part of other 

uncertainty contributors. The error associated with an improperly performed strain load test 

is beyond the scope of this document. 

Other measurement influences may contribute to the uncertainty of the calibration of a scale. 

The servicing technician must be knowledgeable of possible error contributors and must 

determine reasonable estimates of the uncertainty contribution of each by applying the 

guidance contained in this document. 

5. Calculation of the combined uncertainty 

The combined uncertainty, ‘uc’, of the weighing system calibration will be calculated as the 

root-sum-squared of the influence factors. 

2 2

3

2

2

2

1

2 2 2 2 2 2 .... c s p u w u sub u dr u v u u u ui u = u + s + + + + + + + + +

 where ui are 

any other uncertainty components that the scale technician wishes to include in the 

uncertainty calculations. 

Remember: 

- All of the uncertainty components must be in like terms of the mass units. Differing 

units, such as °C and mg, cannot be combined. Convert the impact of the non-mass units 

to the appropriate mass units to calculate the uncertainty. 

- All uncertainty components must be in terms of standard (one standard deviation) 

uncertainties. 

6. Calculating the expanded uncertainty 

The expanded uncertainty, ‘U’, is calculated by multiplying the value obtained for uc by the 

coverage factor, ‘k’, for the confidence interval to be used. By convention, as defined in 

NIST Technical Note 1297, “Guidelines for Evaluating and Expressing the Uncertainty of 

NIST Measurement Results”, a coverage factor of k = 2 is used for a confidence interval of 

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Scale Calibrations Performed in an Uncontrolled Environment 

approximately 95 %. The use of k values other than k = 2, requires an explanation of the 

reason for the deviation from convention. 

7. Evaluating the expanded uncertainty 

There are several things to consider when evaluating the uncertainty. First, does the final 

expanded uncertainty make sense? The expanded uncertainty must be at least ‘k’ times

greater than the largest uncertainty component included. 

Second, does the calculated expanded uncertainty seem appropriate for the device tested? An 

uncertainty value seems unreasonable and should be investigated if the expanded uncertainty 

is calculated to be significantly less than one balance division. Likewise, if the calculated 

uncertainty is excessively large, that value should be investigated. Investigation should 

include verifying the use of proper evaluation techniques. Ensure that arithmetical errors 

were not the cause of the questionable uncertainty value. 

Third, does the calculated uncertainty meet the requirements for weighing with that weighing 

device? Different quality systems have different requirements concerning the allowable 

measurement uncertainty; requirements may state that the uncertainty of a measuring device 

may not exceed 33 %, 25 %, or even as little as 10 %, of the tolerance of the object being 

tested. Will the calculated uncertainty meet the requirements that are in place? If not,

reexamine the entire calibration process for uncertainty contributors that can be reduced. 

Reduction may be accomplished by selection of more accurate standards, repair of the 

weighing device to obtain a smaller standard deviation, or perhaps making multiple 

measurements to determine the true repeatability of a device rather than using an estimated 

repeatability. Evaluate each uncertainty contributor, beginning with the most significant, to 

determine how it can legitimately be reduced until the required uncertainty level is obtained 

or until the decision is made that the weighing device cannot meet the quality requirements. 

Relocation or replacement of the scale may be required. 

Remember, the reported calibration uncertainty is valid only for the environment in which it 

is determined. Temporarily improving the environment for the calibration, then returning it 

to the prior conditions for normal weighing operations invalidates the uncertainty statement. 

Engineering changes made to reduce calibration uncertainty must remain in place during 

normal weighing operations for the reduced uncertainty to remain valid. 

8. Reporting Uncertainty

Reporting the expanded uncertainty value is no longer a matter of simply stating that a 

measurement result is ‘x ± y’ where ‘x’ is the reported value and ‘y’ is the expanded 

uncertainty. The Guide to the Expression of Uncertainty in Measurement requires that you 

identify the various components of the uncertainty. You must also explain why that 

component was included and how it was evaluated. Specifically, the GUM provides a test of 

the stated uncertainty statement: “Has enough information been provided in a sufficiently 

clear manner that the result can be updated in the future if new information or data became 

available?” Another test is to ask: “Would another individual, not associated with the 

measurement process, be able to understand how the stated uncertainty was calculated and 

what was included; and then properly apply it to his/her own uncertainty calculation?” 

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It must be understood that the measurement uncertainty must be calculated for each scale test 

load. The needs of the customer will determine whether a table format is used to report the 

uncertainty at each load, or the maximum uncertainty value of all test loads is reported as a 

single value covering the entire range of the weighing system. In either case, the uncertainty 

for each test load must be calculated and documented. 

It is very important that uncertainty calculations be understandable so that the user can 

properly include the uncertainty components in their process uncertainty calculations. Some

components of the uncertainty will need to be addressed differently from other components, 

and sufficient information must be available to make that possible. For example, when 

calculating the uncertainty of his production process, the customer would likely not wish to 

include reproducibility if the reproducibility value from a control chart of the measuring 

process was used to calculate the uncertainty of the scale calibration. To include the 

reproducibility value a second time would needlessly increase the expanded uncertainty of 

the production measurement process. Sufficient information must be provided to the user 

that it will be clear that the reproducibility should be included only once, in the original scale 

calibration uncertainty. 

The importance of a clear, understandable and defensible uncertainty statement, based on a 

properly documented set of instructions, cannot be emphasized too strongly. The uncertainty 

statement is a critical portion of the calibration process. Write the uncertainty statement so 

that a future reader can make informed decisions concerning proper use of the uncertainty 

components provided. 

The following tables provide a quick reference guide to the proper equation to use for 

calculating the calibration uncertainty in the most common situations. They also provide an 

example of a form that might be used to document the calibration uncertainty. 

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Quick Guide for Scale Calibration Uncertainties 

Reference information: 

Process (sp) Source (in order of desirability): Distribution 

Type 

Value to 

use: 

1 Calculated standard deviation at the test load from an up-todate control chart Normal As calculated

2 Calculated standard deviation from 10 or more readings of 

same load over a short period of time 

Normal As calculated

3 Estimated from scale division (discrimination test passed) Uniform 0.29 x ‘d’ 

4 Estimated from scale division (discrimination test not 

performed) Uniform 0.577 x ‘d’ 

Standards (us) Sources 

(in order of desirability): 

Distribution 

Type 

Value to 

use: 

1 Reported k=1 uncertainty from Report of Test Normal As calculated

2 Tolerance of weights used Uniform

0.577 x 

tolerance 

Additional equations when using multiple standards 

a Multiple standard masses (if independence is proven) 2 2

3

2

2

2

1 ...... s sm sm sm smi u = u + u + u + + u

b Multiple standard masses (if independence is unknown) ..... . us = usm1 + usm2 + usm3 + + usmi

Other uncertainty sources Distribution 

Type 

Value to 

use: 

Uncertainty due to wind effects (uw) (w=max deviation) Uniform 0.577 w

Uncertainty due to substitution loads (usub) Uniform usub = sp

Uncertainty due to display resolution (udr) (see text) Uniform 0 or udr = sp

Uncertainty due to vibration effects (uv) (v=max deviation) Uniform 0.577 v

Additionally, there may be other known measurement uncertainty contributors. Evaluate the 

calibration process carefully to ensure that all significant contributors are properly included 

in the uncertainty calculations. Consult the text of this Guide for additional guidance.

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Equipment Identification

Manufacturer Model Serial ECN Other

Uncertainty Worksheet 

at _____________________ Test Load 

Uncertainty Influence Description Identifier Estimated 

value 

Distribution 

type 

(Normal/Uniform)

Estimated 

Standard 

Uncertainty

Uncertainty of the standards used us

Standard deviation of the process 

(Determined at ___________ test load) sp

Uncertainty due to u1

Uncertainty due to u2

Uncertainty due to u3

Uncertainty due to u4

Uncertainty due to u5

Uncertainty due to u6

Uncertainty due to u6

Uncertainty due to u7

Combined standard uncertainty 

2 2

3

2

2

2

1

2 2 2 2 2 2 .... c s p u w u sub u dr u v u u u ui u = u + s + + + + + + + + +

Expanded uncertainty 

U = k(uc), k = ________ k = 2 is recommended U = 

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Chapter 5: 

Scale Calibrations in a Semi-Controlled 

Environment 

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Chapter 5: 

Scale Calibrations Performed in a Semi-controlled Environment 

Introduction 

Many general purpose scales are located in a semi controlled environment such as a grocery 

store or warehouse that minimizes many of the environmental influence factors that 

contribute to measurement uncertainty. Additionally, as most of these weighing devices have 

a relatively small capacity, it is possible to perform repeated measurements without severely 

affecting the time required to perform the calibration. The controlled environment and the 

ability to make repeated measurements simplify the process of calculating calibration 

uncertainty. Thus, service personnel are required to make fewer subjective decisions. 

The following outlines the process that scale service personnel should use to calculate the 

uncertainty associated with the scale testing process. 

1. Specify the Process and Equation 

Write down a clear concise statement of what is being measured and the relationship between 

it and the parameters on which it depends. Remember, the scale indication is being tested, 

not the mass of the standard mass artifacts. 

Example: y = (mx + b)±U where; 

y is the scale indication, 

m is the sensitivity of the weighing device, 

x is the applied load, 

b is the zero offset, and 

U is the assigned measurement uncertainty. 

Ideally, b = 0 if the scale indication was properly zeroed, and m = 1 because the scale 

indicates one mass unit for each mass unit applied. 

2, 3 & 4. Uncertainty Identification, Characterization and Quantification 

The uncertainty associated with the calibration of a general purpose scale is comprised of 

many influence factors. A cause and effect diagram is used here to identify the factors and to 

show their relationship to other factors. 

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Chapter 5: 

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Sample Cause and Effect Diagram

Uncertainty

Design Installation Staff & Procedures

Standards Facility

Environment/Location Method of Use

Capacity Accuracy

Sensitivity Readability

Draft Shield

Off center

loading sensitivity

Design

Experience

Repeatability Supports

Experience

Attitude

Procedure selection

Frequency

of test Training

Auto zero tracking

Reported

Uncertainty/

Tolerance

Substitution

load

Handling/

condition

Storage

Internal

standards

Use of error

weights

Temperature

stability Product

contamination

Drafts

RFI/EMI Vibration

Static

Electricity

Range of use

Type product

weighed

Gross/net weight

Auto zero-tracking

on/off

Repair

Shock

loading

Dynamic vs static

weighing

Factors

affecting

Uncertainty

Drift

Levelness

* not all inclusive

Draft

Shield

In addition to the influences shown on the diagram, other possible causes of uncertainty are 

standard mass errors, thermal equilibration of standards, scale warm-up, power-line noise, 

etc. 

Evaluate each uncertainty influence for its significance. Consider an uncertainty component 

significant if it can cause a change in the value of the second most significant digit, leading 

zeros excluded, when included in the uncertainty calculations. For example, if the expanded 

uncertainty value is currently 0.052 kg and including the value of an uncertainty contributor 

causes the new value to be 0.053 kg, that contributor is considered significant. If no change 

results in the second significant digit, the ‘2’, the contributor is not considered significant for 

that particular calibration. When an uncertainty contributor is considered significant, a value 

for that uncertainty component must be included in the uncertainty calculations for that 

calibration process. Whether included in the uncertainty calculations or not, the GUM 

requires that the process of performing the evaluation, and the results of the evaluation, be 

documented for each known uncertainty contributor. 

Many of the influence factors contained in the Sample Cause and Effect Diagram are 

addressed in the following paragraphs. The individual(s) performing the test must decide 

how to deal with each influence, whether by avoidance, correction or inclusion as part of the 

uncertainty. 

The service staff must evaluate the situation surrounding the scale and must eliminate or 

minimize as many of the measurement influence factors as possible. Engineering changes 

put in place to minimize a measurement influence must remain in place at all times for the 

calibration uncertainty to remain valid.

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Levelness of the scale should be checked and corrected if necessary. 

Drafts should be eliminated by the installation of draft shields, air diffusers or by 

redirecting air vents, or closing doors. These draft elimination measures must be in 

place at all times (including during use) for the calibration uncertainty statement to 

remain valid during daily use. 

 Off-center loading errors should be evaluated and corrected before recording final 

calibration data. 

Thermal equilibrium of the scale should be ensured by keeping the scale in an 

energized state (i.e., turned on) for sufficient time to ensure that all circuitry and 

hardware has reached a stable temperature. 

Vibration sources must be identified and eliminated or minimized to limit the vibration 

levels to which the weighing device is exposed during daily use, as well as during 

calibration. 

Zero-tracking features should be disabled to avoid undetectable zero indication errors. 

Zero-tracking is designed to eliminate minor changes in scale indication due to drift of 

the weighing system and may be turned 'on' to maintain the 'zero' indication of the 

weighing device. Due to the correction limits included in the system programming 

regarding quantity and motion detection, detrimental effects on measurement results are 

typically not evident. However, service personnel may, while attempting to load the pan 

in a slow, gentle manner encounter times that the zero tracking circuitry will cause an 

offset in the indication. While this would be an unusual occurrence, service personnel 

must be aware of the possibility of such errors and take steps to prevent them from

occurring. 

Mass Standards must be brought into close proximity to the scale to ensure thermal 

equilibrium with the scale and surrounding environment. Thermal stabilization times for 

weights typically used for a scale in this environment are not specified, but sufficient 

time must be allowed to ensure that condensation is not present on the weights. 

Standard mass error effects can be minimized by using the mass values determined 

during calibration of the mass artifacts. For most scales used in a semi-controlled 

environment this will not be necessary as the tolerances of the scales and standard 

masses have been set to minimize the effects of the standard mass errors. The concept is 

presented here as an option should correction be desirable. The standard mass 

uncertainty for general purpose scales is normally based on the tolerance of the mass 

standards. 

However, if the mass corrections of the standard masses are used, the reported 

calibration uncertainty of the mass calibration must be used as us. The complete 

calculation of us is covered in a later section of this chapter. 

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Chapter 5: 

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Scale indication drift errors should be corrected to the maximum extent possible. 

Though slight drift in the indicated value is somewhat normal, significant indication drift 

may indicate thermal instability of the weighing system. Allowing a longer temperature 

stabilization period for the weighing device and standard mass artifacts can minimize 

drift due to thermal instability. Exercising the weighing system may also minimize drift. 

Repeatedly applying and removing loads within the range of the device causes the 

electronic and mechanical components of the system to acquire a thermal equilibrium of 

use. If indication drift is excessive and cannot be eliminated by the means described, 

other corrective action must be taken. 

RFI/EMI susceptibility tests should be performed. Changes in indication due to 

random RFI/EMI influences can be difficult to detect during normal use and may be 

interpreted as repeatability errors or drift. Corrective action should be taken if RFI/EMI 

susceptibility is detected. 

Note: RFI/EMI sources include two-way radios, cell phones and other electronic devices. Laboratories

are available specializing in detecting and measuring RFI/EMI. 

Power-line noise or variations can cause random display indications to occur. The 

specific cause of these random indications may be difficult to determine, but will affect 

the repeatability of the weighing system. Where possible, it is best that scales be

powered by a dedicated power circuit or be powered by a AC line conditioner to prevent 

these measurement influences. 

Operator errors result when operators are inadequately trained. All operators of

weighing devices should have proper training and be knowledgeable about the weighing 

instrument and the process in which it is used. Inadequately trained personnel may 

record data with significant errors, improperly influencing critical process decisions. 

Weighing system operators must be equipped with correct and complete work 

instructions to minimize the likelihood of operator error. 

Careful planning or suitable corrective action will minimize or eliminate many scale 

measurement uncertainty sources. However, there are several uncertainty influences that 

cannot be eliminated. The results of these influences must be included in the calibration 

uncertainty calculations. Each are discussed in the following paragraphs. 

Process standard deviation (sp) 

The source of a value for sp can be based on scale repeatability, scale reproducibility or the 

size of a scale division. Each are discussed below. 

sp from measured repeatability 

Repeatability is a measure of a scale’s ability to produce the same indication every time

the same weight, under identical conditions, is placed on the sensing device. 

Repeatability is normally presented as a standard deviation and is normally determined 

by repeated measurements of a specific weight. At least seven measurement results of a 

weight are required to calculate the repeatability or standard deviation of the scale. 

Increasing the number of measurement results provides greater confidence in the value. 

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A minimum of thirty measurement results is preferred when calculating a standard 

deviation. The calculated standard deviation obtained in this manner is a measure of the 

weighing system’s ability to repeat measurement results over a short time interval and 

does not represent the long-term reproducibility. 

Due to scale resolution, it is possible to make seven or more measurements with every 

measurement resulting in the same value. A true scale standard deviation of zero is not 

statistically possible, though the standard deviation may be less than one display 

increment (d). In this situation, assuming that a control chart of a check standard is not 

available, the standard deviation of the scale can be estimated as d d

s p 0.577

3 = =

(0.577 times the value of ‘d’) as described in the GUM. 

If it is possible to perform a discrimination test as described in NIST Handbook 44, 

N.1.5 and T.N.7.1 or OIML R 76-1, A.4.8, another equation may be used. That equation 

is: d d

s p 0.29

2 3

1 = = (0.29 times the value of ‘d’) since passing the discrimination test 

proves that the true value most probably lies near the center of the scale display interval. 

This estimation equation should only be used when the calculated standard deviation is 

zero and the scale has passed a discrimination test as described in NIST Handbook 44, 

N.1.5 and T.N.7.1 or OIML E 76-1, A.4.8. 

sp from measured reproducibility 

When desiring to ascertain the uncertainty of the scale during everyday weighing 

operations, a check standard and control chart should be used to determine the long-term 

repeatability of the scale, under varying operational conditions. This is called 

reproducibility, also evaluated as a standard deviation. Reproducibility is a measure of 

long-term repeatability and is used in place of the short-term repeatability when 

calculating measurement uncertainty. The value will be included in the combined 

uncertainty as sp. 

When using values from control charts to determine the sp value, attention must be given 

to the range of measurement results over which that value has been determined. When 

readings from a wide range of measurement results are included in the calculation of the 

sp value and the resulting standard deviation is acceptable, an uncertainty calculated 

from that standard deviation is valid for the range over which the measurement results 

were obtained. However, sometimes the calculated uncertainty is greater than desired or 

greater than is acceptable. Repair of the weighing device may not suitably correct the 

problem. For those situations, it might be desirable to divide the data into measuring 

ranges and to calculate the standard deviation for the narrower measuring ranges. These 

standard deviations would be applicable to the measuring range over which the data was 

obtained. Especially in a calibration facility, a scale may have multiple sp values, each 

covering a different portion of the scale measuring range. The existence of multiple sp

values requires the calculation of expanded uncertainty values for each affected 

measuring range. 

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sp from manufacturer’s specifications 

The value for sp is best determined from repeated readings of a weight, whether shortterm or long-term. The manufacturer’s specification sheet is not a recommended source 

for determining the uncertainty contribution due to repeatability. Manufacturers'

specifications are established for a specific set of conditions that may not be 

representative of the actual environment in which the scale is tested and used. 

Manufacturers' specifications are an excellent tool when comparing the expected 

performance of one weighing device to another similar device, but they are established 

to indicate the expected performance of a family of weighing systems, not a specific 

weighing system. As the calibration uncertainty is being estimated for a specific

weighing system, it is best to establish the calibration uncertainty contribution due to 

repeatability from measured data, not from an expected performance parameter. 

Manufacturers' specifications should not be used to estimate calibration uncertainty. 

Uncertainty due to display resolution (udr) 

When the repeatability (sp) is estimated from display resolution, an additional uncertainty 

component for indicator resolution must be included. This value (udr) will be equal to 

d dudr 0.29

2 3

1 = = (0.29 times the value of ‘d’). This uncertainty component should not be 

included in the uncertainty calculations if repeatability is measured by making repeated 

measurements. The effect of the display resolution uncertainty component will have been 

part of the measured values and an additional uncertainty component is not required. 

Uncertainty of the standards (us) 

The uncertainty contribution of the standard masses used to perform the scale calibration 

may be estimated from one of two sources, the tolerance to which the masses have been 

tested, or the uncertainty associated with the calibration of the masses. Specific guidance 

follows. 

us from standard weight tolerances 

When calibrating large capacity scales, the nominal values of the standard weights are 

typically used. In this situation, a portion of the tolerance to which the weights were 

tested is used as the standard uncertainty of the standards value, us, when calculating the 

calibration uncertainty. Only a portion of the tolerance is used as the tolerance is 

considered to follow a uniform probability distribution. The standard uncertainty of the 

standards is calculated as 

3

tolerance

us = or (0.577 times the tolerance of the applied 

weights). When using multiple weights, the tolerance used in the us calculation will be 

the sum of the tolerances for all of the weights used. 

us from calibration uncertainties 

Standard mass uncertainties normally are reported as expanded uncertainties, having 

been calculated by combining the various standard uncertainties that influenced the mass 

calibration process. The combined standard uncertainty has then been multiplied by a 

stated coverage factor (k) to obtain the expanded uncertainty. This expanded uncertainty 

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must be divided by the coverage factor to obtain a standard uncertainty for the mass 

artifacts, us. 

us with multiple standard weights 

When multiple weights are used, the most conservative value for the uncertainty of the 

mass calibrations is the sum of the uncertainties associated with each of the weights used 

simultaneously. For this situation u ..... . s = usm1 + usm2 + usm3 + + usmi While this method 

provides the most conservative estimate of standards uncertainty, it also typically 

provides the largest estimate for the standards uncertainty and may result in an 

uncertainty value greater than is allowed by the customer's weighing process. 

When a smaller uncertainty value is required and sufficient information about the weight 

calibrations is available, the root-sum-squared (RSS) value of the individual weight 

uncertainties may be used, but only if true independence of the mass values is proven. 

Values are independent when no portion of the value of one weight affects the value of 

another. (If the same standard mass was used to calibrate each of the weights being 

used, the calibrations are not independent since the uncertainty and error of the 

standard’s value contribute to the error and uncertainty for each of the weights. Weights 

of the same nominal mass, in a kit, are normally calibrated using the same standard and 

the values are not independent.) Information concerning independence must be obtained 

from the calibrating organization. If the user is unsure of the independence of values, it 

should be assumed that the values are not independent, and the sum of the individual 

uncertainties must be used. The RSS value of us will be used only if the actual 

measured values of the mass artifacts were being used in the calibration of the scale. 

The equation for the root-sum-squared method is 2 2

3

2

2

2

1 ...... s sm sm sm smi u = u + u + u + + u . 

5. Calculation of the combined uncertainty 

The combined uncertainty, ‘uc’, of the weighing system calibration will be calculated as the 

root-sum-squared of the influence factors. 

2 2

3

2

2

2

1

2 2 2 .... c s p u dr u u u ui u = u + s + + + + + + where ui are any other uncertainty 

components that the scale technician wishes to include in the uncertainty calculations. 

Remember: 

- All of the uncertainty components must be in terms of the mass units of the scale. 

Differing units, such as °C and mg, cannot be combined. Convert the impact of the nonmass units to the appropriate mass units to calculate the uncertainty. 

- All uncertainty components must be in terms of standard (one standard deviation) 

uncertainties. 

6. Calculating the expanded uncertainty 

The expanded uncertainty, ‘U’, is calculated by multiplying the value obtained for uc by the 

coverage factor, ‘k’, for the confidence interval to be used. By convention, as defined in 

NIST Technical Note 1297, “Guidelines for Evaluating and Expressing the Uncertainty of 

NIST Measurement Results”, a coverage factor of k = 2 is used for a confidence interval of 

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approximately 95 %. The use of k values other than k = 2 requires an explanation of the 

reason for the deviation from convention. 

7. Evaluating the expanded uncertainty 

There are several things to consider when evaluating the uncertainty. 

First, does the final expanded uncertainty make sense? The expanded uncertainty must be at 

least ‘k’ times greater than the largest uncertainty component included. 

Second, does the calculated expanded uncertainty seem appropriate for the tested device? An 

uncertainty value seems unreasonable and should be investigated if the expanded uncertainty 

is calculated to be significantly less than one balance division. Likewise, if the calculated 

uncertainty is excessively large, that value should be investigated. Investigation should 

include verifying the use of proper evaluation techniques. Ensure that arithmetical errors 

were not the cause of the questionable uncertainty value. 

Third, does the calculated uncertainty meet the requirements of the tests that will be 

performed using the weighing device? Different quality systems have different requirements 

concerning the allowable measurement uncertainty; requirements such that the uncertainty of

a measuring device may not exceed 33 %, 25 %, or even as little as 10 %, of the tolerance of 

the object being tested. Will the calculated uncertainty meet any such requirements that may 

be in place? If not, examine the entire calibration process for uncertainty contributors that 

can be reduced. Reduction may be accomplished by selection of more accurate standards, 

repair of the weighing device to obtain a smaller standard deviation, or perhaps making 

multiple measurements to determine the true repeatability of a device rather than using an 

estimated repeatability. Evaluate each uncertainty contributor, beginning with the most 

significant, to determine how it can legitimately be reduced until the required uncertainty 

level is obtained or until the decision is made that the weighing device cannot meet the 

quality requirements and must be replaced or that the weighing system must be moved to a 

more hospitable environment. 

Remember, the reported calibration uncertainty is valid only for the environment in which it 

is determined. Temporarily improving the environment for the calibration, then returning it 

to the prior conditions for normal weighing operations invalidates the uncertainty statement. 

Engineering changes made to reduce calibration uncertainty must remain in place during 

normal weighing operations. 

8. Reporting Uncertainty

Reporting the expanded uncertainty value is no longer a matter of simply stating that a 

measurement result is ‘x ± y’ where ‘x’ is the reported value and ‘y’ is the expanded 

uncertainty. The Guide to the Expression of Uncertainty in Measurement requires that you 

identify the various components of the uncertainty. You must also explain why that 

component was included and how it was evaluated. Specifically, the GUM provides a test of 

the stated uncertainty statement: “Has enough information been provided in a sufficiently 

clear manner that the result can be updated in the future if new information or data became 

available?” Another test is to ask: “Would another individual, unassociated with the 

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measurement process, be able to understand how the stated uncertainty was calculated and 

what was included, and then properly apply it to his/her own uncertainty calculation?” 

It must be understood that the measurement uncertainty must be calculated for each scale test 

load. The needs of the customer will determine whether a table format is used to report the 

uncertainty at each load, or the maximum uncertainty value of all test loads is reported as a 

single value covering the entire range of the weighing system. In either case, the uncertainty 

for each test load must be calculated and documented. 

It is very important that uncertainty calculations be understandable so that the user can 

properly include the uncertainty components in their process uncertainty calculations. Some

components of the uncertainty will need to be addressed differently from other components, 

and sufficient information must be available to make that possible. For example, when 

calculating the uncertainty of his production process, the customer would likely not wish to 

include reproducibility if the reproducibility value from a control chart of the measuring 

process was used to calculate the uncertainty of the balance calibration. To include the 

reproducibility value a second time would needlessly increase the expanded uncertainty of 

the production measurement process. Sufficient information must be provided to the user so 

that it will be clear that the reproducibility should be included only once, in the originally 

reported scale calibration uncertainty. 

The importance of a clear, understandable and defensible uncertainty statement, based on a 

properly documented set of instructions, cannot be emphasized too strongly. The uncertainty 

statement is a critical portion of the calibration process. Write the uncertainty statement so 

that a future reader can make informed decisions concerning proper use of the uncertainty 

components provided. 

The following tables provide a quick reference guide to the proper equation to use for 

calculating the calibration uncertainty in the most common situations. They also provide an 

example of a form that might be used to document the calibration uncertainty. 

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Quick Guide for Scale Calibration Uncertainties 

Reference information: 

Process (sp) Source (in order of desirability): Distribution 

Type 

Value to 

use: 

1 Calculated standard deviation at the test load from an up-todate control chart Normal As calculated

2 Calculated standard deviation from 10 or more readings of 

same weight over a short period of time

Normal As calculated

3 Estimated from scale division (discrimination test passed) Uniform 0.29 x ‘d’ 

4 Estimated from scale division (discrimination test not 

performed) Uniform 0.577 x ‘d’ 

Standards (us) Sources 

(in order of desirability): 

Distribution 

Type 

Value to 

use: 

1 Reported k=1 uncertainty from Report of Test Normal As calculated

2 Tolerance of weight used Uniform

0.577 x 

tolerance 

Additional equations when using multiple standards 

a Multiple standard masses (if independence is proven) 2 2

3

2

2

2

1 ...... s sm sm sm smi u = u + u + u + + u

b Multiple standard masses (if independence is unknown) ..... . us = usm1 + usm2 + usm3 + + usmi

Other uncertainty sources Distribution 

Type 

Value to 

use: 

Uncertainty due to display resolution (udr) (see text) Uniform 0 or udr = sp

Additionally, there may be other known measurement uncertainty contributors. Evaluate the 

calibration process carefully to ensure that all significant contributors are properly included 

in the uncertainty calculations. Consult the text of this Guide for additional guidance.

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Equipment Identification

Manufacturer Model Serial ECN Other

Uncertainty Worksheet 

at _____________________ Test Load 

Uncertainty Influence Description Identifier Estimated 

value 

Distribution 

type 

(Normal/Uniform)

Estimated 

Standard 

Uncertainty

Uncertainty of the standards used us

Standard deviation of the process 

(Determined at ___________ test load) sp

Uncertainty due to u1

Uncertainty due to u2

Uncertainty due to u3

Uncertainty due to u4

Uncertainty due to u5

Uncertainty due to u6

Uncertainty due to u6

Uncertainty due to u7

Combined standard uncertainty 

2 2

5

2

4

2

3

2

2

2

1

2 2 2 .... c s p u dr u u u u u ui u = u + s + + + + + + + + uc = 

Expanded uncertainty 

U = k(uc), k = ________ k = 2 is recommended U = 

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Chapter 6: 

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Chapter 6: 

Sample Calculations and 

Recommendations for Reducing 

Uncertainty 

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Chapter 6: 

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Introduction

Evaluating the uncertainty of a scale and balance calibration process can be extremely complex 

depending on the location, capacity, design, readability, amount of calibrated load available, air 

currents and a host of other factors. It is not possible to provide an example that will address all 

situations, but the examples given, and the accompanying discussion, provide some insight into the 

process to be completed for every scale or balance calibration

Sample Uncertainty Calculation #1 (Relates to Chapter 3) 

The following example of an uncertainty calculation for a 200 g laboratory balance calibration will 

provide insight into the process that must be used to analyze a balance calibration and evaluate the

associated uncertainty. (The two test loads, 50 g and 200 g, are used for demonstration purposes 

only. Consult the appropriate procedural documents for full test requirements.) 

This balance is a Class I balance with a maximum load of 205 grams and a readability of 1 mg. It is 

located in a facility that deals with materials that are not hazardous, but are unpleasant to work with 

due to an offensive odor. Due to the unpleasant odor, the balance is located inside a fume hood. 

A 100 g to 1 mg set of Class E1 weights is used to perform the calibration and the nominal values of 

the weights will be used. When preparing for the calibration, the balance user switched the fume

hood off so that the service personnel did not have to listen to the drone of the fan. In addition, 

other equipment was removed from the fume hood. Typically, a small drying oven, and a small 

sample shaker are also located inside the fume hood near the balance. All are connected to a 

common power source. The environmental conditions are stable throughout the calibration process. 

NIST Handbook 44 tolerances will be used even though this balance is not used for commercial sale 

of goods and is not technically within the regulatory jurisdiction of any Weights and Measures 

program. No other tolerance guidance is available. 

The Class I balance has a tolerance of 

Class I Maintenance Tolerances 

Range tested (d) Tolerance (d) HB 44, Table 6

0 to 50 000 ± 1 

50 001 to 200 000 ± 2 

200 001 + ± 3 

For the sample calculation, the tolerance at the applied load will be 

Test Load Applied Tolerance 

50 000d (50 g) ± 1d (1 mg) 

200 000d (200 g) ± 2d (2 mg) 

These test points were selected since they cause indications at the upper extreme of the individual 

tolerance ranges. Consult the appropriate procedural document for full test requirements. 

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Due to the small weights employed in this calibration, multiple measurements are performed to 

measure the repeatability of the balance at a 100 g load. Ten readings were recorded: the ten values 

are identical, therefore the calculated sp = zero (0). 

How will the uncertainty of the calibration be determined? 

1. Specify the process and equation 

The indicated values of the balance loads are a function of the applied load and the associated 

uncertainty. A very basic statement of the measurement equation would be: 

Indication = applied load ± Uncertainty or y = mx + b ± U. Uncertainty equals 

2 2

3

2

2

2

1

2 2 2 * .... s p u u u ui U = u + s + + + + +

2. Identify and characterize the uncertainty sources 

The uncertainty sources that are significant for this calibration are: 

the uncertainty of the standard weights, 

the standard deviation of the balance 

3. Quantify the identified uncertainty components 

The uncertainty of the standard weights is taken to be equal to the sum of the tolerances of the 

OIML Class E1 weights used at each applied load. This calculates to be: ± 0.15 mg at 200 g, and ± 

0.030 mg at 50 grams. 

The standard deviation of the 10 measurement results obtained at 100 g calculates to a standard 

deviation of zero, since all of the indications were identical. A standard deviation of zero is not 

statistically possible, but is a result of having recorded only 10 indicated readings. If all possible 

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measurement results were obtained at 100 g, there would be an infinite number of readings. 

Statistics shows that the standard deviation, sp, would be approximately equal to some portion of 

one scale division, or 1 mg. 

4. Convert the measurement influences to standard deviation equivalents 

Using the text and the Quick Guide table from Chapter 3, each of the quantified measurement 

influences is converted to one standard deviation or standard uncertainty equivalent. 

The uncertainty of the standard weights is taken to be equal to the tolerance of the OIML Class E1

weights. This uncertainty component fits a uniform probability distribution, so use of the equation 

0.577 x tolerance is appropriate. Thus, the standard uncertainty, us, due to the tolerance of the 

calibrated weights is: 

0.577 x 0.15 = 0.08655 mg at 200 g, and 

0.577 x 0.030 = 0.01731 mg at 50 g. 

Because the measured standard deviation was equal to zero and this is statistically not possible, the 

Guide to the Expression of Uncertainty in Measurement provides guidance for this situation. 

Basically, the writers of the GUM state that even though there is no indicated difference in the 

readings due to the inability of the weighing device to detect the difference between two values, 

e.g., 1.55 and 2.45, each cause an indication of 2, there is still a variability associated with the 

indications. The writers of the GUM recommend that the standard deviation, sp, be estimated as: 

0.29 x 1 mg = 0.29 mg. 

5. Calculate the combined standard uncertainty, uc, using the equation 

2 2

c s p u = u + s

For this example 

2 2

u c 200 g = 0.08655 mg + 0.29 mg

u c 200 g = 0.3026398891 mg

2 2

u c 50 g = 0 .01731 mg + 0.29 mg

u c 50 g = 0 .2905161546 mg

6. Calculate the Expanded Uncertainty 

Multiply the combined standard uncertainty, uc, by the coverage or 'k' factor to calculate the 

expanded uncertainty. 

 

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For U200 g, 

 

Description Identifier Estimated

value 

Distribution 

type 

Estimated Standard 

Uncertainty

Uncertainty of the weights used us 0.15 mg Uniform 0.08655 mg

Standard deviation of the process 

(repeatability or reproducibility) 

sp 1 mg Uniform 0.29 mg

Combined standard uncertainty 

2 2

c s p u = u + s

0.3026398891 mg

Expanded uncertainty

U = k(uc), k= 2 (rounded to two significant digits) 0.61 mg (at 200 g) 

and 

For U50 g, 

Description Identifier Estimated

value 

Distribution 

type 

Estimated Standard 

Uncertainty

Uncertainty of the weights used us 0.030 mg Uniform 0.01731 mg

Standard deviation of the process 

(repeatability or reproducibility) 

sp 1 mg Uniform 0.29 mg

Combined standard uncertainty 

2 2

c s p u = u + s

0.2905161546 mg

Expanded uncertainty

U = k (uc), k= 2 (rounded to two significant digits) 0.58 mg (at 50 g) 

7. Evaluate U for appropriateness 

Several criteria should be evaluated. 

Does the expanded uncertainty make sense? 

Since the largest single standard uncertainty is 0.29 mg, it would not make sense to have an 

expanded uncertainty of 0.1 mg, when using a k=2 expanded uncertainty, because 2 x 0.29 mg = 

0.58 mg. The expanded uncertainty must always be at least k times the largest single uncertainty 

contribution included in the calculations. 

Likewise, an expanded uncertainty of 50 mg would not make sense, since no single factor 

approached 25 mg. The service technician must think!

Does the calculated uncertainty seem reasonable for the balance being tested? 

If the balance has 1 mg divisions, an expanded uncertainty of 28 mg or 28 divisions would probably 

not be reasonable. Repair of the device and verification of the calculations is recommended. If the 

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balance has 10 mg divisions, 28 mg might be appropriate, but the calculations should be reviewed 

for correctness. The service technician must think!

Does the expanded uncertainty meet the criteria for the calibration process? 

NIST Handbook 44 Fundamental considerations states that "if the standard is to be used without 

correction, its error should be not greater than one-third of the smallest tolerance to be applied 

when the standard is used." Interpreting this statement to indicate that the uncertainty of the 

calibration process should not be greater than one-third of the smallest tolerance to be applied to the 

device being tested, is the expanded uncertainty less than one-third of the tolerance for the balance 

being tested?

Applied Load Expanded 

Uncertainty

HB 44 

Tolerance 

1/3 Tolerance U < 1/3 Tolerance? 

200 g 0.61 mg 2 mg 0.66667 mg Yes 

50 g 0.58 mg 1 mg 0.33333 mg No 

Does the expanded uncertainty meet the requirements of the user's process? 

The answer to this question must come from the scale operator. 

 

If the answer to either of the last two questions is 'NO', ways of decreasing the calibration 

uncertainty must be sought. 

First, the uncertainty calculations must be reviewed for correctness. 

Second, possible changes to the calibration process must be evaluated. Begin with the largest 

uncertainty contributor, working to the least significant. 

For this sample calculation, the largest uncertainty contributor is the estimated standard deviation 

of the balance. The standard deviation was estimated using a formula based on the readability of 

the balance. A better method for determining the standard deviation would be to make between 

20 and 90 additional measurements so that sample size was significantly increased improving the 

likelihood that the calculated standard deviation would be representative of the true balance 

performance. These additional measurements also reduce the likelihood of the calculated 

standard deviation being zero. 

The ideal method of determining the standard deviation of a weighing device, such as was 

described for this example, is to have in place a process measurement assurance program

(PMAP). Control charts, used in a PMAP, will provide a process standard deviation that includes 

all process measurement variables, including the effects of surrounding equipment, operator 

training, and environmental effects. When this example was described, it was mentioned that the 

fume hood fan was turned off, and that other equipment with great potential influence on the 

measurement were removed from the fume hood for the calibration. This created a false 

environment for calibration that will not be present during use which makes the calibration 

uncertainty not applicable to normal balance use. The user is required to calculate an uncertainty 

for the balance in his process, for which the reported calibration uncertainty would be only one 

component. 

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8. Reporting Uncertainty 

The uncertainty statement of the balance calibration should include sufficient information that a 

person not associated with the calibration will understand what uncertainty components were 

included, how and why. The following is a sample: 

"The k=2 Expanded Uncertainty of the balance calibration was as described in the following 

table. 

Applied Load Calibration

Uncertainty (k = 2)

200 g 0.61 mg

50 g 0.58 mg

The uncertainty values were determined by using the root-sum-squared (RSS) combination of 

the significant measurement influences. Included are: 

the estimated standard deviation of the scale, estimated from the size of the scale division (d), 

and the uncertainty of the calibrated mass standards, estimated from the allowable calibration 

tolerance of the weights,

The RSS combined value was multiplied by a coverage factor (k) of 2 to obtain a value fitting 

the probability distribution for a 95 % confidence interval." 

Typically, the balance user must use the reported calibration expanded uncertainty as one

component of the normal weighing operation process uncertainty. 

However, if the calibration uncertainty calculation is based on the process control chart data 

obtained during normal use conditions, the user does not need to perform additional uncertainty 

calculations for his process, provided that the PMAP (Process Measurement Assurance Program) 

measurements are continued and properly evaluated. 

Sample Uncertainty Calculation #2: (Relates to Chapter 4) 

The following example uncertainty calculation, at a single test load of 20 000 kg, for a portion of the 

test of a 50 000 kg Class IIIL scale in an uncontrolled environment will discuss some of the thought 

processes that must be used to analyze a scale calibration and evaluate the associated uncertainty. 

This scale has a capacity of 50 000 kg, with 10 kg divisions. The service personnel have only 

10 000 kg of NIST Class F calibrated weights available to perform the calibration. Thus, 10 000 kg 

of substitution weight will be used as part of the calibration process at this sample load. On the day 

of the test, the location of the scale and the breezes blowing at the time cause the scale indications to 

be unstable, cycling randomly over a total range of approximately 40 kg, but centered around the 

applied nominal value. The temperature of the scale varies 10 °C during the calibration process as 

the sun warms the scale deck and, to save time, no discrimination test is performed. The scale 

operator does not have a measurement assurance program in place, so no charted check standard 

values are available to evaluate the standard deviation of the scale during normal weighing 

operations. Additionally, the scale operator insists that the scale be down for a minimum amount of 

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time because this is the only scale available, and each hour the scale is shut down causes the loss of 

many dollars of revenue. Taking repeated measurements to determine the standard deviation of the 

scale will take eight hours, and the scale operator is unable to absorb that loss of revenue. 

The tolerance being applied to this scale is for a Class IIIL scale and is based on Table 6 of NIST 

Handbook 44. The example scale has 5 000 divisions (5 000d) and a test load will be applied that 

causes an indication near 40 % of the indication range. (This single test load is used for 

demonstration purposes only. Consult the appropriate procedures for full test requirements.) 

Class IIIL Maintenance Tolerance 

Range tested (d) Tolerance (d) calculated from HB 44, Table 6

0 to 500 ± 1 

501 to 1 000 ± 2 

1 001 to 1 500 ± 3 

1 501 to 2 000 ± 4 

Add 1d for each additional 500d or 

fraction thereof 

For the sample calculation, the tolerance at the applied load will be 

Test Load Applied Tolerance 

2 000d (20 000 kg) ± 4d (40 kg) 

How will the uncertainty of the calibration be determined? 

1. Specify the process and equation 

The indicated values of the scale loads are a function of the applied load and the associated 

uncertainty. A very basic statement of the measurement equation would be: 

Indication = applied load ± Uncertainty, or y = mx + b ± U. Uncertainty equals 

2 2

3

2

2

2

1

2 2 2 2 2 2 * .... s p u w u sub u t u u u u i U = u + s + + + + + + + +

2. Identify and characterize the uncertainty sources 

The uncertainty sources that are significant for this calibration are: 

the uncertainty of the standard weights, 

the standard deviation of the scale, 

the instability of the indicated value due to wind effects, 

the use of substitution load during the process, and 

the varying temperature.

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3. Quantify the identified uncertainty components 

The uncertainty of the standard weights is taken to be equal to the tolerance of the NIST Class F 

weights or 1 kg. 

The temperature varies over a range of 10 °C. 

The indicated value varies ± 20 kg. 

The operator will not allow the scale to be closed for repeated measurements, the standard 

deviation is estimated as 0.577 times a scale division, or 5.77 kg. 

The weight of the substitution load can only be determined to the readability of the scale, so the 

probable uncertainty due to substitution load will be initially estimated as ± 1 scale division. 

4. Convert the measurement influences to standard deviation equivalents 

Using the text and the Quick Guide table from Chapter 4, each of the quantified measurement 

influences is converted to one standard deviation or standard uncertainty equivalents. 

The uncertainty of the standard weights is taken to be equal to the tolerance of the NIST Class F 

weights, or 1 kg. This uncertainty component fits a uniform probability distribution, so use of the 

equation 0.577 x tolerance, or 0.577 x 1 kg, is appropriate. Thus, the standard uncertainty due to the 

tolerance of the calibrated weights, us, is 0.577 kg 

The operator will not allow the scale to be closed for repeated measurements, so an estimate is 

made of the standard deviation, sp, based on the 10 kg scale division as: 0.577 x 10 kg = 5.77 kg 

The indicated value varies in a range of 40 kg or an average indication ± 20 kg. 

The service technician estimates a value for the varying indication based on experience and the 

range over which the indication is varying, but the lack of a stable indication contributes to the 

uncertainty of that estimated indication. As the indication is varying 'plus' and 'minus' a fixed value 

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it will be considered to fit a uniform probability distribution. Using guidance from the table, the 

standard uncertainty due to the wind effects, uw, is calculated as: 0.577 x 20 kg = 11.54 kg. 

The weight of the substitution load can only be determined to the readability of the scale. Assuming 

that the weight of the substitution load is adjusted to exactly the same indication as was generated

by the calibrated load, the standard uncertainty due to the substitution load can be estimated as 

being equal to the repeatability of the scale. The repeatability or standard deviation of the balance 

was estimated to be 5.77 kg. One occurrence of this factor must be included for each substitution 

load used. Only one substitution is used for the sample calculation although HB44 allows up to 

three substitutions in a calibration process. 

Thus, the value to be included for substitution load, usub, is 5.77 kg. 

The standard uncertainty due to the unstable scale temperature will be calculated by multiplying the 

scale manufacturer's temperature coefficient for the scale by the range over which the temperature is 

varying, 10 °C. For this example a temperature coefficient of 10 x 10-6/°C will be used. In addition, 

since this coefficient is from a vendor's specification, it will typically be considered to fit a uniform 

probability distribution. Thus, the standard uncertainty due to varying temperature, ut, is: 0.577 x 

10 °C x (10 x 10-6 /°C) x 20 000 kg = 1.154 kg. 

5. Calculate the combined standard uncertainty, uc, using the equation 

2 2 2 2 2

c s p u w u sub u t u = u + s + + +

For this example 2 2 2 2 2

u c = 0.577 kg + 5.77 kg + 11 .54 kg + 5.77 kg + 1.154 kg

Therefore, u c = 14 .1923 kg

6. Calculate the Expanded Uncertainty 

Multiply the combined standard uncertainty, uc, by the coverage or 'k' factor to calculate the 

expanded uncertainty. 

Therefore, U = 2 * 14 .1923 kg

U = 28 .3846 kg 

which is presented as 28 kg when properly rounded to two significant digits. 

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Description Identifier Estimated

value 

Distribution 

type 

Estimated Standard 

Uncertainty

Uncertainty of the weights used us 1 kg Uniform 0.577 kg 

Standard deviation of the process 

(repeatability or reproducibility) 

sp 10 kg Uniform 5.77 kg 

Uncertainty due to wind effects uw 20 kg Uniform 11.54 kg 

Uncertainty due to substitution 

loads 

usub 10 kg Uniform 5.77 kg 

Uncertainty due to temperature 

variations 

ut 10 °C Uniform 1.154 kg

Combined standard uncertainty 

2 2 2 2 2

c s p u w u sub ut u = u + s + + + 14.1923 kg 

Expanded uncertainty 

U = k(uc), k= 2 (rounded to two significant digits) 28 kg 

7. Evaluate U for appropriateness 

Several criteria should be evaluated. 

Does the expanded uncertainty make sense? 

Since the largest single standard uncertainty is 11.54 kg, as in the example, it would not make sense 

to have an expanded uncertainty of 20 kg, when using a k=2 expanded uncertainty, as 

2 x 11.54 = 23.08. The expanded uncertainty must always be at least k times the largest single 

uncertainty contribution included in the calculations. 

Likewise, an expanded uncertainty of 280 kg would not make sense, as no single factor approached 

140 kg. The service technician must think! 

Does the calculated uncertainty seem reasonable for the scale being tested? 

If the scale has 1 kg divisions and the expanded uncertainty is calculated to be 28 kg or 28 divisions, 

there is probably something wrong, either with the scale or the uncertainty calculations. If the scale 

had 100 kg divisions, 28 kg might be appropriate, though the calculations should be reviewed for 

correctness. The service technician must think!

Does the expanded uncertainty meet the criteria for the calibration process? 

NIST Handbook 44, Appendix A, "Fundamental Considerations" states that "if the standard is to be 

used without correction, its error should be not greater than one-third of the smallest tolerance to 

be applied when the standard is used." This statement has also been interpreted to mean that the 

uncertainty of the calibration process should not be greater than one-third of the smallest tolerance 

to be applied to the device being tested. Is the expanded uncertainty less than one-third of the 

tolerance for the scale being tested? In this sample calculation the calculated uncertainty at 20 000 

kg is 28 kg, and the tolerance at 20 000 kg is 40 kg, thus one-third of the tolerance equals 13.33 kg. 

28 kg is greater than 13.33 kg so the correct response to the question is "NO". 

Does the expanded uncertainty meet the requirements of the user's process? 

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Chapter 6: 

Sample Calculations and Recommendations for Reducing Uncertainty 

The answer to this question must come from the scale operator. The scale operator must be 

knowledgeable about the uncertainty requirements for his situation, whether for NIST HB44 

tolerances or other regulatory requirements. 

 

If the answer to either of the last two questions is 'NO', ways of decreasing the calibration 

uncertainty must be sought. 

 

First, the uncertainty calculations must be reviewed for correctness. 

Are there any mathematical errors? 

Second, evaluate possible changes to the calibration process. 

Begin with the largest uncertainty contributor, working to the least significant. 

In the sample calculation, the effect of the wind and air currents on the scale performance are the 

largest contributor. Rescheduling the test on a calm day would have an associated cost but 

eliminating the effect of wind and air currents by re-scheduling could reduce the expanded 

uncertainty to 17 kg by reducing the uw value in the calculation from 11.54 kg to 0 kg. 

The next most significant uncertainty contributor is the estimated standard deviation of the scale. 

If a discrimination test were performed, the equation sp = 0.29d could be used, reducing the 

standard deviation estimate, sp, by a factor of almost two. The expanded uncertainty would 

further decrease to 8.6 kg which would meet the requirements of HB 44. Potentially, performing 

a series of measurements and calculating the standard deviation of seven or more measurement 

results could reduce the standard deviation even more, though the potential does exist that the 

true standard deviation is larger than the estimate. But, there is an associated cost in labor and 

loss of use of the scale during the time required for the additional test. 

A related uncertainty contributor is the use of substitution loads, since the contribution of the 

substitution process is based on the standard deviation of the scale. If 20 000 kg of calibrated 

load were used to test the scale instead of using 10 000 kg of substitution load, the uncertainty 

would be further reduced to 6.5 kg. 

By addressing the three largest uncertainty contributors the expanded uncertainty has been 

reduced by a factor of four, from 28 kg to 6.5 kg. While this may not be typical of all scale 

calibration uncertainties, significant reductions are often possible. The cost vs. benefit must be 

evaluated for each situation, based on the requirements and use of the weighing device. 

Third, repair, relocation or replacement of the scale might be considered. 

Is the large uncertainty due to location or maintenance issues that can be corrected? Large 

uncertainties may be the result of an influence that has a constant effect on the measurement 

results, e.g., a vibration source that is always present when the scale is being used or calibrated. 

But they can also be random events that are unnoticed except for their influence on the scale 

operation. This can be especially true in an industrial environment where heavy equipment, rock 

crushers, trains, ovens, machinery, etc. are in constant use near the scale. In this situation, 

moving the scale must be considered as one option. The service technician must be aware of 

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Chapter 6: 

Sample Calculations and Recommendations for Reducing Uncertainty 

potential sources of measurement errors and take appropriate action to minimize their influence 

on the measurement results. 

The process changes required to reduce the calibration uncertainty must be a collaborative effort 

between the service personnel and the scale operator in order for the resulting reduced calibration 

uncertainty to remain valid. 

And remember, uncertainty calculations must be performed for each calibration load as different 

tolerances will apply to the calibrated weights, and different test loads may incorporate fewer or

additional substitution loads.

8. Reporting Uncertainty 

The uncertainty statement of the scale calibration should include sufficient information that a person 

not associated with the calibration will understand what uncertainty components were included, how 

and why. The following is a sample: 

"The k=2 Expanded Uncertainty of the described scale calibration was ± 28 kg at 20 000 kg. 

This value was determined by using the root-sum-squared (RSS) combination of the 

significant measurement influences. Included are: 

the standard deviation of the scale estimated from the size of the scale division (d); 

the uncertainty of the calibrated mass standards estimated from the allowable calibration 

tolerance of the weights; 

the effects of instability due to air currents; 

the potential effects of temperature variations during the test; and 

the effect of using substitution load estimated from the repeatability of the scale. 

The RSS combined value was multiplied by a coverage factor (k) of 2 to obtain a value 

fitting the probability distribution for a 95 % confidence interval." 

(When multiple loads are applied with differing calculated uncertainty values, it may be necessary 

to use a table format as part of the uncertainty statement.) 

Typically, the scale operator must use the reported calibration expanded uncertainty(ies) as one

component of the normal weighing operations process uncertainty(ies). However, if service 

personnel were provided a measured sp value, calculated from the results of measurements of a 

check standard used in the normal weighing process, the reported expanded uncertainty would be 

the uncertainty associated with the normal weighing process. But this is true only if the sp value was 

determined by measurement results for a check standard. 

Reducing Process Variability 

Measurement uncertainty is largely dependent on the environment in which a scale or balance is 

used or proper use of the weighing device (operator effects). The single greatest uncertainty 

contributor for a weighing device is typically the variability or repeatability of the device. Except 

for some unique situations involving balances with very small capacities, mass standards can be 

selected so that the uncertainty due to the standards is relatively insignificant. Thus, the area of 

concentration when attempting to reduce uncertainty must be those factors that contribute to the 

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Chapter 6: 

Sample Calculations and Recommendations for Reducing Uncertainty 

variability of the weighing device. The factors which most affect variability are normally 

environmental in nature, involving vibration, temperature and humidity. Additional measurement 

influences affecting variability include operator training, skill and attitude, air currents, product 

contamination, etc. Appropriate steps must be taken to minimize these measurement influences if 

improved balance or scale uncertainties are to result. 

Changes Needed in Current Calibration Procedures to Address Uncertainty 

The current selection of available scale and balance calibration procedures includes: 

- OIML R 76-1, “Nonautomatic weighing instruments, Part1: Metrological and technical 

requirements-Tests”, 

- ASTM E 319-85, “Standard Practice for the Evaluation of Single-Pan Mechanical Balances”, 

- ASTM E 898-88, “Standard Method of Testing Top-Loading, Direct-Reading Laboratory Scales 

and Balances”, 

- ASTM E 1270-88, “Standard Test Method for Equal-Arm Balances”; 

- NIST Handbook 44, “Specifications, Tolerances, and Other Technical Requirements for 

Weighing and Measuring Devices”, and 

- NIST Handbook 112, "Examination Procedure Outlines for Commercial Weighing and 

Measuring Devices." 

These procedures do not address the issue of calibration uncertainty in any manner except for a 

requirement in R 76-1, 3.7.1 Weights, that 

"The standard weights or standard masses used for the verification of an instrument shall not have 

an error greater than 1/3 of the maximum permissible error of the instrument for the applied load." 

and a similar requirement in HB 44, Appendix A, Fundamental Considerations, that 

"The error in a standard used by a weights and measures official should be known and corrected 

for when the standard is used; or if the standard is to be used without correction, its error should be 

not greater than one-third of the smallest tolerance to be applied when the standard is used."

This lack of direction in the procedures has led to a general disregard of the issue of uncertainty in 

scale and balance calibrations for many years. Some regulatory bodies have attempted to address this 

issue but, in most cases, have not provided sufficient guidance to enable users to properly and 

accurately assess the uncertainty of their weighing devices. 

It should be noted that the common practice of stating the tolerance of the applied standard masses as 

the uncertainty of a scale or balance calibration is not acceptable because it does not comply with the 

requirement of ISO/IEC 17025 for a documented uncertainty analysis and does not include all known 

measurement influences. 

Though it is desired that a thorough discussion of uncertainty contributors be included in these 

procedures, including specific methodologies for evaluation and calculation of the calibration 

uncertainty, it is strongly recommended that the responsible standards writing organizations include, 

as a minimum, a rudimentary analysis of the measurement influences that contribute to process 

uncertainties. Without an uncertainty analysis resulting in a stated measurement uncertainty, the 

measurement results are just numbers on a page. They have no chance of being traceable, as 

traceability requires an unbroken chain of comparisons, all having stated uncertainties. 

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Appendix A 

Common Terminology and Definitions

Appendix A 

Common Terminology and Definitions 

Accuracy Class (VIM 5.19): class of measuring instruments that meet certain metrological 

requirements that are intended to keep errors within specified limits. 

Note: Usually denoted by a number or symbol adopted by convention and called the class index.

Accuracy of measurement (VIM 3.5): closeness of agreement between the result of a measurement 

and a true value of the measurand. 

Calibration (VIM 6.11): Set of operations that establish, under specified conditions, the relationship 

between values of quantities indicated by a measuring instrument or measuring system, or values

represented by a material measure or a reference material, and the corresponding values realized by 

standards. 

Notes: 

1) the result of a calibration permits either the assignment of values of measurands to the indications or the 

determination of corrections with respect to indications. 

2) A calibration may also determine other metrological properties such as the effect of influence quantities. 

3) The result of a calibration may be recorded in a document, sometimes called a calibration certificate or a 

calibration report. 

Combined standard uncertainty (GUM): standard uncertainty of the result of a measurement 

when that result is obtained from the values of a number of other quantities, equal to the positive 

square root of a sum of terms, the terms being the variances or covariances of these other quantities 

weighted according to how the measurement result varies with changes in the quantities. 

(Author): The combined standard uncertainty and the method of calculation is explained in this 

Recommended Practice Guide and is typically the result of the RSS combining of the values from the 

uncertainty contributors. The combined standard uncertainty is the value multiplied by the coverage 

factor (k) to obtain the expanded uncertainty. 

Corner load or ‘off-center’ error (ASTM E 898): differences in indicated weight when a sample 

weight is shifted to various positions on the weighting area of the sample pan. 

Coverage factor (k factor) (GUM): numerical factor used as a multiplier of the combined standard 

uncertainty in order to obtain an expanded uncertainty 

Note: a coverage factor, k, is typically in the range of 2 to 3. k = 2 is the standard coverage factor to be used. It defines 

a confidence interval of approximately 95 %. The use of other coverage factors requires justification of the reason for 

use. 

Discrimination (threshold) (VIM 5.11): The largest change in a stimulus that produces no 

detectable change in the response of a measuring instrument, the change in the stimulus taking place 

slowly and monotonically. 

Expanded uncertainty (GUM): quantity defining an interval about the result of a measurement that 

may be expected to encompass a large fraction of the distribution of values that could reasonably be 

attributed to the measurand 

Notes: 

1) the fraction may viewed as the coverage probability or level of confidence of the interval

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Appendix A 

Common Terminology and Definitions

2) to associate a specific level of confidence with the interval defined by the expanded uncertainty requires explicit 

or implicit assumptions regarding the probability distribution characterized by the measurement result and its 

combined standard uncertainty. The level of confidence that may be attributed to this interval can be known 

only to the extent to which such assumptions may be justified. 

Experimental Standard deviation (VIM 3.8): for a series of n measurements of the same

measurand, the quantity ‘s’ characterizing the dispersion of the results and given by the formula: 

1

( )

1

2

−

−

=

∑=

n

x x

s

n

i

i

 with xi being the result of the ith measurement and x being the arithmetic mean of 

the n results considered. 

Independent Variables (Author): a property of two or more values where the value of each is not 

related to or dependent on any portion of another. 

Influence Quantity (VIM 2.7): quantity that is not the measurand, but that affects the result of the

measurement. 

Example: Ambient air temperature 

Linearity error (ASTM E 898): the degree to which a graph of weight values indicated by a scale 

vs. the true values of the respective test weights approximates a straight line. 

(Author): error caused by deviations of an indicator from a constant sensitivity, with changing load. 

Mass Correction: value algebraically added to the nominal mass value to obtain the calibrated mass 

value of a mass artifact. 

(Author’s interpretation of definition given in ASTM E 617.) 

Measurand (VIM 2.6): particular quantity subject to measurement.

Normal distribution (paraphrased from the on-line NIST Engineering Statistical Handbook): 

Distribution function for which one value (the mean) has a greater probability of occurrence than all 

other values. A normal distribution can be estimated with the sample mean and sample standard 

deviation. 

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Appendix A 

Common Terminology and Definitions

The normal distribution is commonly described as being a bell shaped distribution. Note that the 

mean, μ, is the most probable value. 

Probability (The American Heritage® Dictionary of the English Language: Fourth Edition): 3. The

likelihood that a given event will occur. 

Statistics: A number expressing the likelihood that a specific event will occur, expressed as the ratio 

of the number of actual occurrences to the number of possible occurrences. 

Rectangular distribution: See Uniform distribution.

Repeatability (VIM 5.27): closeness of agreement between the results of successive measurements 

of the same measurand carried out under the same conditions of measurement. 

Note: ‘Same conditions’ includes the same procedure, same observer, same measuring instrument, same location, with 

repetition over a short period of time.

Reproducibility (VIM 3.7): Closeness of agreement between the results of measurements of the 

same measurand carried out under changed conditions of measurement. 

Resolution (VIM 5.12): Smallest difference between indications of a displaying device that can be 

meaningfully distinguished. 

Note: for a digital displaying device this is the change in the indication when the least significant digit changes by one 

step. 

Scale Division, value of (d) (HB 44): the value of the scale division, expressed in units of mass, is 

the smallest subdivision of the scale for analog indication or the difference between two 

consecutively indicated values for digital indication. 

Sensitivity (VIM 5.10): change in the response of a measuring instrument divided by the 

corresponding change in the stimulus, for example: 10 grams per division. 

Service Personnel: used in this guide to identify the skilled personnel who repair, calibrate or 

service scales and balances, primarily in the field environment. 

Stability (VIM 5.14): the ability of a measuring instrument to maintain constant its metrological 

characteristics with time. 

Standard: For the purposes of this document, 'standard' refers to the calibrated weight artifact used 

to apply a known load to a weighing device. The word 'standard' is used in other circumstances to 

describe a prescriptive document containing specifications to which a device must conform.

Standard Uncertainty (GUM): Uncertainty of the result of a measurement expressed as one 

standard deviation. 

Strain-Load Test (HB 44): The test of a scale beginning with the scale under load, applying known 

test weights to determine accuracy over a portion of the weighing range, not beginning at a zero 

indication. 

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Appendix A 

Common Terminology and Definitions

Substitution Load: quantity of material applied to a scale to cause a duplicate scale indication to 

that generated by known test weights. The known test weights can then be added with the 

substitution load, effectively multiplying the known load. Additional uncertainty results from the use 

of substitution loads. The substitution process can be repeated several times (limited to no more than 

three substitutions by HB 44, Table 4), which requires additional uncertainty to be included for each 

substitution. 

Tolerance (NIST HB 44): A value fixing the limit of allowable error or departure from true 

performance or value. 

Traceability (VIM 6.10): property of the result of a measurement of the value of a standard whereby 

it can be related to stated references, usually national or international standards, through an unbroken 

chain of comparisons all having stated uncertainties. 

Note: the unbroken chain of comparisons is called a traceability chain.

Uncertainty of measurement (VIM 3.9): parameter, associated with the result of a measurement, 

that characterizes the dispersion of the values that could reasonably be attributed to the measurand. 

Notes: 

1) The parameter may be, for example, a standard deviation (or a given multiple of it), or a half-width of an interval 

having a stated level of confidence. 

2) Uncertainty of measurement comprises, in general, many components. Some of these components may be 

evaluated from the statistical distribution of the results of series of measurements and can be characterized by 

experimental standard deviations. The other components, which can also be characterized by standard deviation, 

are evaluated from assumed probability distributions based on experience or other information. 

3) It is understood that the result of the measurement is the best estimate of the value of the measurand, and that all 

components of uncertainty, including those arising from systematic effects, such as components associated with 

corrections and reference standards, contribute to the dispersion. 

Uncertainty (The American Heritage® Dictionary of the English Language, Fourth Edition): 

“Statistical: The estimated amount or percentage by which an observed or calculated value may 

differ from the true value. 

SYNONYMS: doubt, dubiety, skepticism, suspicion, mistrust. These nouns refer to the condition of 

being unsure about someone or something.” 

Uniform distribution (paraphrased from the on-line NIST Engineering Statistical Handbook): 

Distribution function where any value within the range of the function is equally probable. 

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Appendix A 

Common Terminology and Definitions

Sometimes called a rectangular distribution due to its rectangular shape. This distribution function is 

considered the most conservative estimate of uncertainty, i.e., it gives the largest standard deviation. 

Verification scale division, value of (e) (HB 44): a value expressed in units of weight (mass) and 

specified by the manufacturer of a device, by which the tolerance values and the accuracy class 

applicable to the device are determined. The verification scale division, e, may be different from the 

displayed scale division, d, for certain devices used for weight classifying or weighing in predetermined amounts and certain class I and II scales. 

(The value of ‘e’ may be set by the denomination of the smallest possible weight, typically 1 milligram, especially when 

the value of ‘d’ is less than 1 milligram.) 

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Appendix A 

Common Terminology and Definitions

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Appendix B 

Bibliography

Appendix B 

Bibliography 

American Society for Testing and Materials (ASTM) E 319 - 85, "Standard Practice for the 

Evaluation of Single-Pan Mechanical Balances", ASTM, 1997 

American Society for Testing and Materials (ASTM) E 898 - 88, "Standard Method of Testing TopLoading, Direct-Reading Laboratory Scales and Balances", ASTM, 2000 

American Society for Testing and Materials (ASTM) E 1270 - 88, "Standard Test Method for Equal 

Arm Balances", ASTM, 1997 

American Society for Testing and Materials (ASTM) E 617 - 97, "Standard Specification for 

Laboratory Weights and Precision Mass Standards", ASTM, 1997 

American National Standards Institute/National Conference of Standards Laboratories International 

(ANSI/NCSLI) Z540-2-1997 “U.S. Guide to the Expression of Uncertainty in Measurement”, 

ANSI/NCSLI, 1997 

Arthur Reichmuth, "Influence Sources and Prevention in Weighing", Mettler Toledo, Oct 1999 

Arthur Reichmuth, "Non-linearity of Laboratory Balances and Its impact on Uncertainty", 2000 

NCSL Workshop and Symposium, 2000 

Deutscher Kalibrierdienst, DKD-R-7-1, Issue 98, “Calibration of Non-Automatic Electronic 

Weighing Instruments”, DKD, 1998

EURACHEM/CITAC Guide, “Quantifying Uncertainty in Analytical Measurement”, 

EURACHEM/CITAC, 2000 

Georgia Harris, “Ensuring Accuracy and Traceability of Weighing Instruments”, ASTM 

Standardization News, 21(4): 44-51, 1993 

Georgia Harris, "Implementation of the Guide to the Expression of Uncertainty in Measurement for 

State Calibration Laboratories", 1994 NCSL Workshop & Symposium, 1994 

Houghton Mifflin Company, The American Heritage® Dictionary of the English Language, Fourth 

Edition., Houghton Mifflin Company, 2000 

International Organization for Standardization, “Guide to the Expression of Uncertainty in 

Measurement”, (GUM), International Organization for Standardization, 1993 

International Organization for Standardization, "General requirements for the Competence of Testing 

and Calibration Laboratories", ISO/IEC 17025:, International Organization for Standardization, 1999 

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Appendix B 

Bibliography

International Organization for Standardization, "International Vocabulary of Basic and General 

Terms in Metrology (VIM)", International Organization for Standardization, 1993 

Jerry Everhart, "Determining Mass Measurement Uncertainty", Cal Lab, May/June 1995 

Mettler-Toledo, "Fundamentals of Mass Determination", Mettler-Toledo AG, 1991 

National Conference on Weights and Measures Publication 14, "National Type Evaluation Program, 

Administrative Procedures, Technical Policy, Checklists, and Test Procedures", National Conference 

on Weights and Measures, 2002 

National Conference of Standards Laboratories Recommended Practice RP-12 "Determining & 

Reporting Measurement Uncertainties," National Conference of Standards Laboratories International, 

1995 

National Institute of Standards and Technology Technical Note 1297, 1994 Edition, “Guidelines for 

Evaluating and Expressing the Uncertainty of NIST Measurement Results”, National Institute of

Standards and Technology, 1994 

National Institute of Standards and Technology Handbook 44, "Specifications, Tolerances, and Other 

Technical Requirements for Weighing and Measuring Devices," NIST, 2002 

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, 

2001 

National Institute of Standards and Technology Handbook 112, "Examination Procedure Outlines for 

Commercial Weighing and Measuring Devices", National Institute of Standards and Technology,

2002 

National Institute of Standards and Technology Handbook 105-1, " Specifications and Tolerances for 

Reference Standards and Field Standard Weights and Measures, 1. Specifications and Tolerances for 

Field Standard Weights (NIST Class F)", National Institute of Standards and Technology, 1990 

Organisation Internationale de Métrologie Légale (OIML), OIML R 76-1, “Nonautomatic weighing 

instruments, Part 1: Metrological and technical requirements-Tests”, OIML, Edition 1992(E) 

Organisation Internationale de Métrologie Légale (OIML), OIML R 111 Committee Draft OIML/CD 

R 111, "Weights of Classes E1, E2, F1, F2, M1, M2, M3", OIML, 2002 

W. J. Youden, "NIST Special Publication 672: Experimentation and Measurement", National 

Institute of Standards and Technology, 1997 

Walter E. Kupper, Dr.scn.techn, “High Accuracy Mass Measurement, From Micrograms to Tons”, 

ISA Transactions, (Vol 29, 4, 1990)

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