EURAMET Calibration Guide No. 18 

Version 4.0 (10/2015) 

I-CAL-GUI-018/v4.0/2015-10-01

Authorship and Imprint 

This document was developed by the EURAMET e.V., Technical Committee for Mass and Related 

Quantities. 

Version 4.0 was developed thanks to the cooperation of Stuart Davidson (NPL, UK), Klaus Fritsch 

(Mettler Toledo, Switzerland), Matej Grum (MIRS, Slovenia), Andrea Malengo (INRIM, Italy), Nieves 

Medina (CEM, Spain), George Popa (INM, Romania), Norbert Schnell (Sartorius, Germany). 

Version 4.0 (11/2015) 

Version 3.0 (03/2011) 

Version 2.0 (09/2010) 

Version 1.0 (01/2009) 

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ISBN 978-3-942992-40-4 

Guidance Publications 

This document gives guidance on measurement practices in the specified fields of measurements. By 

applying the recommendations presented in this document laboratories can produce calibration results 

that can be recognized and accepted throughout Europe. The approaches taken are not mandatory and 

are for the guidance of calibration laboratories. The document has been produced as a means of 

promoting a consistent approach to good measurement practice leading to and supporting laboratory 

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EURAMET Technical Committee for Mass and Related Quantities (see www.euramet.org). 
EURAMET Calibration Guide No. 18 

Version 4.0 (11/2015) 

EURAMET Calibration Guide No. 18 

Version 4.0 (11/2015) 

Guidelines on the Calibration of 

Non-Automatic Weighing Instruments 

Purpose 

This document has been produced to enhance the equivalence and mutual recognition of 

calibration results obtained by laboratories performing calibrations of non-automatic weighing 

instruments. 

 
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Content 

1 INTRODUCTION ................................................................................................................... 4

2 SCOPE .................................................................................................................................. 4

3 TERMINOLOGY AND SYMBOLS ......................................................................................... 5

4 GENERAL ASPECTS OF THE CALIBRATION .................................................................... 5

4.1 Elements of the calibration ............................................................................................. 5

4.1.1 Range of calibration ......................................................................................................... 5

4.1.2 Place of calibration ........................................................................................................... 5

4.1.3 Preconditions, preparations ............................................................................................. 6

4.2 Test load and indication ................................................................................................. 6

4.2.1 Basic relation between load and indication ...................................................................... 6

4.2.2 Effect of air buoyancy ....................................................................................................... 6

4.2.3 Effects of convection ........................................................................................................ 8

4.2.4 Buoyancy correction for the reference value of mass ...................................................... 9

4.3 Test loads ..................................................................................................................... 10

4.3.1 Standard weights ............................................................................................................ 10

4.3.2 Other test loads .............................................................................................................. 10

4.3.3 Use of substitution loads ................................................................................................ 11

4.4 Indications .................................................................................................................... 12

4.4.1 General ........................................................................................................................... 12

4.4.2 Resolution ...................................................................................................................... 12

5 MEASUREMENT METHODS ............................................................................................. 13

5.1 Repeatability test .......................................................................................................... 13

5.2 Test for errors of indication ........................................................................................... 14

5.3 Eccentricity test ............................................................................................................ 15

5.4 Auxiliary measurements ............................................................................................... 16

6 MEASUREMENT RESULTS ............................................................................................... 16

6.1 Repeatability ................................................................................................................. 17

6.2 Errors of indication ....................................................................................................... 17

6.2.1 Discrete values ............................................................................................................... 17

6.2.2 Characteristic of the weighing range .............................................................................. 17

6.3 Effect of eccentric loading ............................................................................................ 18

7 UNCERTAINTY OF MEASUREMENT ................................................................................ 18

7.1 Standard uncertainty for discrete values ...................................................................... 19

7.1.1 Standard uncertainty of the indication ............................................................................ 19

7.1.2 Standard uncertainty of the reference mass .................................................................. 21

7.1.3 Standard uncertainty of the error ................................................................................... 24

7.2 Standard uncertainty for a characteristic ...................................................................... 25

7.3 Expanded uncertainty at calibration ............................................................................. 25

7.4 Standard uncertainty of a weighing result .................................................................... 25

7.4.1 Standard uncertainty of a reading in use ....................................................................... 27

7.4.2 Uncertainty of the error of a reading .............................................................................. 27

7.4.3 Uncertainty from environmental influences .................................................................... 28

7.4.4 Uncertainty from the operation of the instrument ........................................................... 29

7.4.5 Standard uncertainty of a weighing result ...................................................................... 31

7.5 Expanded uncertainty of a weighing result ................................................................... 32

7.5.1 Errors accounted for by correction ................................................................................. 32
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7.5.2 Errors included in uncertainty ......................................................................................... 32

7.5.3 Other ways of qualification of the instrument ................................................................. 33

8 CALIBRATION CERTIFICATE ............................................................................................ 34

8.1 General information ...................................................................................................... 34

8.2 Information about the calibration procedure ................................................................. 34

8.3 Results of measurement .............................................................................................. 35

8.4 Additional information ................................................................................................... 35

9 VALUE OF MASS OR CONVENTIONAL VALUE OF MASS .............................................. 36

9.1 Value of mass ............................................................................................................... 36

9.2 Conventional value of mass ......................................................................................... 36

10 REFERENCES .................................................................................................................... 37

APPENDIX A: ADVICE FOR ESTIMATION OF AIR DENSITY ................................................. 38

A1 Formulae for the density of air ...................................................................................... 38

A1.1 Simplified version of CIPM-formula, exponential version ............................................... 38

A1.2 Average air density ......................................................................................................... 38

A2 Variations of parameters constituting the air density .................................................... 39

A2.1 Barometric pressure ....................................................................................................... 39

A2.2 Temperature ................................................................................................................... 39

A2.3 Relative humidity ............................................................................................................ 39

A3 Uncertainty of air density .............................................................................................. 40

APPENDIX B: COVERAGE FACTOR k FOR EXPANDED UNCERTAINTY OF 

MEASUREMENT ....................................................................................................................... 41

B1 Objective ...................................................................................................................... 41

B2 Normal distribution and sufficient reliability .................................................................. 41

B3 Normal distribution, no sufficient reliability ................................................................... 42

B4 Determining k for non-normal distributions ................................................................... 42

APPENDIX C: FORMULAE TO DESCRIBE ERRORS IN RELATION TO THE INDICATIONS 43

C1 Objective ...................................................................................................................... 43

C2 Functional relations ...................................................................................................... 43

C3 Terms without relation to the readings ......................................................................... 49

APPENDIX D: SYMBOLS .......................................................................................................... 50

APPENDIX E: INFORMATION ON AIR BUOYANCY ................................................................ 52

E1 Density of standard weights ......................................................................................... 52

E2 Air buoyancy for weights conforming to OIML R111 .................................................... 52

APPENDIX F: EFFECTS OF CONVECTION ............................................................................. 54

F1 Relation between temperature and time....................................................................... 54

F2 Change of the apparent mass ...................................................................................... 56

APPENDIX G: MINIMUM WEIGHT ............................................................................................ 58

APPENDIX H: EXAMPLES ........................................................................................................ 60

H1 Instrument of 220 g capacity and scale interval 0,1 mg ............................................... 60

H2 Instrument of 60 kg capacity, multi-interval .................................................................. 75

H3 Instrument of 30 000 kg capacity, scale interval 10 kg .................................................. 91

H4 Determination of the error approximation function ..................................................... 109
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1 INTRODUCTION 

Non-automatic weighing instruments (NAWI) are widely used to determine the value of a 

load in terms of mass. For some applications specified by national legislation, NAWI are 

subject to legal metrological control – i.e. type approval, verification etc. – but there is an 

increasing need to have their metrological quality confirmed by calibration, e.g. where 

required by ISO 9001 or ISO/IEC 17025 standards. 

2 SCOPE 

This document contains guidance for the static calibration of self-indicating, nonautomatic weighing instruments (hereafter called “instrument”), in particular for 

1. measurements to be performed, 

2. calculation of the measuring results, 

3. determination of the uncertainty of measurement, 

4. contents of calibration certificates. 

The object of the calibration is the indication provided by the instrument in response to 

an applied load. The results are expressed in units of mass. The value of the load 

indicated by the instrument will be affected by local gravity, the load temperature and 

density, and the temperature and density of the surrounding air. 

The uncertainty of measurement depends significantly on properties of the calibrated 

instrument itself, not only on the equipment of the calibrating laboratory; it can to some 

extent be reduced by increasing the number of measurements performed for a 

calibration. This guideline does not specify lower or upper boundaries for the uncertainty 

of measurement. 

It is up to the calibrating laboratory and the client to agree on the anticipated value of the 

uncertainty of measurement that is appropriate in view of the use of the instrument and 

in view of the cost of the calibration. 

While it is not intended to present one or few uniform procedures the use of which would 

be obligatory, this document gives general guidance for establishing of calibration 

procedures the results of which may be considered as equivalent within the EURAMET 

Member Organisations. 

Any such procedure must include, for a limited number of test loads, the determination of 

the error of indication and of the uncertainty of measurement assigned to these errors. 

The test procedure should as closely as possible resemble the weighing operations that 

are routinely being performed by the user – e.g. weighing discrete loads, weighing 

continuously upwards and/or downwards, use of tare balancing function. 

The procedure may further include rules how to derive from the results advice to the 

user of the instrument with regard to the errors, and assigned uncertainty of 

measurement, of indications which may occur under normal conditions of use of the 

instrument, and/or rules on how to convert an indication obtained for a weighed object 

into the value of mass or conventional value of mass of that object. 

The information presented in this guideline is intended to serve, and should be observed 

by 

1. bodies accrediting laboratories for the calibration of weighing instruments, 

2. laboratories accredited for the calibration of non-automatic weighing 

instruments, 
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3. test houses, laboratories, or manufacturers using calibrated non-automatic 

weighing instruments for measurements relevant for the quality of production 

subject to QM requirements (e.g. ISO 9000 series, ISO 10012, ISO/IEC 17025). 

3 TERMINOLOGY AND SYMBOLS 

The terminology used in this document is mainly based on existing documents 

 JCGM 100 [1] for terms related to the determination of results and the 

uncertainty of measurement, 

 OIML R76 [2] (or EN 45501 [3]) for terms related to the functioning, to the 

construction, and to the metrological characterisation of non-automatic weighing 

instruments, 

 OIML R111 [4] for terms related to the standard weights, 

 JCGM 200 [5] for terms related to the calibration. 

Such terms are not explained in this document, but where they first appear, references 

will be indicated. 

Symbols whose meanings are not self-evident, will be explained where they are first 

used. Those that are used in more than one section are collected in Appendix D. 

4 GENERAL ASPECTS OF THE CALIBRATION 

4.1 Elements of the calibration 

Calibration consists of 

1. applying test loads to the instrument under specified conditions, 

2. determining the error or variation of the indication, and 

3. evaluating the uncertainty of measurement to be attributed to the results. 

4.1.1 Range of calibration 

Unless requested otherwise by the client, a calibration extends over the full weighing 

range [2] (or [3]) from zero to the maximum capacity Max . The client may specify a 

certain part of a weighing range, limited by a minimum load Min and the largest load to 

be weighed Max, or individual nominal loads, for which he requests calibration. 

On a multiple range instrument [2] (or [3]), the client should identify which range(s) shall 

be calibrated. The paragraph above may be applied to each range separately. 

4.1.2 Place of calibration 

Calibration is normally performed in the location where the instrument is being used. 

If an instrument is moved to another location after the calibration, possible effects from 

1. difference in local gravity acceleration, 

2. variation in environmental conditions, 

3. mechanical and thermal conditions during transportation 

are likely to alter the performance of the instrument and may invalidate the calibration. 

Moving the instrument after calibration should therefore be avoided, unless immunity to 

these effects of a particular instrument, or type of instrument has been clearly 

demonstrated. Where this has not been demonstrated, the calibration certificate should 

not be accepted as evidence of traceability. 
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4.1.3 Preconditions, preparations 

Calibration should not be performed unless 

1. the instrument can be readily identified, 

2. all functions of the instrument are free from effects of contamination or damage, 

and functions essential for the calibration operate as intended, 

3. presentation of weight values is unambiguous and indications, where given, are 

easily readable, 

4. the normal conditions of use (air currents, vibrations, stability of the weighing 

site etc.) are suitable for the instrument to be calibrated, 

5. the instrument is energized prior to calibration for an appropriate period, e.g. as 

long as the warm-up time specified for the instrument, or as set by the user, 

6. the instrument is levelled, if applicable, 

7. the instrument has been exercised by loading approximately up to the largest 

test load at least once, repeated loading is advised. 

Instruments that are intended to be regularly adjusted before use should be adjusted 

before the calibration, unless otherwise agreed with the client. Adjustment should be 

performed with the means that are normally applied by the client, and following the 

manufacturer’s instructions where available. Adjustment could be done by means of 

external or built-in test loads. 

The most suitable operating procedure for high resolution balances (with relative 

resolution better 1 × 10-5 of full scale) is to perform the adjustment of the balance 

immediately before the calibration and also immediately before use. 

Instruments fitted with an automatic zero-setting device or a zero-tracking device [2] (or 

[3]) should be calibrated with the device operative or not, as set by the client. 

For on site calibration the user of the instrument should be asked to ensure that the 

normal conditions of use prevail during the calibration. In this way disturbing effects such 

as air currents, vibrations, or inclination of the measuring platform will, so far as is 

possible, be inherent in the measured values and will therefore be included in the 

determined uncertainty of measurement. 

4.2 Test load and indication 

4.2.1 Basic relation between load and indication 

In general terms, the indication of an instrument is proportional to the force exerted by 

an object of mass m on the load receptor 

I = ks mg  1 a  (4.2.1-1) 

with g local gravity acceleration 

 a density of the surrounding air 

 density of the object 

ks adjustment factor 

The terms in the brackets account for the reduction of the force due to the air buoyancy 

of the object. 

4.2.2 Effect of air buoyancy 

It is state of the art to use standard weights that have been calibrated to the 
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conventional value of mass mc

1, for the adjustment and/or the calibration of weighing 

instruments. In principle, at the reference air density 0 = 1,2 kg/m3, the balance should 

indicate the conventional mass mc of the test object. 

The adjustment is performed at an air density ρas and is such that the effects of g and of 

the actual buoyancy of the adjustment weight having conventional mass mcs are included 

in the adjustment factor ks. Therefore, at the moment of the adjustment, the indication s I

is 

s mcs I  (4.2.2-1) 

This adjustment is performed under the conditions characterized by the actual values of

gs , s  c , and as  0 , identified by the suffix “s”, and is valid only under these 

conditions. For another body of conventional mass mc with    s , weighed on the 

same instrument but under different conditions: g  gs and a  as the indication is in 

general (neglecting terms of 2nd or higher order) [6] 

      a as  s c s a 0 s I  m g / g 1    1  1      /  (4.2.2-3) 

If the instrument is not displaced, there will be no variation of g , so 1 s g g  . This is 

assumed hereafter. 

The indication of the balance will be exactly the conventional mass of the body, only in 

some particular cases, the most evident are 

 ρa = ρas = ρ0. 

 the weighing is performed at ρa = ρas and the body has a density ρ = ρs. 

The formula simplifies further in situations where some of the density values are equal 

a) weighing a body in the reference air density: a  0 , then 

    c a as s I  m 1    /  (4.2.2-4) 

b) weighing a body of the same density as the adjustment weight:   s , then 

again (as in case a)) 

    c 1  a  as  s I  m   / (4.2.2-5) 

c) weighing in the same air density as at the time of adjustment: a  as , then 

     c a 0 s I  m 1    1  1  (4.2.2-6) 

Figure 4.2-1 shows examples for the magnitude of the relative changes 

 

1

The conventional value of mass mc of a body has been defined in [4] as the numerical value of mass m of a weight of reference 

density c = 8000 kg/m3 which balances that body at 20 °C in air of density 0 : 

mc  m1  0  /1  0  c  (4.2.2-2) 

with 0 = 1,2 kg/m3 = reference value of the air density
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  c c c I / m  I  m / m for an instrument adjusted with standard weights of s  c , when 

calibrated with standard weights of different but typical density. 

Line  is valid for a body of   7 810 kg/m3, weighed in a  as (as for case c above) 

Line × is valid for a body of   8 400 kg/m3, weighed in a  as (as for case c above) 

Line  is valid for a body of   s  c after adjustment in as  0 (as for case b above) 

It is obvious that under these conditions, a variation in air density has a far greater effect 

than a variation in the body density. 

Further information on air density is given in Appendix A, and on air buoyancy related to 

standard weights in Appendix E. 

4.2.3 Effects of convection 

Where weights have been transported to the calibration site they may not be at the same 

temperature as the instrument and its environment. The temperature difference T is 

defined as the difference between the temperature of a standard weight and the 

temperature of the surrounding air. Two phenomena should be noted in this case: 

 An initial temperature difference T0 may be reduced to a smaller value T by 

acclimatisation over a time t ; this occurs faster for smaller weights than for 

larger ones. 

 When a weight is put on the load receptor, the actual difference T will 

produce an air flow about the weight leading to parasitic forces which result in 

an apparent change mconv on its mass. The sign of mconv is normally 

opposite to the sign of T , its value being greater for large weights than for 

small ones. 

The relations between any of the quantities mentioned: T0 , t , T , m and mconv

are nonlinear, and they depend on the conditions of heat exchange between the weights 

and their environment – see [7]. 
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Figure 4.2-2 gives an impression of the magnitude of the apparent change in mass in 

relation to a temperature difference, for some selected weight values. 

This effect should be taken into account by either letting the weights acclimatise to the 

extent that the remaining change mconv is negligible in view of the uncertainty of the 

calibration required by the client, or by considering the possible change of indication in 

the uncertainty budget. The effect may be significant for weights of high accuracy, e.g. 

for weights of class E2 or F1 in R 111 [4]. 

More detailed information is given in Appendix F. 

4.2.4 Buoyancy correction for the reference value of mass 

To determine the errors of indication of an instrument, standard weights of known 

conventional value of mass mcCal are applied. Their density  Cal is normally different 

from the reference value  c and the air density aCal at the time of calibration is 

normally different from  0 . 

The error E of indication is 

ref E  I  I (4.2.4-1) 

where ref I is the reference value of the indication of the instrument, further called 

reference value of mass, mref. Due to effects of air buoyancy, convection, drift and others 

which may lead to minor correction termsmx , mref is not exactly equal to mcCal , the 

conventional value of the mass 

.. mref  mcCal  mB  m (4.2.4-2) 

The correction for air buoyancy mB is affected by values of s and as , that were valid 

for the adjustment but are not normally known. It is assumed that weights of the 

reference density s  c have been used. From (4.2.2-3) the general expression for the 

correction is 
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      mB  mcCal aCal  0 1  Cal 1 c  aCal  as c (4.2.4-3) 

For the air density as two situations are considered. If the instrument has been 

adjusted immediately before the calibration, then as  aCal . This simplifies (4.2.4-3) to 

   mB  mcCal aCal  0 1  Cal 1 c (4.2.4-4) 

If the instrument has been adjusted independent of the calibration, in unknown air 

density as , it is not possible to perform the correction for the last term of equation 

(4.2.4-3), which intrinsically forms part of the error of indication. The correction to be 

applied should also be (4.2.4-4) [10]. 

The suffix “Cal” will from now on be omitted unless where necessary to avoid confusion. 

4.3 Test loads 

Test loads should preferably consist of standard weights that are traceable to the SI unit 

of mass. However, other test loads may be used for tests of a comparative nature – e.g. 

test with eccentric loading, repeatability test – or for the mere loading of an instrument – 

e.g. preloading, tare load that is to be balanced, substitution load. 

4.3.1 Standard weights 

The traceability of weights to be used as standards shall be demonstrated by calibration 

[8] consisting of 

1. determination of the conventional value of mass mc and/or the correction mc

to its nominal value mN : mc  mc  mN , together with the expanded uncertainty 

of the calibration U95 , or 

2. confirmation that mc is within specified maximum permissible errors mpe : 

  mN  mpe U95 ≤ mc ≤   mN  mpe U95

The standards should further satisfy the following requirements to an extent 

appropriate to their accuracy: 

3. density s sufficiently close to c = 8 000 kg/m3, 

4. surface finish suitable to prevent a change in mass through contamination by 

dirt or adhesion layers, 

5. magnetic properties such that interaction with the instrument to be calibrated is 

minimized. 

Weights that comply with the relevant specifications of the International 

Recommendation OIML R 111 [4] should satisfy all these requirements. 

The maximum permissible errors, or the uncertainties of calibration of the standard 

weights, shall be compatible with the scale interval d [2] (or [3]) of the instrument and/or 

the needs of the client with regard to the uncertainty of the calibration of the instrument. 

4.3.2 Other test loads 

For certain applications mentioned in 4.3, 2nd sentence, it is not essential that the 

conventional value of mass of a test load is known. In these cases, loads other than 

standard weights may be used, with due consideration of the following 

1. shape, material, composition should allow easy handling, 
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2. shape, material, composition should allow the position of the centre of gravity to 

be readily estimated, 

3. their mass must remain constant over the full period they are in use for the 

calibration, 

4. their density should be easy to estimate, 

5. loads of low density (e.g. containers filled with sand or gravel), may require 

special attention in view of air buoyancy. Temperature and barometric pressure 

may need to be monitored over the full period the loads are in use for the 

calibration. 

4.3.3 Use of substitution loads 

A test load, of which the conventional value of mass must be known, should be made up 

entirely of standard weights. But where this is not possible, or where the standard 

weights are not sufficient to calibrate the normal range of the instrument or the range 

agreed with the customer, any other load which satisfies 4.3.2 may be used for 

substitution. The instrument under calibration is used as a comparator to adjust the 

substitution load Lsub so that it brings about approximately the same indication I as the 

corresponding load LSt made up of standard weights. 

A first test load LT1 made up of standard weights mref is indicated as 

   St mref I L  I (4.3.3-1) 

After removing LSt a substitution load Lsub1 is put on and adjusted to give approximately 

the same indication 

   sub1 mref I L  I (4.3.3-2) 

so that 

    sub1 ref sub1 ref ref 1 L  m  I L  I m  m  I (4.3.3-3) 

The next test load LT2 is made up by adding mref

T2 sub1 ref 2 ref 1 L  L  m  m  I (4.3.3-4) 

mref is again replaced by a substitution load of ≈ Lsub1 with adjustment to ≈   LT2 I . 

The procedure may be repeated, to generate test loads LT3, ...,LTn

Tn  ref   1   2    n1 L nm I I  I (4.3.3-5a) 

With each substitution step however, the uncertainty of the total test load increases 

substantially more than if it were made up of standard weights only, due to the effects of 

repeatability and resolution of the instrument. – cf. also 7.1.2.62. 

If the test load LT1 is made up of more than one standard weight, it is possible to first use 

 

2

 Example: for an instrument with Max = 5000 kg, d= 1 kg, the standard uncertainty of 5 t standard weights of accuracy class M1 

– based on their nominal value, and using (7.1.2-3) – is around150 g, while the standard uncertainty of a test load made up of 1 t 

standard weights and 4 t substitution load, using (7.1.2-16a), will be about 1,2 kg. In this example, uncertainty contributions due to 

buoyancy and drift were neglected. Equally, it was assumed that the uncertainty of the indication only comprises the rounding error 

at no-load and at load. 
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the standard weights to create N individual test loads mref,k (k = 1,..., N) with the 

condition 

ref,1 ref 2 mref mref LT1 m m ...  ,   ,N   . (4.3.3-6) 

Afterwards, LT1 is substituted by a substitution load Lsub1 , and then the test loads mref,k

can again be added consecutively. The individual test loads shall be referred to as LTn,k

with 

  Tn,k  1 ref  ref ,k   1   2    n1 L n m m I I  I . (4.3.3-5b) 

4.4 Indications 

4.4.1 General 

Any indication I related to a test load is basically the difference of the indications L I

under load and 0 I at no-load, before the load is applied 

0 I I I  L  (4.4.1-1a) 

It is preferable to record the no-load indications together with the load indications for any 

test measurement. In the case that the user of the instrument takes into account the zero 

return of any loading during normal use of the instrument, e.g. in the case of a 

substantial drift, the indication can be corrected according to equation (4.4.1-1b)3. 

However, recording the no-load indications may be redundant where a test procedure 

calls for a balance to be zeroed before a test load is applied. 

For any test load, including no-load, the indication I of the instrument is read and 

recorded only when it can be considered as being stable. Where high resolution of the 

instrument, or environmental conditions at the calibration site prevent stable indications, 

an average value should be estimated and recorded together with information about the 

observed variability (e.g. spread of values, unidirectional drift). 

During calibration tests, the original indications should be recorded, not errors or 

variations of the indication. 

4.4.2 Resolution 

Indications are normally obtained as integer multiples of the scale interval d. 

At the discretion of the calibration laboratory and with the consent of the client, means to 

obtain indications in higher resolution than in d may be applied, e.g. where compliance 

to a specification is checked and smallest uncertainty is desired. Such means may be 

1. switching the indicating device to a smaller scale interval dT<d (“service mode”). 

In this case, the indications are obtained as integer multiple of dT. 

2. applying small extra test weights in steps of 5 T d  d or d 10 to determine 

more precisely the load at which an indication changes unambiguously from I

to I  d (“changeover point method”). In this case, the indication I’ is recorded 

 

3

In case of linear drift the corrected reading is given by 

  2 L 0 0i I  I  I  I (4.4.1-1b) 

where I0 and I0i are the no-load indications before and after the load is applied. 
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together with the amount L of the n additional small test weights necessary to 

increase I by one d . 

The indication L I is 

2 d 2 ndT I I d L I '

L         (4.4.2-1) 

Where the changeover point method is applied, it is advised to apply it for the indications 

at zero as well as for the indications at load. 

5 MEASUREMENT METHODS 

Tests are normally performed to determine 

 the repeatability of indications, 

 the errors of indications, 

 the effect of eccentric application of a load on the indication. 

A Calibration Laboratory deciding on the number of measurements for its routine 

calibration procedure should consider that, in general, a larger number of measurements 

tends to reduce the uncertainty of measurement but increase the cost. 

Details of the tests performed for an individual calibration may be fixed by agreement of 

the client and the Calibration Laboratory, in view of the normal use of the instrument. 

The parties may also agree on further tests or checks which may assist in evaluating the 

performance of the instrument under special conditions of use. Any such agreement 

should be consistent with the minimum numbers of tests as specified in the following 

sections. 

5.1 Repeatability test 

The test consists of the repeated deposition of the same load on the load receptor, 

under identical conditions of handling the load and the instrument, and under constant 

test conditions. 

The test load(s) need not be calibrated nor verified, unless the results serve for the 

determination of errors of indication as per 5.2. The test load should, as far as possible, 

consist of one single body. 

The test is performed with at least one test load LT which should be selected in a 

reasonable relation to Max and the resolution of the instrument, to allow an appraisal of 

the instrument performance. For instruments with a constant scale interval d a load of 

about Max  L  Max T 0,5 is quite common; this is often reduced for instruments where 

LT would amount to several 1000 kg. For multi-interval instruments [2] (or [3]) a load 

below and close to Max1may be preferred. For multiple range instruments, a load below 

and close to the capacity of the range with the smallest scale interval may be sufficient. 

A special value of LT may be agreed between the parties where this is justified in view 

of a specific application of the instrument. 

The test may be performed at more than one test point, with test loads LTj , L 1 j  k

with L k = number of test points. 

Prior to the test, the indication is set to zero. The load is to be applied at least 5 times, or 
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at least 3 times where LT ≥ 100 kg. 

Indications Li I are recorded for each deposition of the load. After each removal of the 

load, the indication should be checked, and may be reset to zero if it does not show 

zero; recording of the no-load indications i I 0 may be advisable as per 4.4.1. In addition, 

the status of the zero-setting or zero-tracking device if fitted should be recorded. 

5.2 Test for errors of indication 

This test is performed with L k ≥ 5 different test loads LTj , 1 ≤ j ≤ L k , distributed fairly 

evenly over the normal weighing range or at individual test points agreed upon as per 

4.1.1. Examples for target values 

 L k = 5: zero or Min; 0,25 Max; 0,5 Max; 0,75 Max;Max. Actual test loads may 

deviate from the target value up to 0,1 Max, provided the difference between 

consecutive test loads is at least 0,2 Max, 

 L k = 11: zero or Min, 10 steps of 0,1 Max up to Max. Actual test loads may 

deviate from the target value up to 0,05 Max, provided the difference between 

consecutive test loads is at least 0,08 Max. 

The purpose of this test is an appraisal of the accuracy of the instrument over the whole 

weighing range. 

Where a significantly smaller range of calibration has been agreed, the number of test 

loads may be reduced accordingly, provided there are at least 3 test points including 

Min and Max, and the difference between two consecutive test loads is not greater 

than 0,15Max. 

It is necessary that test loads consist of appropriate standard weights or of substitution 

loads as per 4.3.3. 

Prior to the test, the indication is set to zero. The test loads LTj are normally applied 

once in one of these manners 

1. increasing by steps with unloading between the separate steps – corresponding 

to the majority of uses of the instruments for weighing single loads, 

2. continuously increasing by stepswithout unloading between the separate steps; 

this may include creep effects in the results but reduces the amount of loads to 

be moved on and off the load receptor as compared to 1, 

3. continuously increasing and decreasing by steps – procedure prescribed for 

verification tests in [2] (or [3]), same comments as for 2, 

4. continuously decreasing by steps starting from Max - simulates the use of an 

instrument as hopper weigher for subtractive weighing, same comments as for 

2. 

On multi-interval instruments – see [2] (or [3]), the methods above may be modified for 

load steps smaller than Max , by applying increasing and/or decreasing tare loads, 

taring the instrument, and applying a test load close to but not larger than Max1 to obtain 

indications with d1. 

On a multiple range instrument [2] (or [3]), the client should identify which range(s) shall 

be calibrated (see 4.1.1, 2nd paragraph). 
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Further tests may be performed to evaluate the performance of the instrument under 

special conditions of use, e.g. the indication after a tare balancing operation, the 

variation of the indication under a constant load over a certain time, etc. 

The test, or individual loadings, may be repeated to combine the test with the 

repeatability test under 5.1. 

Indications Lj I are recorded for each load. In the case that the loads are removed, the 

zero indication should be checked, and may be reset to zero if it does not show zero; 

recording of the no-load indications j I 0 may be advisable as per 4.4.1. 

5.3 Eccentricity test 

The test comprises placing a test load Lecc in different positions on the load receptor in 

such a manner that the centre of gravity of the applied load takes the positions as 

indicated in Figure 5.3-1 or equivalent positions, as closely as possible. 

Fig. 5.3-1 Positions of load for test of eccentricity 

1. Centre 

2. Front left 

3. Back left 

4. Back right 

5. Front right 

There may be applications where the test load cannot be placed in or close to the centre 

of the load receptor. In this case, it is sufficient to place the test load at the remaining 

positions as indicated in Figure 5.3-1. Depending on the platter shape, the number of the 

off-centre positions might deviate from figure 5.3-1. 

The test load Lecc should be about Max 3 or higher, or Min    Max  Min 3 or 

higher for a reduced weighing range. 

Advice of the manufacturer, if available, and limitations that are obvious from the design 

of the instrument should be considered – e.g. see OIML R76 [2] (or EN 45501 [3]) for 

special load receptors. 

For a multiple range instrument [2] (or [3]) the test should only be performed in the range 

with the largest capacity identified by the client (see 4.1.1, 2nd paragraph). 

The test load need not be calibrated or verified, unless the results serve to determine the 

errors of indication as per 5.2. 

The test can be carried out in different manners: 

1. Prior to the test, the indication is set to zero. The test load is first put on position 

1, is then moved to the other 4 positions in arbitrary order. Indications Li I are 

recorded for each position of the load. 

2. The test load is first put on position 1, then the instrument is tared. The test load 

is then moved to the other 4 positions in arbitrary order. Indications Li I are 

recorded for each position of the load. 

3. Prior to the test, the indication is set to zero. The test load is first put on position 

1, removed, and then put to the next position, removed, etc. until it is removed 
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from the last position. Indications Li I are recorded for each position of the load. 

After each removal of the load, the indication should be checked, and may be 

reset to zero if it does not show zero; recording of the no-load indications i I 0

may be advisable as per 4.4.1. 

4. The test load is first put on position 1, then the instrument is tared. The test load 

is then moved to the next position and moved back to position 1, etc. until it is 

removed from the last position. The center indication L1 I is recorded individually 

for all off-centre indications Li I . 

Method 3 and 4 are suggested for instruments that show a substantial drift during the 

time of the eccentricity test. 

For methods 2 and 4 zero-setting or zero-tracking devices must be switched off during 

the complete eccentricity test. 

5.4 Auxiliary measurements 

The following additional measurements or recordings are recommended, in particular 

where a calibration is intended to be performed with the lowest possible uncertainty. 

In view of buoyancy effects – cf. 4.2.2: 

The air temperature in reasonable vicinity to the instrument should be measured, at 

least once during the calibration. Where an instrument is used in a controlled 

environment, the span of the temperature variation should be noted, e.g. from a 

thermograph, from the settings of the control device etc. 

Barometric pressure or, by default, the altitude above sea-level of the site may also 

be useful. 

In view of convection effects – cf. 4.2.3: 

Special care should be taken to prevent excessive convection effects, by observing 

a limiting value for the temperature difference between standard weights and 

instrument, and/or recording an acclimatisation time that has been executed. A 

thermometer kept inside the box with standard weights may be helpful, to check the 

temperature difference. 

In view of effects of magnetic interaction: 

On high resolution instruments a check is recommended to see if there is an 

observable effect of magnetic interaction. A standard weight is weighed together 

with a spacer made of non-metallic material (e.g. wood, plastic), the spacer being 

placed on top or underneath the weight to obtain two different indications. 

If the difference between these two indications is significantly different from zero, 

this should be mentioned as a warning in the calibration certificate. 

6 MEASUREMENT RESULTS 

The procedures and formulae in chapters 6 and 7 provide the basis for the evaluation of 

the results of the calibration tests and therefore require no further description on a test 

report. If the procedures and formulae used deviate from those given in the guide, 

additional information may need to be provided in the test report. 

It is not intended that all of the formulae, symbols and/or indices are used for 

presentation of the results in a Calibration Certificate. 
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The definition of an indication I as given in 4.4 is used in this section. 

6.1 Repeatability 

From the n indications ji I for a given test load LTj , the standard deviation j s is 

calculated 

  

  

n

i

j ji j I I

n

s

1

2

1

1 (6.1-1) 

with 





n

i

j ji I

n

I

1

1 (6.1-2) 

Where only one test load has been applied, the index j may be omitted. 

6.2 Errors of indication 

6.2.1 Discrete values 

For each test load LTj , the error of indication is calculated as follows 

j j m j E I   ref (6.2-1) 

Where an indication j I is the mean of more than one reading, j I is understood as 

being the mean value as per (6.1-2). 

The reference value of mass mref could be approximated to its nominal value mNj

mrefj  mNj (6.2-2) 

or, more accurately, to its actual conventional value mc

  mrefj  mcj  mNj  mcj (6.2-3) 

If a test load is made up of more than one weight, mNj is replaced by  j mN and mcj

is replaced by  j mc in the formulae above. 

Furthercorrections as per (7.1.2-1) might apply. 

6.2.2 Characteristic of the weighing range 

In addition, or as an alternative to the discrete values j I , E j , a characteristic, or 

calibration curve may be determined for the weighing range, which allows estimation of 

the error of indication for any indication I within the weighing range. 

A function 

E  f  I (6.2-4) 

may be generated by an appropriate approximation which should, in general, be based 

on the “least squares” approach 
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       2 2

j j Ej v f I minimum (6.2-5) 

with 

j v = residual 

f = approximation function 

The approximation should further 

 take account of the uncertainties   u E j of the errors, 

 use a model function that reflects the physical properties of the instrument, e.g. 

the form of the relation between load and its indication I  g  L , 

 include a check as to whether the parameters found for the model function are 

mathematically consistent with the actual data. 

It is assumed that for any mNj the error E j remains the same if the actual indication j I

is replaced by its nominal value j I N . The calculations to evaluate (6.2-5) can therefore 

be performed with the data sets mNj , E j , or j I N , E j . 

Appendix C offers advice for the selection of a suitable approximation formula and for 

the necessary calculations. 

6.3 Effect of eccentric loading 

From the indications i I obtained in the different positions of the load as per 5.3, the 

differences ecc I are calculated. 

For method 1 and 2 as per 5.3 

ecci Li L1 I  I  I (6.3-1) 

For method 3 as per 5.3 

  ecci Li 0i L1 I  I  I  I (6.3-2) 

For method 4 as per 5.3 

i Li L i I I I  ecc   1 (6.3-3) 

where for each off-centre indication Li I the respective centre indication L i I 1 is taken for 

the calculation. 

7 UNCERTAINTY OF MEASUREMENT 

In this and the following sections, there are uncertainty terms assigned to small 

corrections, which are proportional to a specified mass value or to a specified indication. 

For the quotient of such an uncertainty divided by the related value of mass or indication, 

the abbreviated notation urel will be used. 

Example: let 

u  m  mucorr  corr (7-1) 
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with the dimensionless term ucorr, then 

u   m  ucorr rel  corr (7-2) 

Accordingly, the related variance will be denoted by   corr

2

rel

u m and the related 

expanded uncertainty by   Urel mcorr . 

For the determination of uncertainty, second order terms have been considered 

negligible, but when first order contributions cancel out, second order contributions 

should be taken into account (see JCGM 101 [9], 9.3.2.6). 

7.1 Standard uncertainty for discrete values 

The basic formula for the calibration is 

mref E  I  (7.1-1) 

with variance 

     ref

2 2 2

u E  u I  u m (7.1-2) 

Where substitution loads are employed, see 4.3.3, mref is replaced by LTn or LTn,k in 

both expressions. 

The terms are further expanded hereafter. 

7.1.1 Standard uncertainty of the indication 

To account for sources of variability of the indication, (4.4.1-1) is amended by correction 

terms xx I as follows 

I I I I I I I ...  L   digL   rep   ecc  0  dig0  (7.1.1-1) 

Further correction terms may be applied in special conditions (temperature effects, drift, 

hysteresis,..), which are not considered hereafter. 

All these corrections have the expectation value zero. Their standard uncertainties are 

7.1.1.1 Rdig0 accounts for the rounding error of no-load indication. Limits are  d0 2 or 2 T  d

as applicable; rectangular distribution is assumed, therefore 

  2 3 dig0 d0 u I  (7.1.1-2a) 

or 

  2 3 dig0 dT u I  (7.1.1-2b) 

respectively. 

Note 1: cf. 4.4.2 for significance of T d . 

Note 2: on an instrument which has been type approved to OIML R76 [2] (or EN 45501 

[3]), the rounding error of a zero indication after a zero-setting or tare balancing 

operation is limited to  d0 4 , therefore 
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  4 3 dig0 d0 u I  (7.1.1-2c) 

7.1.1.2 L I  dig accounts for the rounding error of indication at load. Limits are 2 I  d or 2 T  d

as applicable; rectangular distribution to be assumed, therefore 

u  IdigL  dI 2 3 (7.1.1-3a) 

or 

u  IdigL  dT 2 3 (7.1.1-3b) 

Note: on a multi-interval instrument, I d varies with I.

7.1.1.3 rep I accounts for the repeatability of the instrument; normal distribution is assumed, 

estimated as 

  j u I  s I  rep (7.1.1-5) 

where  j s I is determined in 6.1. 

If the indication I is a single reading and only one repeatability test has been 

performed, this uncertainty of repeatability can be considered as representative for the 

whole range of the instrument. 

Where an indication j I is the mean of N indications performed with the same test load 

during the error of indication test, the corresponding standard uncertainty is 

u   Irep  s I j N (7.1.1-6) 

Where several j s (   j j s  s I in abbreviated notation) values have been determined with 

different test loads, the greater value of j s for the two test points enclosing the 

indication whose error has been determined, should be used. 

For multi-interval and multiple range instruments where a repeatability test was carried 

out in more than one interval/range, the standard deviation of each interval/range may 

be considered as being representative for all indications of the instrument in the 

respective interval/range. 

Note: For a standard deviation reported in a calibration certificate, it should be clear 

whether it is related to a single indication or to the mean of N indications. 

7.1.1.4 ecc I accounts for the error due to off-centre position of the centre of gravity of a test 

load. This effect may occur where a test load is made up of more than one body. Where 

this effect cannot be neglected, an estimate of its magnitude may be based on these 

assumptions: 

 the differences ecc I determined by (6.3-1) are proportional to the distance of 

the load from the centre of the load receptor, 

 the differences ecc I determined by (6.3-1) are proportional to the value of the 

load, 
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 the effective centre of gravity of the test loads is not further from the centre of 

the load receptor than half the distance between the load receptor centre and 

the eccentricity load positions, as per figure 5.3-1. 

Based on the largest of the differences determined as per 6.3, ecc I is estimated to be 

I  I  L I  ecc   ecci max 2 ecc (7.1.1-9) 

Rectangular distribution is assumed, so the standard uncertainty is 

  2 3 ecc ecc max Lecc u I I I    i (7.1.1-10) 

or, in relative notation 

  2 3 rel ecc ecc max Lecc u I I    i (7.1.1-11) 

7.1.1.5 The standard uncertainty of the indication is normally obtained by 

      2

ecc

2

rep rel

2 2 2

0

2

u I d 12 d 12 u I u I I   I     (7.1.1-12) 

Note 1: the uncertainty u I is constant only where s is constant and no eccentricity 

error has to be considered. 

Note 2: the first two terms on the right hand side may have to be modified in special 

cases as mentioned in 7.1.1.1 and 7.1.1.2. 

7.1.2 Standard uncertainty of the reference mass 

From 4.2.4 and 4.3.1 the reference value of mass is 

mref  mN  mc  mB  mD  mconv  m (7.1.2-1) 

The rightmost term stands for further corrections which may be necessary to apply under 

special conditions.These are not considered hereafter. 

The corrections and their standard uncertainties are 

7.1.2.1 mc is the correction to mN to obtain the conventional value of mass mc ; given in the 

calibration certificate for the standard weights, together with the uncertainty of calibration 

U and the coverage factor k . The standard uncertainty is 

u  m U k  c (7.1.2-2) 

Where the standard weight has been calibrated to specified tolerances Tol , e.g. to the 

mpe given in OIML R111 [4], and where it is used its nominal value mN , then mc = 0, 

and rectangular distribution is assumed, therefore 

u  mc  Tol 3 (7.1.2-3) 

Where a test load consists of more than one standard weight, the standard uncertainties 

are summed arithmetically not by a sum of squares, to account for assumed correlation. 

For test loads partially made up of substitution loads see 7.1.2.6. 
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7.1.2.2 mB is the correction for air buoyancy as introduced in 4.2.4. The value depends on the 

density  of the calibration weight and on the assumed range of air density  a at the 

laboratory. 

   B N a 0 c m  m    1  1  (7.1.2-4) 

with relative standard uncertainty 

         2 2 4

a 0

2

a c

2

B

2

urel m  u  1  1      u   (7.1.2-5a)4

As far as values for  , u   ,  a and   u a , are known, these values should be used 

to determine   urel mB . 

The density  and its standard uncertainty may, in the absence of such information, be 

estimated according to the state of the art or based on information provided by the 

manufacturer. Appendix E1 offers internationally recognized values for common 

materials used for standard weights. 

The air density  a and its standard uncertainty can be calculated from temperature and 

barometric pressure if available (the relative humidity being of minor influence), or may 

be estimated from the altitude above sea-level. 

Where conformity of the standard weights to OIML R111 [4] is established, and no 

information on  and  a is at hand, recourse may be taken to section 10 of OIML 

R1115. No correction is applied, and the relative uncertainties are 

If the instrument is adjusted immediately before calibration 

  4 3 rel mB mpe mN u   (7.1.2-5c) 

If the instrument is not adjusted before calibration 

urel  mB    0,10 c  mpe 4mN  3 (7.1.2-5d) 

If some information can be assumed for the temperature variation at the location of the 

instrument, equation (7.1.2-5d) can be substituted by: 

  107 10 133 10 K   4 3 0 c N

4 6 2 2

urel mB  ,   ,  T   mpe m       (7.1.2-5e) 

where T is the maximum variation of environmental temperature that can be assumed 

for the location (see appendixes A2.2 and A3 for details). 

From the requirement in footnote 5, the limits of the value of  can be derived: e.g. for 

class E2:   c  200 kg/m3, and for class F1:   c  600 kg/m3. 

 

4

 A more accurate formula for (7.1.2-5a) would be [10] 

           2 4

a 0 a 0 a1 0

2

a c

2

B

2 urel m  u  1  1          2    u   (7.1.2-5b)

where a1 is the air density at the time of the calibration of the standard weights. This formula is useful when the instrument is 

located at high altitude above sea level, otherwise the uncertainty could be overestimated. 

5

The density of the material used for weights shall be such that a deviation of10 % from the specified air density (1.2 kg/m3) does 

not produce an error exceeding one quarter of the maximum permissible error. 
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Note: Due to the fact that the density of materials used for standard weights is 

normally closer to  c than the OIML R111 limits would allow, the last 3 

formulae may be considered as upper limits for   rel mB u  . Where a simple 

comparison of these values with the resolution of the instrument   d / Max

shows they are small enough, a more elaborate calculation of this uncertainty 

component based on actual data may be superfluous. 

7.1.2.3 mD corresponds to the possible drift of mc since the last calibration. A limiting value D

is best assumed, based on the difference in mc evident from consecutive calibration 

certificates of the standard weights. 

D may be estimated in view of the quality of the weights, and frequency and care of 

their use, to at least a multiple of their expanded uncertainty   U mc

 (7.1.2-10) 

where Dk is a chosen value between 1 and 3. 

In the absence of information on drift the value of D will be chosen as the mpe

according to OIML R 111 [4]. 

It is not advised to apply a correction but to assume even distribution within  D

(rectangular distribution). The standard uncertainty is then 

u  m D 3  D  (7.1.2-11) 

Where a set of weights has been calibrated with a standardised expanded relative 

uncertainty   Urel mc , it may be convenient to introduce a relative limit value for drift 

Drel  D mN and a relative uncertainty for drift 

urel  mD  Drel 3 (7.1.2-12) 

7.1.2.4 mconv corresponds to the convection effects as per 4.2.3. A limiting value mconv may 

be taken from Appendix F, depending on a known difference in temperature T and on 

the mass of the standard weight. 

It is not advised to apply a correction but to assume even distribution within  mconv . 

The standard uncertainty is then 

u  mconv  mconv 3 (7.1.2-13) 

It appears that this effect is only relevant for weights of classes F1 or better. 

7.1.2.5 The standard uncertainty of the reference mass is obtained from – cf. 7.1.2 

          conv

2 2

B

2

c

2

ref

2

u m  u m  u m  u mD  u m (7.1.2-14) 

with the contributions from 7.1.2.1 to 7.1.2.4. 

  U mc D k  D 
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7.1.2.6 Where a test load is partially made up of substitution loads as per 4.3.3, and the test 

loads are defined per (4.3.3-5a), the standard uncertainty for the sum 

Tn  ref   1   2    n1 L nm I I  I is given by the following expression 

           1

2

2

2

1

2

ref

2 2

T

2 2 n      n u L n u m u I u I  u I (7.1.2-15a) 

with   u mref from 7.1.2.5, and  j u I from 7.1.1.5 for   L j I I  T

Where a test load is partially made up of substitution loads as per 4.3.3, and the test 

loads are defined per (4.3.3-5b), the standard uncertainty for the sum 

  T , ref ref , 1 2 1 1 n k    k        n L n m m I I  I is given by the following expression 

               1

2

2

2

1

2 2

T , ref ref ,

2 1 2 n k    k     n u L n u m u m u I u I  u I (7.1.2-15b) 

with   u mref from 7.1.2.5, and  j u I from 7.1.1.5 for   L j I I  T

Note: The uncertainties  j u I also have to be included for indications where the 

substitution load has been adjusted in such a way that the corresponding I

becomes zero. 

Depending on the kind of the substitution load, it may be necessary to add further 

uncertainty contributions 

 for eccentric loading as per 7.1.1.4 to some or all of the actual indications 

  L j I T

 for air buoyancy of the substitution loads, where these are made up of low 

density materials (e.g. sand, gravel) and the air density varies significantly over 

the time the substitution loads are in use. 

Where  j u I = const, the expression (7.1.2-15a) simplifies to 

u   L n u m  n u I  n

2

ref

2 2

T

2   2 1 (7.1.2-16a) 

and the expression (7.1.2-15b) simplifies to 

u   L     n u m  u m  n u I  n k k

2 2

T , ref ref ,

2   1   2  1 (7.1.2-16b) 

7.1.3 Standard uncertainty of the error 

The standard uncertainty of the error is, with the terms from 7.1.1 and 7.1.2, as 

appropriate, calculated from 

         

       conv

2 2

B

2

c

2

ecc

2

rep

2

dig

2

dig 0

2 2

u m u m u m u m

u E u I u I u I u I

D

I

   

   

   

    (7.1.3-1a) 

or, where relative uncertainties apply, from 

         

         conv

2 2

ref

2

B rel

2

c rel

2

rel

2

ecc

2

rep rel

2

dig

2

dig0

2 2

u m u m u m m u m

u E u I u I u I u I I

D

I

   

   

   

   

 (7.1.3-1b) 

In the case of using substitution loads 

            n k I u L n k u E u I u I u I u I T ,

2

ecc

2

rep

2

dig

2

dig0

2

,

2          (7.1.3-1c) 
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where n is related to the number of substitution steps and k is the number of standard 

weights. 

All input quantities are considered to be uncorrelated, therefore covariances are not 

considered. 

The index “ j ”has been omitted. 

In view of the general experience that errors are normally very small compared to the 

indication, or may even be zero, in (7.1.3-1b) the values for mref and I may be replaced 

by NI . 

The terms in (7.1.3-1b) may then be grouped into a simple formula which better reflects 

the fact that some of the terms are absolute in nature while others are proportional to the 

indication 

  2 2 2 2

u E     I (7.1.3-2) 

7.2 Standard uncertainty for a characteristic 

Where an approximation is performed to obtain a formula E  f  I for the whole 

weighing range as per 6.2.2, the standard uncertainty of the error per 7.1.3 has to be 

modified to be consistent with the method of approximation. Depending on the model 

function, this may be 

 a single variance which is added to (7.1.3-1), or 

 a set of variances and covariances which include the variances in (7.1.3-1). 

The calculations should also include a check whether the model function is 

mathematically consistent with the data sets E j , j I ,   u E j . 

The minimum 2  approach, which is similar to the least squares approach, is proposed 

for approximations. Details are given in Appendix C. 

7.3 Expanded uncertainty at calibration 

The expanded uncertainty of the error is 

U  E  k u E (7.3-1) 

The coverage factor k should be chosen such that the expanded uncertainty 

corresponds to a coverage probability of 95,45 %. 

Further information on how to derive the coverage factor is given in Appendix B. 

7.4 Standard uncertainty of a weighing result 

Chapter 7.4 and 7.5 provide advice on how the measurement uncertainty of an 

instrument could be estimated in normal usage, thereby taking into account the 

measurement uncertainty at calibration. Where a calibration laboratory offers to its 

clients such estimates which are based upon information that has not been measured by 

the laboratory, the estimates must not be presented as part of the calibration certificate. 

However, it is acceptable to provide such estimates as long as they are clearly 

separated from the calibration results. 

The user of an instrument should be aware of the fact that in normal usage, the situation 

is different from that at calibration in some if not all of these aspects 
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1. the indications obtained for weighed bodies are not the ones at calibration, 

2. the weighing process may be different from the procedure at calibration 

a. generally only one reading is taken for each load, not several readings to 

obtain a mean value, 

b. reading is to the scale interval d , of the instrument, not to a higher 

resolution, 

c. loading is up and down, not only upwards – or vice versa, 

d. load may be kept on load receptor for a longer time, not unloading after 

each loading step – or vice versa, 

e. eccentric application of the load, 

f. use of tare balancing device, etc. 

3. the environment (temperature, barometric pressure etc.) may be different, 

4. for instruments which are not readjusted regularly e.g. by use of a built-in 

device, the adjustment may have changed, due to drift or to wear and tear. 

Unlike items 1 to 3, this effect should therefore be considered in relation to a 

certain period of time, e.g. for one year or the normal interval between 

calibrations, 

5. the repeatability of the adjustment. 

In order to clearly distinguish from the indications I obtained during calibration, the 

weighing results obtained when weighing a load L on the calibrated instrument, these 

terms and symbols are introduced 

 RL = reading when weighing a load L on the calibrated instrumentobtained 

after the calibration. 

 R0 = reading without load on the calibrated instrumentobtained after the 

calibration. 

Readings are taken to be single readings in normal resolution (multiple of d ), with 

corrections to be applied as applicable. 

For a reading taken under the same conditions as those prevailing at calibration, the 

result may be denominated as the weighing result under the conditions of the calibration 

W *

W*  RL  RdigL  Rrep  Recc  R0  Rdig0  E (7.4-1a) 

with the associated uncertainty 

             ecc

2

rep

2

dig

2

dig0

2 2

u W *  u E  u R  u R L  u R  u R (7.4-2a) 

To take account of the remaining possible influences on the weighing result, further 

corrections are formally added to the reading in a general manner resulting in the 

general weighing result 

 (7.4-1b) 

where Rinstr represents a correction term due to environmental influences and Rproc

represents a correction term due to the operation of the instrument. 

W  W *Rinstr Rproc
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The associated uncertainty is 

       proc

2

instr

2 2

u W  u W *  u R  u R (7.4-2b) 

The added terms and the corresponding standard uncertainties are discussed in 7.4.3 

and 7.4.4. The standard uncertainties uW * and uW  are finally presented in 7.4.5. 

Sections 7.4.3 and 7.4.4, 7.4.5 and 7.5, are meant as advice to the user of the 

instrument on how to estimate the uncertainty of weighing results obtained under their 

normal conditions of use. They are not meant to be exhaustive or mandatory. 

7.4.1 Standard uncertainty of a reading in use 

To account for sources of variability of the reading, (7.1.1-1) applies, with I replaced by 

R

R  RL  RdigL  Rrep  Recc  R0  Rdig0  (7.4.1-1) 

The corrections and their standard uncertainties are 

7.4.1.1 Rdig 0 accounts for the rounding error at zero reading. 7.1.1.1 applies with the exception 

that the variant d d T  , is excluded, so 

u  Rdig0  d0 12 (7.4.1-2) 

7.4.1.2 RdigL accounts for the rounding error at load reading. 7.1.1.2 applies with the exception 

that the variant L d  d T is excluded, so 

u  RdigL  dL 12 (7.4.1-3) 

7.4.1.3 Rrep accounts for the repeatability of the instrument. 7.1.1.3 applies, the relevant 

standard deviation s for a single reading is to be taken from the calibration certificate, so 

u  R  s  rep or u R   sR  rep (7.4.1-4) 

Note: The standard deviation not the standard deviation of the mean should be used 

for the uncertainty calculation. 

7.4.1.4 Recc accounts for the error due to off-centre position of the centre of gravity of a load. 

  2 3 rel ecc ecc max Lecc u R I    i (7.4.1-5) 

7.4.1.5 The standard uncertainty of the reading is then obtained by 

       2

 2

ecc max ecc

2 2 2

0

2

u R  d 12  dL 12  s R  I i 2L 3 R (7.4.1-6) 

7.4.2 Uncertainty of the error of a reading 

Where a reading R corresponds to an indication j Ical reported in the calibration 

certificate,   u Ecalj may be taken from there. For other readings, u  E may be calculated 

by (7.1.3-2) if  and  are known, or it results from interpolation, or from an 
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approximation formula as per 7.2. 

The uncertainty u  E is normally not smaller than   u Ecalj for an indication j I that is 

close to the actual reading R , unless it has been determined by an approximation 

formula. 

Note: the calibration certificate normally presents   U95 Ecal from which   u Ecal is 

calculated by dividing   U95 Ecal by the coverage factor k stated in the 

certificate. 

7.4.3 Uncertainty from environmental influences 

The term Rinstr accounts for up to 3 effects Rtemp ,Rbuoy and Radj , which are discussed 

hereafter. Except for the contribution due to buoyancy, they do normally not apply to 

instruments which are adjusted directly before they are actually used. Other instruments 

should be considered as appropriate. No corrections are actually applied, the 

corresponding uncertainties are estimated, based on the user’s knowledge of the 

properties of the instrument. 

7.4.3.1 The term Rtemp accounts for a change in the characteristic of the instrument caused by 

a change in ambient temperature. A limiting value can be estimated to be 

Rtemp  KT TR where T is the maximum temperature variation at the instrument 

location and KTis the sensitivity of the instrument to temperature variation. When the 

balance is controlled by a temperature triggered adjustment by means of the built-in 

weights then T can be reduced to the trigger threshold. 

Normally there is a manufacturer’s specification such as K   I( Max )/ T / Max T    , 

in many cases quoted in 10-6/K. By default, for instruments with type approval under 

OIML R76 [2] (or EN 45501 [3]), it may be assumed     KT  mpe Max MaxTApproval

where TApproval is the temperature range of approval marked on the instrument; for 

other instruments, either a conservative assumption has to be made, leading to a 

multiple (3 to 10 times) of the comparable value for instruments with type approval, or no 

information can be given at all for a use of the instrument at other temperatures than that 

at calibration. 

The range of variation of temperature T (full width) should be estimated in view of the 

site where the instrument is being used, as discussed in Appendix A2.2. 

Rectangular distribution is assumed, therefore the relative uncertainty is 

urel  Rtemp  KTT 12 (7.4.3-1) 

7.4.3.2 The term Rbuoy accounts for a change in the adjustment of the instrument due to the 

variation of the air density; no correction to be applied. 

When the balance is adjusted immediately before use and some assumption for the 

variation in air density with respect to the air density value at the calibration time a can 

be made, the uncertainty contribution could be [10] 

    2 s

c

a

rel bouy 



 u R u  (7.4.3-2) 
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where u(s) is the uncertainty of the density of the reference weight used for adjustment 

(built-in or external). 

When the balance is not adjusted before use and some assumption for the variation in 

density a can be made, the uncertainty contribution could be 

  c 3

a

rel bouy



  

u R  (7.4.3-3) 

If some assumptions can be made for the temperature variation in the location of the 

balance, equation (7.4.3-3) can be approximated by 

  c

0

4 6 2 2

rel bouy

1 07 10 133 10 K



       

   , , T

u R (7.4.3-4) 

where T is the maximum assumed variation for the temperature in the location of the 

balance (see appendixes A2.2 and A3 for details). 

If no assumption about the density variation can be made the most conservative 

approach would be 

  3

01

c

0

rel bouy 

  , u R  (7.4.3-5) 

7.4.3.3 The term Radj accounts for a change in the characteristics of the instrument since the 

time of calibration due to drift, or wear and tear. 

A limiting value may be taken from previous calibrations where they exist, as the largest 

difference EMax in the errors at or near Max between any two consecutive 

calibrations. By default, E  Max should be taken from the manufacturer’s specification 

for the instrument, or may be estimated as EMax   mpe Max for instruments 

conforming to a type approval under OIML R76 [2] (or EN 45501 [3]). Any such value 

can be considered in view of the expected time interval between calibrations, assuming 

fairly linear progress of the change with time. 

Rectangular distribution is assumed, therefore the relative uncertainty is 

     3 urel Radj  E Max Max (7.4.3-6) 

7.4.3.4 The relative standard uncertainty related to errors resulting from environmental effects is 

calculated by 

        adj

2

buoy rel

2

temp rel

2

instr rel

2

urel R  u R  u R  u R (7.4.3-7) 

7.4.4 Uncertainty from the operation of the instrument 

The correction term Rproc accounts for additional errors (RTare ,Rtime and Recc ) which 

may occur where the weighing procedure(s) is different from the one(s) at calibration. No 

corrections are actually applied but the corresponding uncertainties are estimated, 

based on the user’s knowledge of the properties of the instrument. 
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7.4.4.1 The term RTare accounts for a net weighing result after a tare balancing operation [2] (or 

[3]). The possible error and the uncertainty assigned to it should be estimated 

considering the basic relation between the readings involved 

RNet RGross R Tare     (7.4.4-1) 

where the R are fictitious readings which are processed inside the instrument, while 

the visible indication RNet is obtained directly, after setting the instrument indication to 

zero with the tare load on the load receptor. The weighing result, in this case, is, in 

theory 

     WNet  RNet  E Gross  E Tare  Rinstr  Rproc (7.4.4-2) 

consistent with (7.4-1). The errors at gross and tare would have to be taken as errors for 

equivalent R values as above. However, the tare values – and consequently the gross 

values – are not normally recorded. 

The error may then be estimated as 

  ENet  E Net RTare (7.4.4-3) 

where E  Net is the error for a reading RNet and RTare is an additional correction for 

the effect of non-linearity of the error curve E I cal . To quantify the non-linearity, 

recourse may be taken to the first derivative of the function E  f   R , if known, or the 

slope E q between consecutive calibration points may be calculated by 

j j

j j

E I I

E E

I

E

q 

  

 





1

1 (7.4.4-4) 

The largest and the smallest values of the derivatives or of the quotients are taken as 

limiting values for the correction RTare , for which rectangular distribution may be 

assumed. This results in the relative standard uncertainty 

urel   RTare  qE max  qE min 12 (7.4.4-5) 

To estimate the uncertainty uW , R  RNet is considered. For uE it is valid to assume 

u   E  Net  u  E R  Net because there is full correlation between the quantities 

contributing to the uncertainties of the errors of the fictitious grossand tarereadings. 

7.4.4.2 The term Rtime accounts for possible effects of creep and hysteresis, in situations such 

as 

a) loading at calibration continuously upwards, or continuously upwards and 

downwards (method 2 or 3 in 5.2), so the load remains on the load receptor for a 

certain period of time; this is quite significant where the substitution method has 

been applied, usually with high capacity instruments. When in normal use, a discrete 

load to be weighed is put on the load receptor and is kept there just as long as is 

necessary to obtain a reading or a printout, the error of indication may differ from the 

value obtained for the same load at calibration. 

Where tests were performed continuously up and down, the largest difference of 
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errors Ej for any test load m j may be taken as the limiting value for this effect, 

leading to a relative standard uncertainty 

   12 urel Rtime  Ejmax mj (7.4.4-6) 

Where tests were performed only upwards, the error on return to zero E0 , if 

determined, may be used to estimate a relative standard uncertainty 

   3 urel Rtime  E0 Max (7.4.4-7) 

In the absence of such information, the limiting value may be estimated for 

instruments with type approval under OIML R76 [2] (or EN 45501 [3]) as 

E  R  R mpeMax Max (7.4.4-8) 

For instruments without such type approval, a conservative estimate would be a 

multiple (m = 3 to 10 times) of this value. 

The relative standard uncertainty is 

     3 urel Rtime  mpe Max Max , (7.4.4-9a) 

for instruments with type approval and 

     3 urel Rtime  m mpe Max Max (7.4.4-9b) 

for instruments without type approval. 

b) loading at calibration with unloading between load steps, loads to be weighed in 

normal use are kept on the load receptor for a longer period. In the absence of any 

other information – e.g. observation of the change in indication over a typical period 

of time – recourse may be taken to (7.4.4-9) as applicable. 

c) loading at calibration only upwards, discharge weighing is performed in use. This 

situation may be treated as the inverse of the tare balancing operation – see 7.4.4.1 

- combined with point b) above. (7.4.4-5) and (7.4.4-9) apply. 

Note: In case of discharge weighing, the reading R shall be taken as a positive value 

although it may be indicated as negative by the weighing instrument. 

7.4.4.3 Recc accounts for the error due to off-centre position of the centre of gravity of a load. 

(7.4.1-5) applies with the modification that the effect found during calibration should be 

considered in full, so 

   3 rel ecc ecc max Lecc u R I    i (7.4.4-10) 

7.4.5 Standard uncertainty of a weighing result 

The standard uncertainty of a weighing result is calculated from the terms specified in 

7.4.1 to 7.4.4, as applicable. 

For the weighing result under the conditions of the calibration 

u   W * d dL s R u  R   R u E 2 2

ecc

2

rel

2 2 2

0

2  12  12     (7.4.5-1a) 

For the weighing result in general 
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  

          2

time

2

Tare rel

2

adj rel

2

bouy rel

2

temp rel

2

rel

2 2

u R u R u R u R u R R

u W u W*

         

 (7.4.5-1b) 

The many contributions to u  W may be grouped in two terms 2  W and 2  W

  2 2 2 2

u W   W   W R (7.4.5-2) 

where 2  W is the sum of squares of all absolute standard uncertainties, and 2  W is the 

sum of squares of all relative standard uncertainties. 

7.5 Expanded uncertainty of a weighing result 

7.5.1 Errors accounted for by correction 

The complete formula for a weighing result which is equal to the reading corrected for 

the error determined by calibration, is 

W*  R  E  R UW * (7.5.1-1a) 

or 

W  R  E  R UW  (7.5.1-1b) 

as applicable. 

The expanded uncertainty UW  is to be determined as 

U  W *  k uW * (7.5.1-2a) 

or 

U  W  k u W (7.5.1-2b) 

with u  W * or u  W as applicable from 7.4.5. 

For U  W * the coverage factor k should be determined as per 7.3. 

For U  W the coverage factor k will, in most cases be equal to 2 even where the 

standard deviation s is obtained from only few measurements, and/or where kcal  2 

was stated in the calibration certificate. This is due to the large number of terms 

contributing to u  W . 

7.5.2 Errors included in uncertainty 

It may have been agreed by the calibration laboratory and the client to derive a “global 

uncertainty” U W  gl which includes the errors of indication such that no corrections have 

to be applied to the readings in use 

W  R Ugl  W (7.5.2-1) 

Unless the errors are more or less centred around zero, they form a one-sided 

contribution to the uncertainty which can only be treated in an approximate manner. For 

the sake of simplicity and convenience, the “global uncertainty” is best stated in the 

format of an expression for the whole weighing range, instead of individual values stated 
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for fixed values of the weighing result. 

Let E  R be a function, or 0 E be one value representative for all errors stated over the 

weighing range in the calibration certificate. The combination with the uncertainties in 

use may then, in principle, take on one of these forms 

       2 2 Ugl W  k u W  E R (7.5.2-2a) 

   2 2 0 Ugl W  k u W  E (7.5.2-2b) 

      2 2 2 0

gl Max U W  k u W  E R (7.5.2-2c) 

U   W  ku W  ER gl (7.5.2-3a) 

  0 Ugl W  ku W  E (7.5.2-3b) 

  Max U W ku W E0 R

gl   (7.5.2-3c) 

Quite frequently, (7.5.2-3a) is taken as basis for the statement of the global uncertainty. 

Thereby, U  W  k u W is often approximated by the following formula 

    

  Max R

U W

U W Max

U W U W

















 







    

0

0 (7.5.2-3d) 

and E  R is often approximated by ER  a1R as per (C2.2-16) and (C2.2-16a) so that 

    

  Max R a R

U W

U W Max

U gl W U W 1 0

0 

















 







     (7.5.2-3e) 

For further information on alternative generation of the formulae E  R or the 

representative value 0 E see Appendix C. 

In analogy to (7.5.2-3d), for multi-interval instruments U(W)is indicated per interval as 

        ( R Max ) Max Max

U Max U Max U W U Max i

i i

i i

i 1

1

1

1 



   















   (7.5.2-3f) 

and for multiple range instruments U(W)is indicated per range. 

It is important to ensure that U W  gl retains a coverage probability of not less than 95 % 

over the whole weighing range. For U W  gl the coverage factor k will, in most cases be 

equal to 2 even where the standard deviation s is obtained from only few 

measurements, and/or where kcal  2 was stated in the calibration certificate. This is due 

to the large number of terms contributing to uW . 

7.5.3 Other ways of qualification of the instrument 

A client may expect from, or have asked the Calibration Laboratory for a statement of 
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conformity to a given specification, as W  R  Tol with Tol being the applicable 

tolerance. The tolerance may be specified as “Tol  x % of R ”, as “Tol  n d ”, or the 

like. 

Conformity may be declared, in consistency with ISO/IEC 17025 under condition that 

E  R U  W R  TolR (7.5.3-1) 

either for individual values of R or for any values within the whole or part of the 

weighing range. 

Within the same weighing range, conformity may be declared for different parts of the 

weighing range, to different values of Tol. 

If the user defines a relative weighing accuracy requirement, then Appendix G “Minimum 

weight” provides further advice. 

8 CALIBRATION CERTIFICATE 

This section contains advice regarding what information may usefully be provided in a 

calibration certificate. It is intended to be consistent with the requirements of ISO/IEC 

17025, which take precedence. 

8.1 General information 

Identification of the calibration laboratory, 

reference to the accreditation (accrediting body, number of the accreditation), 

identification of the certificate (calibration number, date of issue, number of pages), 

signature(s) of authorised person(s). 

Identification of the client. 

Identification of the calibrated instrument, 

information about the instrument (manufacturer, kind of instrument, Max, d, place of 

installation). 

Warning that the certificate may be reproduced only in full unless the calibration 

laboratory permits otherwise in writing. 

8.2 Information about the calibration procedure 

Date of measurements, 

site of calibration, 

conditions of environment and/or use that may affect the results of the calibration. 

Information about the instrument (adjustment performed: internal or external 

adjustment and in the case of external adjustment what weight has been used, any 

anomalies of functions, setting of software as far as relevant for the calibration, 

etc.). 

Reference to, or description of the applied procedure, as far as this is not obvious 

from the certificate, e.g. constant time interval observed between loadings and/or 

readings. 

Agreements with the client e.g. over limited range of calibration, metrological 

specifications to which conformity is declared. 
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Information about the traceability of the measuring results. 

8.3 Results of measurement 

Indications and/or errors for applied test loads, or errors related to indications – as 

discrete values and/or by an equation resulting from approximation, 

details of the loading procedure if relevant for the understanding of the above, 

standard deviation(s) determined as related to a single indication, 

information about the eccentricity test if performed, 

expanded uncertainty of measurement for the error of indication results. 

Indication of the coverage factor k , with comment on coverage probability, and 

reason for k  2 where applicable. 

Where the indications/errors have not been determined by normal readings - single 

readings with the normal resolution of the instrument - a warning should be given 

that the reported uncertainty is smaller than would be found with normal readings. 

8.4 Additional information 

Additional information about the uncertainty of measurement expected in use, inclusive 

of conditions under which it is applicable, may be attached to the certificate without 

becoming a part of it. 

Where errors are to be accounted for by correction, this formula could be used 

W  R  E  R UW  (8.4-1) 

accompanied by the equation for ER. 

Where errors are included in the “global uncertainty”, this formula could be used 

W  R Ugl  W (8.4-2) 

A statement should be added that the expanded uncertainty of values from the formula 

corresponds to a coverage probability of at least 95 %. 

Optional: 

Statement of conformity to a given specification, and range of validity where applicable. 

This statement may take the form 

W  R  Tol (8.4-3) 

and may be given 

in addition to the results of measurement, or 

as stand-alone statement, with reference to the results of measurement declared to 

be retained at the calibration laboratory. 

The statement may be accompanied by a comment indicating that all measurement 

results enlarged by the expanded uncertainty of measurement, are within the 

specification limits. 

Information about the minimum weight values for various weighing tolerances as per 

appendix G may be provided. 

For clients that are less knowledgeable, advice might be provided where applicable, on 
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the definition of the error of indication, 

how to correct readings in use by subtracting the corresponding errors, 

how to interpret indications and/or errors presented with fewer digits than the scale 

interval d . 

It may be useful to quote the values of UW * for either all individual errors or for the 

function E  R resulting from approximation. 

9 VALUE OF MASS OR CONVENTIONAL VALUE OF MASS 

The quantity W is an estimate of the conventional value of mass mc of the object 

weighed6. For certain applications it may be necessary to derive from W the value of 

mass m , or a more accurate value for mc . 

The density  or the volume V of the object, together with an estimate of their standard 

uncertainty, must be known from other sources. 

9.1 Value of mass 

The mass of the object is 

    m  W 1 a 1  1 c (9.1-1) 

Neglecting terms of second and higher order, the relative standard uncertainty u m rel is 

given by 

       

4

2

2

a

2

c

a

2

2

2

2

rel

1 1



 

 



u

u

W

u W

u m  













   

 (9.1-2) 

For a and   u a (density of air) see Appendix A. 

If V and u  V are known instead of  and u,  may be approximated by W V , 

and urel   may be replaced by u V  rel . 

9.2 Conventional value of mass 

The conventional value of mass of the object is 

    c a 0 c m  W 1    1  1  (9.2-1) 

Neglecting terms of second and higher order, the relative standard uncertainty   urel mc

is given by 

         

4

2

2

a 0

2

c

a

2

2

2

c

2

rel

1 1



  

 



u

u

W

u W

u m   













   

 (9.2-2) 

The same comments as given to (9.1-2) apply. 

 

6

In the majority of cases, especially when the results are used for trade, the value Wis used as the result of the weighing
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10 REFERENCES 

[1] JCGM 100:2008 (GUM) Evaluation of measurement data — Guide to the expression of 

uncertainty in measurement, September 2008 

[2] OIML R 76: Non-automatic Weighing Instruments Part 1: Metrological Requirements - 

Tests, Edition 2006 (E) 

[3] EN 45501: Metrological Aspects of Non-automatic Weighing Instruments, Edition 2015 

[4] OIML R111, Weights of Classes E1, E2, F1, F2, M1, M1-2, M2, M2-3, M3, Edition 2004 

(E) 

[5] JCGM 200:2012 (VIM), International Vocabulary of Metrology – Basic and General 

Concepts and Associated Terms, 3rd edition with minor corrections, 2012 

[6] Comprehensive Mass Metrology, M. Kochsiek, M. Glaser, WILEY-VCH Verlag Berlin 

GmbH, Berlin. ISBN 3-527-29614-X 

[7] M. Gläser: Change of the apparent mass of weights arising from temperature 

differences, Metrologia 36 (1999), p. 183-197 

[8] ILAC P10:01/2013, ILAC Policy on the Traceability of Measurement Results, 2013 

[9] JCGM 101:2008, Evaluation of Measurement Data – Supplement 1 to the "Guide to the 

expression of uncertainty in measurement" – Propagation of Distributions using a Monte 

Carlo method, 1st edition, 2008 

[10] A. Malengo, Buoyancy effects and correlation in calibration and use of electronic 

balances, Metrologia 51 (2014) p. 441–451 

[11] A. Picard, R. S. Davis, M. Gläser, K. Fujii: Revised formula for the density of moist air 

(CIPM-2007), Metrologia 45 (2008), p. 149-155 

[12] R. T. Birge, The Calculation of Errors by the Method of Least Squares, Phys. Rev. 40, 

207 (1932) 

[13] Dictionary of Weighing Terms – A Guide to the Terminology of Weighing, R. Nater, A. 

Reichmuth, R. Schwartz, M. Borys and P. Zervos, Springer, Berlin, Heidelberg, 2009. 

ISBN 978-3-642-02013-1 
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APPENDIX A: ADVICE FOR ESTIMATION OF AIR DENSITY 

Note: In Appendix A, the symbols are T for temperature in K, and t for temperature in °C 

A1 Formulae for the density of air 

The most accurate formula to determine the density of moist air is the one 

recommended by the CIPM [11]7. For the purposes of this guideline, less sophisticated 

formulae which render slightly less precise results are sufficient. 

A1.1 Simplified version of CIPM-formula, exponential version 

From OIML R111 [4], section E3 

 

, t

, p , RH exp , t



  27315

0 34848 0 009 0 061

a (A1.1-1) 

with 

a air density in kg/m3 

p barometric pressure in hPa 

RH relative humidity of air in % 

t air temperature in °C 

The relative uncertainty of this approximation formula is uform / a  2,4×10-4 under the 

following conditions of environment 

600 hPa ≤ p ≤ 1 100 hPa 

20 % ≤ RH ≤ 80 % 

15 °C ≤ t ≤ 27 °C 

Apart from the uncertainty uform , the uncertainties of the estimates for p , RH and t

determine the uncertainty of  a (see section A3). 

A1.2 Average air density 

Where measurement of temperature and barometric pressure is not possible, the mean 

air density at the site can be calculated from the altitude above sea level, as 

recommended in [4] 





 



   SL

0

0

a 0 exp gh

p

   (A1.2-1) 

with p0 = 1 013,25 hPa 

 0 = 1,200 kg/m3 

g = 9,81 m/s2 

hSL = altitude above sea level in metre 

This calculation for air density is intended for 20 °C and RH = 50%. 

 

7

The relative uncertainty of the CIPM-2007 air density formula, without the uncertainties of the parameters is 

5

form a / 2.2 10 u    , the best relative uncertainty achievable, which includes the uncertainty contributions for temperature, 

humidity and pressure measurements, is about 5

a a ( )/ 8 10 u     .The recommended ranges of temperature and pressure 

over which the CIPM-2007 equation may be used are: 600 hPa ≤ p ≤1 100 hPa, 15 oC ≤ t ≤ 27 oC.
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The relative uncertainty of this approximation formula is uform / a  1,2 x10-2. 

A2 Variations of parameters constituting the air density 

In order to evaluate the uncertainties associated to the estimates p , RH and t , some 

advice about their typical variations are given in the following chapter. This information 

may be used when environmental measurements are not going to be performed. 

A2.1 Barometric pressure 

At any given location, the variation is at most p = ±40 hPa about the average8. Within 

these limits, the distribution is not rectangular as extreme values do occur only once in 

several years. It has been found that the distribution is basically normal. Taking into 

account the typical atmospheric pressure variation it is realistic to assume a standard 

uncertainty 

u  p = 10 hPa (A2.1-1) 

The average barometric pressure   SL p h (in hPa) can be evaluated according to the 

International Standard Atmosphere, and may be estimated from the altitude hSL in 

metres above sea level of the location, using the relation 

  exp( 0,00012 m )

1

SL 0 SL

 p h  p h  (A2.1-2) 

with p0 = 1 013,25 hPa 

A2.2 Temperature 

The possible variation T  Tmax Tmin of the temperature at the place of use of the 

instrument may be estimated from information which is easy to obtain 

limits stated by the client from his experience, 

reading from suitable recording means, 

setting of the control instrument, where the room is acclimatized or temperature 

stabilized, 

in case of default, sound judgement should be applied, leading to – e.g. 

17 °C ≤ t ≤ 27 °C for closed office or laboratory rooms with windows, 

T ≤ 5 K for closed rooms without windows in the centre of a building, 

- 10 °C ≤ t ≤ + 30 °C or T ≤ 40 K for open workshops or factory spaces. 

As stated for the barometric pressure, a rectangular distribution is unlikely to occur for 

open workshops or factory spaces where the atmospheric temperature prevails. 

However, to avoid different assumptions for different room situations, the assumption of 

rectangular distribution is recommended, leading to 

u  T  T 12 (A2.2-1) 

A2.3 Relative humidity 

The possible variation RH  RHmax  RHmin of the relative humidity at the place of use 

of the instrument may be estimated from information which is easy to obtain 

limits stated by the client from his experience, 

 

8

Example: at Hannover, Germany, the difference between highest and lowest barometric pressures ever observed over 20 years 

was 77,1 hPa (Information from DWD, the German Meteorological Service).
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reading from suitable recording means, 

setting of the control instrument, where the room is acclimatized, 

in case of default, sound judgement should be applied, leading, for example, to 

30 % ≤ RH ≤ 80 % for closed office or laboratory rooms with windows, 

RH ≤ 30 % for closed rooms without windows in the centre of a building, 

20 % ≤ RH ≤ 80 % for open workshops or factory spaces. 

It should be kept in mind that 

at RH < 40 % electrostatic effects may already influence the weighing result on high 

resolution instruments, 

at RH > 60 % corrosion may begin to occur. 

As has been said for the barometric pressure, a rectangular distribution is unlikely to 

occur for open workshops or factory spaces where the atmospheric relative humidity 

prevails. However, to avoid different assumptions for different room situations, the 

assumption of rectangular distribution is recommended, leading to 

u  RH  RH 12 (A2.3-1) 

A3 Uncertainty of air density 

The relative standard uncertainty of the air density   a a u  /  may be calculated by 

                2

a

form a

2

a

a

2

a

a

2

a

a

a

a

















 















  















  













   

















 u

u RH uu T uu p

u up T RH

 (A3-1) 

with the sensitivity coefficients (derived from the CIPM formula for air density) 

  a a u / p = 1  10-5 Pa-1 for barometric pressure 

  a a u / T = - 4  10-3 K-1 for air temperature 

  a a uRH  /  = - 9  10-3 for relative humidity (the unit for RH in this case is 1, not %) 

These sensitivity coefficients may also be used for equation (A1.1-1). 

Equation (A3-1) can be approximated as (A3-2) based on the following assumptions: 

 the standard uncertainty for pressure variation based on meteorological data, 

that show it is a normal distribution, is 10 hPa 

 the maximum variation for humidity is 100 %. 

 the maximum variation of temperature in the location is included as T 

  4 6 2 2

a

a 1,07 10 1,33 10 K T u        





 (A3-2) 

 
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Examples of standard uncertainty of air density, calculated for different parameters using 

the formula (A.1.1-1) 

u  p

/hPa 

T

/K 

RH

/%  

u  p

up

a

a



  

u  T u

a

T a



 ( ) ( )

a

a u RH uRH





a

form a ( )



u 

a

a ( )



u 

10 2 20 1  10-2 -2,31  10-3 -5,20  10-4 2,4×10-4 1,03  10-2

10 2 100 1  10-2 -2,31  10-3 -2,60  10-3 2,4×10-4 1,06  10-2

10 5 20 1  10-2 -5,77  10-3 -5,20  10-4 2,4×10-4 1,16  10-2

10 5 100 1  10-2 -5,77  10-3 -2,60  10-3 2,4×10-4 1,18  10-2

10 10 20 1  10-2 -1,15  10-3 -5,20  10-4 2,4×10-4 1,53  10-2

10 10 100 1  10-2 -1,15  10-3 -2,60  10-3 2,4×10-4 1,55  10-2

10 20 20 1  10-2 -2,31  10-2 -5,20  10-4 2,4×10-4 2,52  10-2

10 20 100 1  10-2 -2,31  10-2 -2,60  10-3 2,4×10-4 2,53  10-2

10 30 20 1  10-2 -3,46  10-2 -5,20  10-4 2,4×10-4 3,61  10-2

10 30 100 1  10-2 -3,46  10-2 -2,60  10-3 2,4×10-4 3,61  10-2

10 40 20 1  10-2 -4,62  10-2 -5,20  10-4 2,4×10-4 4,73  10-2

10 40 100 1  10-2 -4,62  10-2 -2,60  10-3 2,4×10-4 4,73  10-2

10 50 20 1  10-2 -5,77  10-2 -5,20  10-4 2,4×10-4 5,86  10-2

10 50 100 1  10-2 -5,77  10-2 -2,60  10-3 2,4×10-4 5,87  10-2

T is the maximum variation of temperature and RH is the maximum variation of 

humidity in the location of the balance. 

APPENDIX B: COVERAGE FACTOR k FOR EXPANDED UNCERTAINTY OF 

MEASUREMENT 

Note: in this Appendix the general symbol y is used for the result of measurement, not a 

particular quantity as an indication, an error, a mass of a weighed body etc. 

B1 Objective 

The coverage factor k shall in all cases be chosen such that the expanded uncertainty 

of measurement has a coverage probability of 95,45 %. 

B2 Normal distribution and sufficient reliability 

The value k  2, corresponding to a 95,45% probability, applies where 

a) a normal (Gaussian) distribution can be attributed to the error of indication, 

and 

b) the standard uncertainty uE is of sufficient reliability (i.e. it has a sufficient number 

of degrees of freedom), see JCGM 100 [1]. 

Normal distribution may be assumed where several (i.e. N ≥ 3) uncertainty components, 

each derived from “well-behaved” distributions (normal, rectangular or the like), 

contribute to uE in comparable amounts. 

Sufficient reliability is depending on the degrees of freedom. This criterion is met where 

no Type A contribution to u  E is based on less than 10 observations. A typical Type A 

contribution stems from repeatability. Consequently, if during a repeatability test a load is 

applied not less than 10 times, sufficient reliability can be assumed. 
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B3 Normal distribution, no sufficient reliability 

Where a normal distribution can be attributed to the error of indication, but u  E is not 

sufficiently reliable, then the effective degrees of freedom eff have to be determined 

using the Welch-Satterthwaite formula 



 N

i i

ui E

u E

1

4

4

eff ( )

( )



 (B3-1) 

whereui  E are the contributions to the standard uncertainty as per (7.1.3-1a), and i is 

the degrees of freedom of the standard uncertainty contribution ui  E . Based on eff the 

applicable coverage factor k is read from the extended table of [1], Table G.2 or the 

underlying t-distribution described in [1], Annex C.3.8 may be used to determine the 

coverage factor k . 

B4 Determining k for non-normal distributions 

In any of the following cases, the expanded uncertainty is Uy   ku y . 

It may be obvious in a given situation that uy contains one Type B uncertainty 

component u y 1 from a contribution whose distribution is not normal but, e.g., 

rectangular or triangular, which is significantly greater than all the remaining 

components. In such a case, uy is split up in the (possibly dominant) part 1 u and R u = 

square root of  2

u j with j ≥ 2, the combined standard uncertainty comprising the 

remaining contributions, see [1]. 

If R u ≤ 0,3 1 u , then 1 u is considered to be “dominant“ and the distribution of y is 

considered to be essentially identical with that of the dominant contribution. 

The coverage factor is chosen according to the shape of distribution of the dominant 

component 

for trapezoidal distribution with   0,95, 

(  = edge parameter, ratio of smaller to larger edge of trapezoid) 

1   0,051   1  6 2 2 k       (B4-1) 

for a rectangular distribution (  = 1), k = 1,65, 

for a triangular distribution (  = 0), k = 1,90, 

for U-shaped distribution, k = 1,41. 

The dominant component may itself be composed of 2 dominant components u y 1 , 

u   y 2 , e.g. 2 rectangular making up one trapezoid, in which case R u will be determined 

from the remaining u j with j  3. 
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APPENDIX C: FORMULAE TO DESCRIBE ERRORS IN RELATION TO THE INDICATIONS 

C1 Objective 

This Appendix offers advice on how to derive, from the discrete values obtained at 

calibration and/or given in a calibration certificate, errors and associated uncertainties for 

any other reading R within the calibrated weighing range. 

It is assumed that the calibration yields n sets of data j I N , E j , U j , or alternatively mNj

, j I , U j , together with the coverage factor k and an indication of the distribution of E

underlying k . 

In any case, the nominal indication j I N is considered to be j m j I N  N . 

It is further assumed that for any mNj the error E j remains the same if j I is replaced by 

j I N , it is therefore sufficient to look at the data j I N , E j , u j , and to omit the suffix N for 

simplicity. 

C2 Functional relations 

C2.1 Interpolation 

There are several polynomial formulae for interpolation9 between tabulated values and 

equidistant values which are relatively easy to employ. However, the test loads may not, 

in many cases, be equidistant, which leads to quite complicated interpolation formulae if 

applying a single formula to cover the whole weighing range. 

Linear interpolation between two adjacent points may be performed by 

       k k k k k k E R  E  R  I E  E I  I 1 1 (C2.1-1) 

         k k k k k k U E R U  R  I U U I  I 1 1 (C2.1-2) 

for a reading R with k   k1 I R I . A higher order polynomial would be needed to 

estimate the possible interpolation error – this is not further elaborated. 

C2.2 Approximation 

Approximation should be performed by calculation or by algorithms based on the 

”minimum 2  ” approach, that is, the parameters of a function f are determined so that 

        minimum 2 2 2

j j j j Ej  p v p f I (C2.2-1) 

with 

p j = weighting factor (basically proportional to 2 1 u j ), 

 j = residual, 

f = approximation function containing npar parameters to be determined, 

j = 1...n, 

 

9

An interpolation formula is understood to yield exactly the given values between which interpolation takes place. An approximation 

formula will normally not yield the given values exactly. 
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n = number of test points. 

From the observed chi-squared value obs

2  , if the following condition is met [12] 

 obs   2 (C2.2-2a) 

with the degrees of freedom   n  npar , it is justified to assume the form of the model 

function E I  f (I) to be mathematically consistent with the data underlying the 

approximation. 

An alternative option for testing the goodness of the fit is to assume that the maximum 

value of the weighted differences will have to fulfil 

 

    max  1 













 

j

j j

U f I

f I E

 (C2.2-2b) 

that is, the expanded uncertainty must include the residual for each point j. This 

condition is much more restrictive than equation (C2.2-2a). 

C2.2.1 Approximation by polynomials 

Approximation by a polynomial yields the general function 

  a

a ... 2

0 1 2

n E R  f R  a  a R  a R   an R (C2.2-3) 

The degree na of the polynomial should be chosen such that npar  na 1 n 2 . 

The calculation is best performed by matrix calculation. 

Let ( x ) par X n n be a matrix whose n rows are (1, j I , 2

j I ,..., a n

j I ), 

( x 1) par a n be a column vector whose components are the coefficients 

0 a , 1 a , ... , a an to be determined of the approximation 

polynomial, 

  n x 1 e be a column vector whose components are the E j , 

   n x n U e be the variance-covariance matrix of e . 

U  e is given by 

U   e  U m UI Umod ref Cal (C2.2-3a) 

where 

  U mref is the covariance matrix associated with the reference values mref

(4.2.4-2). Considering reasonably high correlation among the referencevalues 

  T

ref ref mref m m U  s s (C2.2-3b) 

where mref s is the column vector of the uncertainties   u mref (equ. 7.1.2-14),
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  Cal U I is a diagonal matrix whose elements are   jj j u u I 2  ,

U  mod is an additional covariance matrix, which is given by 

U  I

2

m mod  s (C2.2-3c) 

where I is the identity matrix and sm is an uncertainty due to the model. This contribution 

is considered in order to take into account the model inadequacy. 

Initially smis set to zero, if the 2 test (C2.2-2a) fails, sm is enlarged in an iterative way, 

until the 2 test result is satisfied. 

If U(ICal) is the dominant contribution, the covariances maybe be neglected and Ue

can be approximated to a diagonal matrix whose elements are 

  2

m

2

u u E s jj  j  (C2.2-3d) 

The weighting matrix P is 

 1 P  U e (C2.2-4) 

and the coefficients a0 , 1 a ,... are found by solving the normal equations 

X PXa  X Pe  0 T T (C2.2-5) 

with the solution 

a   X PX X Pe T T 1 (C2.2-6) 

The n residuals   j j E j v  f I  are comprised in the vector 

v  Xaˆ  e (C2.2-7) 

and 2  obs is obtained by 

v Pv T  2  obs (C2.2-8) 

Provided the condition of (C2.2-2) is met, the variances and covariances for the 

coefficients ai are given by the matrix 

   1

U a  X PX T ˆ (C2.2-9) 

Where the condition (C2.2-2) is not met, one of these procedures may be applied 

a: repeat the approximation with an approximating polynomial of higher 

degree na , as long as na 1 n 2 , 

b: repeat the approximation after increasing Umod. 

The results of the approximation, aˆ and Uaˆ  may be used to determine the 

approximated errors and the associated uncertainties for the n points j I . 
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The errors Eapprj are comprised in the vector 

e  Xaˆ appr (C2.2-10) 

with the uncertainties given by 

      T

u Eapprj diag XU aˆ X 2  . (C2.2-11) 

They also serve to determine the error and its associated uncertainty for any other 

indication – called a reading R to discriminate from the indications j I – within the 

calibrated weighing range. 

Let 

r be a column vector whose elements are  

T a 1 , 2 3 n ,R,R ,R ,... R , 

r be a column vector whose elements are the derivatives 

 T 1

a

2 a 0 1 2 3 , n  , , R, R ,... n R . 

The error is 

E   R r aˆ T

appr  (C2.2-12) 

and the uncertainty is given by 

u   E   r aˆ U  R r aˆ  r U  aˆ r T T T T

appr     2 (C2.2-13) 

The first term on the right-hand side simplifies, as all 3 matrices are only one 

dimensional, to 

  ˆ   R   ˆ a a R a R ... n a R  u   R n

n

2 2 1

a

2

1 2 3

T a

a 2 3        

T T r a U r a (C2.2-14) 

with       2

ecc

2

rel

2 2 2

0

2

u R  d 12  d R 12  s R  u R R as per (7.1.1-12). 

C2.2.2 Approximation by a straight line 

Many modern electronic instruments are well designed, and corrected internally to 

achieve good linearity. Therefore errors mostly result from incorrect adjustment, and the 

error increases in proportion to R . For such instruments it may be appropriate to restrict 

the polynomial to a linear function, provided it is sufficient in view of condition (C2.2-2). 

The standard solution is to apply (C2.2-3) with na  1

E  R  f R  a0  a1R (C2.2-15) 

One variation to this is to set a0  0 and to determine only 1 a . This can be justified by 

the fact that due to zero-setting – at least for increasing loads – the error E  R  0 is 

automatically zero 

E  R  f R  a1R (C2.2-16) 
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Another variation is to define the coefficient a (= 1 a in (C2.2-16)) as the mean of all 

relative errors j j j q  E I . This allows inclusion of errors of net indications after a tare 

balancing operation if these have been determined at calibration 

a  E j I j n (C2.2-17) 

The calculations, except for the variation (C2.2-17), may be performed using the matrix 

formulae in C2.2.1. 

Other possibilities are given hereafter. 

C2.2.2.1 Linear regression as per (C2.2-15) may be performed by software. 

Correspondence between results is typically 

”intercept”  a0

”slope”  1 a

However, simple pocket calculators may not be able to perform linear regression based 

on weighted error data, or linear regression with 0 a0  . 

C2.2.2.2 To facilitate programming the calculations by computer in non-matrix notation, the 

relevant formulae are presented hereafter. 

If condition (C2.2-2a) is intended to be fulfilled, the method starts with the first linear 

regression using 

  p j u E j

2  1 (C2.2-18a) 

If (C2.2-2a) is not yet fulfilled, then the standard deviation of the fit can be determined as 

   

  par

2

n n

f I E

std fit j

j j









 (C2.2-18b) 

As a second step new weighting factors have to determined as 

    2 2p 1 u E std fit 

j  j  (C2.2-18c) 

With these new weighting factors a new linear regression has to be determined. 

Following this method, the linear regression fulfils condition (C2.2-2a). 

If condition (C2.2-2b), which is more restrictive, is intended to be fulfilled, it is very likely 

that an additional uncertainty component, sm, has to be included in (C2.2-18a. Initially 

smis set to zero, then sm is enlarged in an iterative way until the condition in (C2.2-2b) is 

satisfied. A proposal to increase the step to enlarge sm may be to consider 1/10 of the 

resolution of the instrument. 

In the following expressions for simplicity, all indices ” j ” have been omitted from I , E , 

p . 

a) linear regression for (C2.2-15) 
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 2 2

2

0

  

   



  p pI pIpE pI pI pIE

a (C2.2-15a) 

 2 2 1

  

   



  p pI pIp pIE pE pI

a (C2.2-15b) 

 2

0 1

2    p a  a I  E (C2.2-15c) 

   2 2

2

0

2

  



  p pI pI

pI

u a (C2.2-15d) 

   2 2 1

2

  



  p pI pI

p

u a (C2.2-15e) 

   2 2 0 1 cov

  



   p pI pI

pI

a ,a (C2.2-15f) 

(C2.2-15) applies for the approximated error of the reading R , and the uncertainty of the 

approximation   u Eappr is given by 

         1 0 1

2 2

0

2 2 2

appr 1

2

u E  a u R  u a  R u a  2Rcov a , a (C2.2-15g) 

b) linear regression with a0 = 0 

   2a1 pIE pI (C2.2-16a) 

 2

1

2    p a I  E (C2.2-16b) 

    2

1

2

u a 1 pI (C2.2-16c) 

(C2.2-16) applies for the approximated error of the reading R , and the assigned 

uncertainty   u Eappr is given by 

     1

2 2 2 2

appr 1

2

u E  a u R  R u a (C2.2-16d) 

c) mean gradients 

In this variant the uncertainties are     j j j j u E I  u E I and   j j u Ej p I 2 2  . 
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a   pE I  p (C2.2-17a) 

  2 2

   p a  E I (C2.2-17b) 

u   a  1  p

2 (C2.2-17c) 

(C2.2-17) applies for the approximated error of the reading R which may be also a net 

indication, and the uncertainty of the approximation   u Eappr is given by 

u   E a u R R u a 2 2 2 2

appr

2   (C2.2-17d) 

C3 Terms without relation to the readings 

While terms that are not a function of the indication do not offer any estimated value for 

an error to be expected for a given reading in use, they may be helpful to derive the 

”global uncertainty” mentioned in 7.5.2. 

C3.1 Mean error 

The mean of all errors is 



 

n

j

E j nE E

1

0 1 (C3.1-1) 

with the standard deviation 

    appr

1

2

1

1 E E u

n

s E

n

j

 j    

 (C3.1-2) 

Note: the data point I  0 , E  0 shall be included as 1I , E1 . 

Where E is close to zero, only s E 2 may be added in (7.5.2-2a). In other cases, in 

particular where E  u  W , (7.5.2-3a) should be used, with u  W increased by 

uappr  s  E . 

C3.2 Maximum error 

The ”maximum error” shall be understood as the largest absolute value of all errors 

max Emax  E j (C3.2-1) 

C3.2.1 With max

0 E  E , (7.5.2-3a) would certainly describe a ”global uncertainty” which would 

cover any error in the weighing range with a higher coverage probability than 95 %. The 

advantage is that the formula is simple and straightforward. 

C3.2.2 Assuming a rectangular distribution of all errors over the – fictitious! – range  Emax , 0 E

could be defined as the standard deviation of the errors 

max 3 0 E  E (C3.2-2) 

 to be inserted into (7.5.2-2a). 
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APPENDIX D: SYMBOLS 

Symbols that are used in more than one section of the main document are listed and explained hereafter. 

Symbol Definition 

D drift, variation of a value with time 

E error (of an indication) 

I indication of an instrument 

ref I reference value of the indication of an instrument 

KT sensitivity of the instrument to the temperature variation 

L load on an instrument 

Max maximum weighing capacity 

Max1 upper limit of the weighing range with the smallest scale interval 

Max upper limit of specified weighing range, Max  Max

Min value of the load below which the weighing result may be subject to an 

excessive relative error (from [2] and [3]) 

Min lower limit of specified weighing range, Min  Min

R indication (reading) of an instrument not related to a test load 

Rmin minimum weight 

Rmin,SF minimum weight for a safety factor >1 

Req user requirement for relative weighing accuracy 

T temperature (in K) 

Tol specified tolerance value 

U expanded uncertainty 

Ugl global expanded uncertainty 

W weighing result, weight in air 

d scale interval, the difference in mass between two consecutive 

indications of the indicating device 

1 d smallest scale interval 

dT effective scale interval < d , used in calibration tests 

g local gravity acceleration 

k coverage factor 

ks adjustment factor 

m mass of an object 

mc conventional value of mass, preferably of a standard weight 

mN nominal value of mass of a standard weight 

mref reference weight (“true value“) of a test load 

mpe maximum permissible error (of an indication, a standard weight etc.) in 

a given context 

n number of items, as indicated in each case 

p barometric pressure 

s standard deviation 

t temperature (in °C) 

u standard uncertainty 
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urel standard uncertainty related to a base quantity 

 number of degrees of freedom 

 density 

 0 reference density of air,  0 = 1,2 kg/m3 

a air density 

c reference density of a standard weight, c = 8 000 kg/m3 

Suffix related to 

B air buoyancy (at calibration) 

D drift 

L at load 

N nominal value 

St standard (mass) 

T test 

adj adjustment 

appr approximation 

buoy air buoyancy (weighing result) 

cal calibration 

conv convection 

corr correction 

dig digitalisation 

ecc eccentric loading 

gl global, overall 

i , j numbering 

instr weighing instrument 

max maximum value from a given population 

min minimum value from a given population 

proc weighing procedure 

ref reference 

rel relative 

rep repeatability 

s actual at time of adjustment 

sub substitution load 

tare tare balancing operation 

temp temperature 

time time 

0 zero, no-load 

 
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APPENDIX E: INFORMATION ON AIR BUOYANCY 

This Appendix gives additional information to the air buoyancy correction treated in 

7.1.2.2. 

E1 Density of standard weights 

Where the density  of a standard weight, and its standard uncertainty u   are not 

known, the following values may be used for weights of R111 classes E2 to M2 (taken 

from [4], Table B7). 

Alloy/material Assumed density 

 in kg/m3 

Standard 

uncertainty 

u in kg/m3 

Nickel silver 8 600 85 

Brass 8 400 85 

stainless steel 7 950 70 

carbon steel 7 700 100 

iron 7 800 100 

cast iron (white) 7 700 200 

cast iron (grey) 7 100 300 

aluminium 2 700 65 

For weights with an adjustment cavity filled with a considerable amount of material of 

different density, [4] gives a formula to calculate the overall density of the weight. 

E2 Air buoyancy for weights conforming to OIML R111 

As quoted in a footnote to 7.1.2.2, OIML R111 requires the density of a standard weight 

to be within certain limits that are related to the maximum permissible error mpe and a 

specified variation of the air density. The mpe are proportional to the nominal value for 

weights of ≥ 100 g. This allows an estimate of the relative uncertainty   urel mB . The 

corresponding formulae (7.1.2-5c) for the case that the instrument is adjusted 

immediately before calibration and (7.1.2-5d) for the case when the instrument is not 

adjusted before calibration have been evaluated in Table E2.1, in relation to the 

accuracy classes E2 to M1. 

For weights of mN  50 g the mpe are tabled in R111, the relative value mpe mN is 

increasing with decreasing mass. For these weights, Table E2.1 contains the absolute 

standard uncertainties     u mB urel mB mN . 

The values in Table E2.1 can be used for an estimate of the uncertainty contribution if air 

buoyancy is not corrected for. 
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Table E2.1: Standard uncertainty of air buoyancy correction for standard weights 

conforming to R 111 

Calculated according to 7.1.2.2 for the case the instrument is adjusted immediately before 

calibration (7.1.2-5c), uA and the case where the instrument is not adjusted before calibration 

(7.1.2-5d), uB.

 Class E2 Class F1 Class F2 Class M1

mN 

in g 

mpe

in mg A u 

in mg 

B u in 

mg 

mpe

in mg A u 

in mg 

B u in 

mg 

mpe

in mg A u 

in mg 

B u in 

mg 

mpe

in mg A u 

in mg 

B u in 

mg 

50 0,100 0,014 0,447 0,30 0,043 0,476 1,00 0,14 0,58 3,0 0,43 0,87 

20 0,080 0,012 0,185 0,25 0,036 0,209 0,80 0,12 0,29 2,5 0,36 0,53 

10 0,060 0,009 0,095 0,20 0,029 0,115 0,60 0,09 0,17 2,0 0,29 0,38 

5 0,050 0,007 0,051 0,16 0,023 0,066 0,50 0,07 0,12 1,6 0,23 0,27 

2 0,040 0,006 0,023 0,12 0,017 0,035 0,40 0,06 0,08 1,2 0,17 0,19 

1 0,030 0,004 0,013 0,10 0,014 0,023 0,30 0,04 0,05 1,0 0,14 0,15 

0,5 0,025 0,004 0,008 0,08 0,012 0,016 0,25 0,04 0,04 0,8 0,12 0,12 

0,2 0,020 0,003 0,005 0,06 0,009 0,010 0,20 0,03 0,03 0,6 0,09 0,09 

0,1 0,016 0,002 0,003 0,05 0,007 0,008 0,16 0,02 0,02 0,5 0,07 0,07 

Relative mpe and relative standard uncertainties   urel mB in mg/kg for weights of 100 g and greater 

Class E2 Class F1 Class F2 Class M1

mpe/

mN

mg/kg 

urel A urel B mpe/ 

mN

mg/kg 

 urel A urel 

B

mpe/ 

mN

mg/kg 

 urel A urel 

B

mpe/ 

mN

mg/kg 

 urel A urel 

B

 100 1,60 0,23 8,89 5,00 0,72 9,38 16,0 2,31 11,0 50,0 7,22 15,88 

 
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APPENDIX F: EFFECTS OF CONVECTION 

In 4.2.3 the generation of an apparent change of mass mconv by a difference in 

temperature T between a standard weight and the surrounding air has been explained 

in principle. More detailed information is presented hereafter, to allow an assessment of 

situations in which the effect of convection should be considered in view of the 

uncertainty of calibration. 

All calculations of values in the following tables are based on [7]. The relevant formulae, 

and parameters to be included, are not reproduced here. Only the main formulae, and 

essential conditions are referenced. 

The problem treated here is quite complex, both in the underlying physics and in the 

evaluation of experimental results. The precision of the values presented hereafter 

should not be overestimated. 

F1 Relation between temperature and time 

An initial temperature difference T0 is reduced with time t by heat exchange 

between the weight and the surrounding air. The rate of heat exchange is fairly 

independent of the sign of T0 , therefore warming up or cooling down of a weight 

occurs in similar time intervals. 

Figure F1.1 gives some examples of the effect of acclimatisation. Starting from an initial 

temperature difference of 10 K, the actual T after different acclimatisation times is 

shown for 4 different weights. The weights are supposed to rest on three fairly thin PVC 

columns in “free air”. In comparison, T is also shown for a 1 kg weight resting on the 

same columns but enclosed in a bell jar which reduces the air flow of convection, so it 

takes about 1,5 times to 2 times as much time to achieve the same reduction of T , as 

for the 1 kg piece without the jar. 

References in [7]: formula (21), and parameters for cases 3b and 3c in Table 4. 

 
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Figure F1.1: Acclimatisation of standard weights 

Tables F1.2 and F1.3 give acclimatisation times t for standard weights that may have 

to be waited if the temperature difference is to be reduced from a value T1 to a lower 

value T2 . The conditions of heat exchange are the same as in Figure F1.1: Table F1.2 

as for “m = 0,1 kg” to “m = 50 kg”; Table F1.3 as for “m = 1 kg enclosed”. 

Under actual conditions the waiting times may be shorter where a weight stands directly 

on a plane surface of a heat conducting support; they may be longer where a weight is 

partially enclosed in a weight case. 

References in [7]: formula (26), and parameters for cases 3b, 3c in Table 4. 

Table F1.2 Time intervals for reduction in steps of temperature differences 

Weights standing on 3 thin PVC columns in free air 

Acclimatisation time in min for T to be reached from the next higher T , case 

3b 

T / K

m/kg 20 K to 

15 K 

15 K to 

10 K 

10 K to 

7 K 

7 K to 

5 K 

5 K to 

3 K 

3 K to 

2 K 

2 K to 

1 K 

50 149,9 225,3 212,4 231,1 347,9 298,0 555,8 

20 96,2 144,0 135,2 135,0 219,2 186,6 345,5 

10 68,3 101,9 95,3 94,8 153,3 129,9 239,1 

5 48,1 71,6 66,7 66,1 106,5 89,7 164,2 

2 30,0 44,4 41,2 40,6 65,0 54,4 98,8 

1 20,8 30,7 28,3 27,8 44,3 37,0 66,7 

0,5 14,3 21,0 19,3 18,9 30,0 24,9 44,7 

0,2 8,6 12,6 11,6 11,3 17,8 14,6 26,1 

0,1 5,8 8,5 7,8 7,5 11,8 9,7 17,2 

0,05 3,9 5,7 5,2 5,0 7,8 6,4 11,3 

0,02 2,3 3,3 3,0 2,9 4,5 3,7 6,4 

0,01 1,5 2,2 2,0 1,9 2,9 2,4 4,2 
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Examples for a 1 kg weight 

to reduce T from 20 K to 15 Kwill take 20,8 min, 

to reduce T from 15 K to 10 K will take 30,7 min, 

to reduce T from 10 K to 5 K will take 28,3 min + 27,8 min = 56,1 min. 

Table F1.3 Time intervals for reduction in steps of temperature differences 

Weights standing on 3 thin PVC columns, enclosed in a bell jar 

Acclimatisation time in min for T to be reached from the next higher T , case 

3c 

T / K

m/kg 20 K to 

15 K 

15 K to 

10 K 

10 K to 

7 K 

7 K to 

5 K 

5 K to 

3 K 

3 K to 

2 K 

2 K to 

1 K 

50 154,2 235,9 226,9 232,1 388,7 342,7 664,1 

20 103,8 158,6 152,4 155,6 260,2 228,9 442,2 

10 76,8 117,2 112,4 114,7 191,5 168,1 324,0 

5 56,7 86,4 82,8 84,3 140,5 123,1 236,5 

2 37,8 57,5 54,9 55,8 92,8 81,0 155,0 

1 27,7 42,1 40,1 40,7 67,5 58,8 112,0 

0,5 20,2 30,7 29,2 29,6 48,9 42,4 80,5 

0,2 13,3 20,1 19,1 19,2 31,7 27,3 51,6 

0,1 9,6 14,5 13,7 13,8 22,6 19,5 36,6 

0,05 6,9 10,4 9,8 9,9 16,1 13,8 25,7 

0,02 4,4 6,7 6,3 6,2 10,2 8,6 16,0 

0,01 3,2 4,7 4,4 4,4 7,1 6,0 11,1 

F2 Change of the apparent mass 

The air flow generated by a temperature difference T is directed upwards where the 

weight is warmer, T  0 , than the surrounding air, and downwards where it is cooler 

T  0 . The air flow causes friction forces on the vertical surface of a weight, and 

pushing or pulling forces on its horizontal surfaces, resulting in a change mconv of the 

apparent mass. The load receptor of the instrument is also contributing to the change, in 

a manner not yet fully investigated. 

There is evidence from experiments that the absolute values of the change are generally 

smaller for T  0 than for T  0 . It is therefore reasonable to calculate the mass 

changes for the absolute values of T , using the parameters for T  0 . 

Table F2.1 gives values for mconv for standard weights, for the temperature differences 

T appearing in Tables F1.2 and F1.3. They are based on experiments performed on a 

mass comparator with turning table for automatic exchange of weights inside a glass 

housing. The conditions prevailing at calibration of “normal” weighing instruments being 

different, the values in the table should be considered as estimates of the effects that 

may be expected at an actual calibration. 

References in [7]: formula (34), and parameters for case 3d in Table 4 
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Table F2.1 Change in apparent mass mconv

Change mconv in mg of standard weights, for selected temperature differences T

T in K 

m in 

kg 

20 15 10 7 5 3 2 1 

50 113,23 87,06 60,23 43,65 32,27 20,47 14,30 7,79 

20 49,23 38,00 26,43 19,25 14,30 9,14 6,42 3,53 

10 26,43 20,47 14,30 10,45 7,79 5,01 3,53 1,96 

5 14,30 11,10 7,79 5,72 4,28 2,76 1,96 1,09 

2 6,42 5,01 3,53 2,61 1,96 1,27 0,91 0,51 

1 3,53 2,76 1,96 1,45 1,09 0,72 0,51 0,29 

0,5 1,96 1,54 1,09 0,81 0,61 0,40 0,29 0,17 

0,2 0,91 0,72 0,51 0,38 0,29 0,19 0,14 0,08 

0,1 0,51 0,40 0,29 0,22 0,17 0,11 0,08 0,05 

0,05 0,29 0,23 0,17 0,12 0,09 0,06 0,05 0,03 

0,02 0,14 0,11 0,08 0,06 0,05 0,03 0,02 0,01 

0,01 0,08 0,06 0,05 0,03 0,03 0,02 0,01 0,01 

The values in this table may be compared with the uncertainty of calibration, or with a 

given tolerance of the standard weights that are used for a calibration, in order to assess 

whether an actual T value may produce a significant change of apparent mass. 

As an example, Table F2.2 gives the temperature differences which are likely to 

produce, for weights conforming to R 111, values of mconv not exceeding certain limits. 

The comparison is based on Table F2.1. 

The limits considered are the maximum permissible errors, and 1 3 thereof. 

It appears that with these limits, the effect of convection is relevant only for weights of 

classes F1 of OIML R111 or better. 

Table F2.2 Temperature limits for specified mconv values 

TA = temperature difference for m  mpe conv

TB = temperature difference for mconv  mpe 3

Differences TA for m  mpe conv and TB for mconv  mpe 3

 Class E2 Class F1

mN in kg mpe in 

mg 

TA in K TB in K mpe in 

mg 

TA in K TB in K 

50 75 12 4 250 >20 12 

20 30 11 3 100 >20 11 

10 15 10 3 50 >20 10 

5 7,5 10 3 25 >20 10 

2 3 9 1 10 >20 9 

1 1,5 7 1 5 >20 7 

0,5 0,75 6 1 2,5 >20 6 

0,2 0,30 5 1 1,0 >20 5 

0,1 0,15 4 1 0,50 >20 4 

0,05 0,10 6 1 0,30 >20 6 

0,02 0,08 10 2 0,25 >20 10 

0,01 0,06 15 3 0,20 >20 15 
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APPENDIX G: MINIMUM WEIGHT 

The minimum weight is the smallest sample quantity required for a weighment to just 

achieve a specified relative accuracy of weighing [13]. 

Consequently, when weighing a quantity representing minimum weight, Rmin, the relative 

measurement uncertainty of the weighing result equals the required relative weighing 

accuracy, Req, so that 

  Req R

U R 

min

min

 (G-1) 

This leads to the following relation that describes minimum weight 

 

Req

U R

R min

min  (G-2) 

It is general practice that users define specific requirements for the performance of an 

instrument (User Requirement Specifications). Normally they define upper thresholds for 

measurement uncertainty values that are acceptable for a specific weighing application. 

Colloquially users refer to weighing process accuracy or weighing tolerance 

requirements. Very frequently users also have to follow regulations that stipulate the 

adherence to a specific measurement uncertainty requirement. Normally these 

requirements are indicated as a relative value, e.g. adherence to a measurement 

uncertainty of 0,1 %. 

For weighing instruments, usually the global uncertainty is used to assess whether the 

instrument fulfils specific user requirements. 

The global uncertainty is usually approximated by the linear equation (7.5.2-3e) 

    

  Max R a R R

U W

U W Max

Ugl W U W gl gl    

















 







     1   0

0

 (G-3) 

The relative global uncertainty thus is a hyperbolic function and is defined as 

   

gl

gl gl

gl,rel 

   

R R

U W

U W

 (G-4) 

For a given accuracy requirement, Req, only weighings with 

Ugl,rel  W  Req (G-5) 

fulfil the respective user requirement. Consequently only weighings with a reading of 

gl

gl





 Req

R

 (G-6) 

have a relative measurement uncertainty smaller than the specific requirement set by 

the user and are thus acceptable. The limit value, i.e. the smallest weighing result that 

fulfils the user requirement is 
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gl

gl

min 



  Req

R

 (G-7) 

and is called “minimum weight”. Based on this value the user is able to define 

appropriate standard operating procedures that assure that the weighings he performs 

on the instrument comply with the minimum weight requirement, i.e. he only weighs 

quantities with higher mass than the minimum weight. 

As measurement uncertainty in use may be difficult to estimate due to environmental 

factors such as high levels of vibration, draughts, influences induced by the operator, 

etc., or due to specific influences of the weighing application such as electrostatically 

charged samples, magnetic stirrers, etc., a safety factor is usually applied. 

The safety factor SF is a number larger than one, by which the user requirement Req is 

divided. The objective is to ensure that the relative global measurement uncertainty is 

smaller than or equal to the user requirement Req, divided by the safety factor. This 

ensures that environmental effects or effects due to the specific weighing application that 

have an important effect on the measurement and thus might increase the measurement 

uncertainty of a weighing above a level estimated by the global uncertainty, still allow – 

with a high degree of insurance – that the user requirement Req is fulfilled. 

Ugl,rel  W  Req / SF (G-8) 

Consequently, the minimum weight based on the safety factor can be calculated as 

Req SF

SF

R  

 

gl

gl

min,SF 



 (G-9) 

The user is responsible for defining the safety factor depending on the degree to which 

environmental effects and the specific weighing application could influence the 

measurement uncertainty. 

Note that the minimum weight refers to the net (sample) weight which is weighed on the 

instrument, i.e. the tare vessel mass must not be considered to fulfil the user 

requirement Req. Therefore, minimum weight is frequently called "minimum sample 

weight". 

Figure G.1: Measurement uncertainty 

Absolute (green line) and relative (blue line) measurement uncertainty of a weighing 
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instrument. The accuracy limit of the instrument, the so-called minimum weight, is the 

intersection point between relative measurement uncertainty and the required weighing 

accuracy. 

APPENDIX H: EXAMPLES 

The examples presented in this Appendix demonstrate in different ways how the rules 

contained in this guideline may be applied correctly. They are not intended to indicate 

any preference for certain procedures as against others for which no example is 

presented. 

Where a calibration laboratory wishes to proceed in full conformity to one of the 

examples, it may make reference to it in its quality manual and in any certificate issued. 

Examples H1, H2 and H3 provide a basic approach for the determination of error and 

uncertainties in calibration. Example H4 provides a more sophisticated approach. 

Note 1: The certificate should contain all the information presented in Hn.1, as far as 

known, and, as applicable, at least what is printed in bold figures in Hn.2 and 

Hn.3, with Hn = H1, H2... 

Note 2: The values in the examples are indicated with more digits that may appear in a 

calibration certificate for illustrative purposes. 

Note 3: For rectangular distributions infinite degrees of freedom are assumed. 

H1 Instrument of 220 g capacity and scale interval 0,1 mg 

Preliminary note: 

The calibration of a laboratory balance is demonstrated. This example shows the 

complete standard procedure for the presentation of measurement results and the 

related uncertainties, as executed by most laboratories. An alternative method for the 

consideration of air buoyancy effects and convection effects is also presented as option 

2 (in italic type). 

 
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First situation: Adjustment of sensitivity carried out independently of calibration 

H1.1/AConditions specific for the calibration

Instrument: Electronic weighing instrument, description and 

identification 

Maximum Weighing 

Capacity Max/ Scale 

interval d

220 g / 0,1 mg 

Temperature coefficient 

KT = 1,5 x 10 -6/K (manufacturer’s manual); only 

necessary for calculation of the uncertainty of a weighing 

result. 

Built-in adjustment device 

Acts automatically after switching-on the balance and 

when T ≥ 3 K. Only necessary for calculation of 

uncertainty of a weighing result. Status: activated 

Adjustment by 

calibrator Not adjusted immediately before calibration. 

Temperature during 

calibration 21 °C measured at the beginning of calibration. 

Barometric pressure and 

humidity (optional) 990 hPa, 50 % RH. 

Room conditions 

Maximum temperature deviation 5 K (laboratory room 

without windows).If used for calculation of the buoyancy 

uncertainty as per formula 7.1.2-5e, it must be presented 

in the calibration certificate. Not relevant for the 

uncertainty of a weighing result, when built-in adjustment 

device is activated ( T ≥ 3 K). In this case the 

maximum temperature variation for the estimation of the 

uncertainty of a weighing result is 3 K. 

Test loads/ 

acclimatization 

Standard weights, class E2, acclimatized to room 

temperature (in option 2 a temperature difference of 2 K 

against room temperature is taken into account). 

H1.2/A Tests and results 

Repeatability

Requirements given in Chapter 

5.1. 

Indication at no load reset to 

zero where necessary; 

indications recorded. 

Test load 100 g (applied 5 times)

100,000 6 g 

100,000 3 g 

100,000 5 g 

100,000 4 g 

100,000 5 g 

Standard deviation s = 0,00011 g 

 
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Eccentricity

Requirements given in 

Chapter 5.3.

Indication set to zero 

prior to test; load put 

in centre first then 

moved to the other 

positions.

Position of the load Test load 100 g

Middle 100,000 6 g 

Front left 100,000 4 g 

Back left 100,000 5 g 

Back right 100,000 7 g 

Front right 100,000 5 g 

Maximum deviation max ecci I 0,000 2 g 

Errors of indication: 

General prerequisites: Requirements given in Chapter 5.2, weights distributed fairly 

evenly over the weighing range. 

Test loads each applied once; discontinuous loading only 

upwards, indication at no load reset to zero if necessary. 

Option 1: Air densities unknown during adjustment and during calibration (i.e. no buoyancy 

correction applied to the error of indication values) 

Load mref Indication I Error of indication E

0,0000 g 0,000 0g 0,000 0 g 

50,0000 g 50,000 4 g 0,000 4 g 

99,9999 g 100,000 6 g 0,000 7 g 

149,9999 g 150,000 9 g 0,001 0 g 

220,0001 g 220,001 4 g 0,001 3 g 

 

Option 2: Air density ρas unknown during adjustment and air density ρacal during calibration 

calculated according to the simplified CIPM formula (A1.1-1) 

 

Measurement values used for calculation: 

Barometric pressure p: 990 hPa 

Relative humidity RH: 50 %RH 

Temperature t: 21 °C 

Air density ρacal: 1,173 kg/m3 

Calculated buoyancy correction δmB according to formula (4.2.4-4). 

Numerical value used for calculation: 

Density of the reference mass ρcal: (7950 + 70) kg/m3 

Buoyancy correction δmB: 2,138 x 10-8 mref

The calculated buoyancy correction δmB of mref of load L following formula (4.2.4-4) is 

negligible as the relative resolution of the instrument is in the order of 10 -6 and thus much 

larger than the buoyancy correction. The above table is effectual. 

H1.3/A Errors and related uncertainties (budget of related uncertainties) 

Conditions common to both options: 

- The uncertainty for the zero position only results from the digitalisation d0 and 

repeatability s.
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- The eccentric loading is taken into account for the calibration according to (7.1.1-

10). 

- The conventional value of the test weights (class E2) is taken into account for the 

calibration results. Therefore U(mc) = U/k is calculated following formula (7.1.2-2). 

- The drift of the weights has been statistically monitored and the factor kD of formula 

(7.1.2-11) was chosen as 1,25. 

- The degrees of freedom for the calculation of the coverage factor k are derived 

following appendix B3 and table G.2 of [1]. In the case of the example, the influence 

of the uncertainty of the repeatability test with 5 measurements is significant. 

- The information about the relative uncertainty U(E)rel = u(E)/L is not mandatory, but 

helps to demonstrate the characteristics of the uncertainties. 

Uncertainty budget for option 1 (no buoyancy correction applied to the error of indication 

values) 

Additional condition: 

The balance is not adjusted immediately before calibration. The procedure according to 

option 1 is applied, with no information about air density. Therefore formula (7.1.2-5d) is 

applied for the uncertainty due to air buoyancy. As an alternative in the table, formula 

(7.1.2-5e) was used, thereby assuming a temperature variation during use of 5 K. 
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Quantity or Influence Load, indication and error in g 

Uncertainties in g Formula

Loadmref /g 0,000 0 50,000 0 99,999 9 149,999 9 220,000 1 

Indication I /g 0,000 0 50,000 4 100,000 6 150,000 9 220,001 4 

Error of indication E /g 0,000 0 0,000 4 0,000 7 0,001 0 0,001 3 7.1-1 

Repeatability u(δIrep)/g 0,000 114 7.1.1-5 

Resolution u(δIdig0) /g 0,000 029 7.1.1-2a 

Resolution u(δIdigL) /g 0,000 000 0,000 029 7.1.1-3a 

Eccentricity u(δIecc)/g 0,000 000 0,000 029 0,000 058 0,000 087 0,000 127 7.1.1-10 

Uncertainty of the indication

u(I)/g 0,000 118 0,000 124 0,000 134 0,000 149 0,000 175 7.1.1-12 

Test loads mc /g 0,000 0 50,000 0 99,999 9 99,999 9 

50,000 0 

200,000 1 

20,000 0 

Conventional mass u(mc)/g 0,000 000 0,000 015 0,000 025 0,000 040 0,000 062 7.1.2-2 

Drift u(mD)/g 0,000 000 0,000 022 0,000 036 0,000 058 0,000 089 7.1.2-11 

Buoyancy u(mB)/g 0,000 000 0,000 447 0,000 889 0,001 330 0,001 960 

7.1.2-5d 

/ Table 

E2.1 

Convection u(mconv)/g Not relevant in this case (weights are acclimatized). 7.1.2-13 

Uncertainty of the reference 

mass u(mref)/g 0,000 000 0,000 448 0,000 890 0,001 332 0,001 963 7.1.2-14 

Standard uncertainty of the 

error u(E) /g 0,000 118 0,000 465 0,000 900 0,001 340 0,001 971 7.1.3-1a 

eff (degrees of freedom) 4 1104 15538 76345 357098 B3-1 

k(95,45 %) 2,87 2,00 2,00 2,00 2,00 [1] 

U(E) = ku(E)/g 0,000 34 0,000 93 0,001 80 0,002 68 0,003 94 7.3-1 

Urel(E)/% ---- 0,001 86 0,001 80 0,001 79 0,001 79 

Alternative: Uncertainty due to buoyancy with formula (7.1.2-5e) instead of (7.1.2-5d), i.e. substituting the 

worst case approach with a value derived from the estimated room temperature variations of 5 K during 

use.

Buoyancy u(mB)/g 0,000 000 0,000 103 0,000 201 0,000 304 0,000 446 7.1.2-5e 

Uncertainty of the reference 

mass u(mref)/g 0,000 000 0,000 107 0,000 205 0,000 312 0,000 459 7.1.2-14 

Standard uncertainty of the 

erroru(E) /g 0,000 118 0,000 164 0,000 245 0,000 346 0,000 491 7.1.3-1a 

eff (degrees of freedom) 4 17 85 338 1377 B3-1 

k(95,45 %) 2,87 2,16 2,03 2,01 2,00 [1] 

U(E) = ku(E)/g 0,000 34 0,000 35 0,000 50 0,000 69 0,000 98 7.3-1 

Urel(E)/% ---- 0,000 70 0,000 50 0,000 46 0,000 45 

It is seen in this example that the uncertainty of the reference mass is reduced 

significantly if an uncertainty contribution for buoyancy is taken into account that is 

based on the estimated room temperature changes during use rather than using the 

most conservative approach provided by (7.1.2-5d). 

It would be acceptable to state in the certificate only the largest value of expanded 

uncertainty for all the reported errors: u(E) = 0,003 94 g (or alternatively 0,000 98 g) 

based on k = 2,00 accompanied by the statement that the coverage probability is at 

least 95 %.
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The certificate shall give the advice to the user that the expanded uncertainty stated in 

the certificate is only applicable, when the error (E) is taken into account. 

Uncertainty budget for option 2 (buoyancy correction applied to the error of indication 

values) 

Additional condition: 

The balance is not adjusted immediately before calibration. The procedure according to 

option 2 is applied, taking into account the determination of the air density and buoyancy 

correction. Therefore, formula (7.1.2-5a) is applied for the uncertainty due to air 

buoyancy. 

Option 2 above has shown that the buoyancy correction δmB is negligible as it is smaller 

than the relative resolution of the instrument, but the result of the calculation is 

nevertheless shown in the table below. Now, the uncertainty of the buoyancy correction 

u(mB) is calculated using formula (7.1.2-5a). Note that the air density during adjustment 

(which occurred independently of the calibration) is unknown, so that the variation of air 

density over time is taken as an estimate for the uncertainty. Consequently, the 

uncertainty of the air density is derived based on assumptions for pressure, temperature 

and humidity variations which can occur at the installation site of the instrument. 

Appendix A3 provides advice to estimate the uncertainty of the air density. The example 

uses the approximation of the uncertainty based on (A3-2) instead of the general 

equation (A3-1), i.e. with temperature being the only free parameter. 

For a temperature variation of 5 K, the calculation with the approximation formula (A3-2) 

leads to a relative uncertainty of u(a)/a = 1,18 × 10-2, which, for an air density at 

calibration of ρa = 1,173 kg/m3, leads to an uncertainty u(a)= 0,014 kg/m3.The same 

result can be obtained if the exact formula for the uncertainty of the air density (A3-1) is 

taken. 

The following numeric values are taken to calculate the relative uncertainty of the 

buoyancy correction, using formula (7.1.2-5a):

Air density ρaCal: (1,173 ± 0,014) kg/m3 

Density of the reference mass ρCal: (7950 ± 70) kg/m3

Formula (7.1.2-5a) leads to the relative uncertainty of the buoyancy correction of 

urel(mB) = 3,203 × 10 -8

The relative uncertainty of the buoyancy correction is negligible as compared to the 

other contributions to the uncertainty of the reference mass but the result of the 

calculation is nevertheless shown in the table below. 

This example has shown that the calculated correction of the error mB and the 

calculated relative uncertainty of the buoyancy correction u(mB) are both negligible. 

This leads to an updated measurement uncertainty budget. 

The uncertainty of convection effects due to non-acclimatized weights u(mconv) for a 

temperature difference of 2 K is shown. The rest of the uncertainty contributions are the 

same as in the table above and are not repeated in the table below. 

 
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Quantity or Influence Load, indication and error in g 

Uncertainties in g Formula 

Load mref /g 0,000 0 50,000 0 99,999 9 149,999 9 220,000 1 

Correction mB /g 0,000 0 0,000 001 0,000 002 0,000 003 0,000 005 4.2.4.3 

Indication I /g 0,000 0 50,000 4 100,000 6 150,0009 220,001 4 

Error of indication E /g 0,000 0 0,000 4 0,000 7 0,001 0 0,001 3 7.1-1 

Buoyancyu(mB)/g 0,000 0 0,00000 2 0,000 003 0,000 005 0,000 007 7.1.2-5a 

Convectionu(mconv)/g 0,000 0 0,000 029 0,000 046 0,000 075 0,000 092 

7.1.2-13 / 

Table 

F2.1

Uncertainty of the 

reference massu(mref)/g 0,000 0 0,000 039 0,000 064 0,000 103 0,000 143 7.1.2-14 

Standard uncertainty of 

the erroru(E) /g 0,000 118 0,000 130 0,000 149 0,000 181 0,000 226 7.1.3-1a 

eff(degrees of freedom) 4 6 11 25 62 B3-1 

k(95,45 %) 2,87 2,52 2,25 2,11 2,05 [1] 

U(E) = ku(E)/g 0,000 34 0,000 33 0,000 33 0,000 38 0,000 46 7.3-1 

Urel(E)/% ----- 0,000 66 0,000 33 0,000 25 0,000 21 

It can be seen from this example that the contribution of buoyancy to the standard 

uncertainty is significant when the most conservative approach following formula (7.1.2-

5d) is chosen. 

If information about the temperature estimated room temperature variations during use is 

available and the uncertainty of the air buoyancy is calculated following formula (7.1.2-

5e), the difference in the uncertainty of the error is less significant. 

H1.4/A Uncertainty of a weighing result (for option 1) 

As stated in 7.4, the following information may be developed by the calibration laboratory 

or by the user of the instrument. The results must not be presented as part of the 

calibration certificate except for the approximated error of indication and the uncertainty 

of the approximated error which can form part of the certificate. Usually the information 

on the uncertainty of a weighing result is presented as an appendix to the calibration 

certificate or is otherwise shown if its contents are clearly separated from the calibration 

results. 

Normal conditions of use of the instrument, as assumed, or as specified by the user may 

include: 

- Built-in adjustment device available and activated (T ≥ 3 K).Variation of room 

temperature T = 5 K. 

- Tare balancing function operated. 

- Loads not always centred carefully. 

The uncertainty of a weighing result is derived using a linear approximation of the error 

of indication according to (C2.2-16). 

The uncertainty of a weighing result is presented for option 1 only (no buoyancy 

correction applied to the error of indication values). The approximated error of indication 

per (C2.2-16) and the uncertainty of the approximated error of indication per (C2.2-16d) 

differ insignificantly between both options as the underlying weighting factors 
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  p j u E j

2  1 differ in the order of a few parts per million, and the errors of indication are 

the same for both options (buoyancy correction smaller than the resolution of the 

instrument). 

The designations R and W are introduced to differentiate from the weighing instrument 

indication I during calibration. 

R: Reading when weighing a load on the calibrated instrument obtained after the 

calibration 

W: Weighing result 

Note that within the following table the reading R and all results are in g. 

Quantity or Influence Reading, weighing result and error in g 

Uncertainties in g or as relative value Formula 

Error of Indication 

Eappr(R) for gross or net 

readings: Approximation 

by a straight line through 

zero 

E R R6

appr 6,709 10   C2.2-16 

Uncertainty of the approximated error of indication 

Standard uncertainty of 

the error u(Eappr)     11 2 12 2

appr

2 u E 4,501 10 u R 1,543 10 R       10 C2.2-16d 

Standard uncertainty of 

the error, neglecting the 

offset 

uE  R6

appr 1,242 10  

Uncertainties from environmental influences 

Temperature drift of 

sensitivity   6

rel temp 1 299 10 u R  ,  7.4.3-1 

Buoyancy   6

rel buoy 1,636 10  u R   7.4.3-4 

Change in characteristics 

due to drift 

Not relevant in this case (built-in adjustment activated and 

drift between calibrations negligible). 7.4.3-5 

Uncertainties from the operation of the instrument 

Tare balancing operation   6

rel Tare 1072 10 u R  , 

7.4.4 

7.4.4-5 

Creep, hysteresis 

(loading time) Not relevant in this case (short loading time). 7.4.4-9a/b 

Eccentric loading   6

rel ecc 1155 10 u R  ,  7.4.4-10 

Uncertainty of a weighing result 

Standard uncertainty, 

corrections to the 

readings u(Eappr) to be 

applied 

    8 2 12 2 u W 1,467 10 g 8,390 10 R      

7.4.5-1a 

7.4.5-1b 

Standard uncertainty, 

corrections to the 

readings u(Eappr) to be 

applied 

    8 2 12 2 U W 2 1,467 10 g 8,390 10 R       7.5.-2b 

Simplified to first order UW  R 4 6 2,422 10 g 4,796 10       7.5.2-3d 

Global uncertainty of a weighing result without correction to the readings 

U   W U W E R gl   appr U W  R 4 5

gl 2,422 10 g 1,150 10       7.5.2-3a 

7.5.2-3e 

 

10The first term is negligible as the uncertainty of the reading u(R) is in the order of some g. Thus the first term is in the order of 10-7

g2

 while the second term represents values up to 15 g2.
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The condition regarding the observed chi-squared value following (C2.2-2a) was 

checked with positive result. The first linear regression is taking into account the 

weighing factors p j

 , equation (C2.2-18b). 

Based on the global uncertainty, the minimum weight value for the instrument may be 

derived as per Appendix G. 

Example: 

Weighing tolerance requirement: 1 % 

Safety factor: 3 

The minimum weight according to formula (G-9), using the above equation for the global 

uncertainty in results is 0,072 9 g; i.e. the user needs to weigh a net quantity of material 

that exceeds 0,072 9 g in order to achieve a relative (global) measurement uncertainty 

for a relative weighing tolerance requirement of 1 % and a safety factor of 3 (equals a 

relative weighing tolerance of 0,33 %). 

Second situation: Adjustment of sensitivity carried out immediately before 

calibration 

H1.1/B Conditions specific for the calibration 

Instrument: Electronic weighing instrument, description and 

identification 

Maximum Capacity 

L/Scale interval d 220 g / 0,000 1 g 

Temperature coefficient 

KT = 1,5 x 10 -6/K (manufacturer’s manual) Only 

necessary for calculation of the uncertainty of a weighing 

result.

Built-in adjustment 

device 

Acts automatically upon: after switch-on of the balance, 

and when T ≥ 3 K. Only necessary for calculation of 

uncertainty of a weighing result.Status: activated. 

Adjustment by 

calibrator 

Adjusted immediately before calibration (built-in 

adjustment weights). 

Temperature during 

calibration 21 °C measured at the beginning of calibration. 

Barometric pressure and 

humidity (optional) 990 hPa, 50 % RH. 

Room conditions 

Max. Temperature deviation 5 K (laboratory room without 

windows). Not relevant, when built-in adjustment device is 

activated (T ≥ 3 K). In this case the maximum 

temperature variation for the estimation of uncertainty of a 

weighing result is 3 K. 

Test loads / 

acclimatization 

Standard weights, class E2, acclimatized to room 

temperature (alternative a temperature difference of 2 K 

against room temperature is taken into account). 

 
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H1.2/B Tests and results 

Option 1: Air densities unknown during adjustment /calibration (i.e. no buoyancy 

correction applied to the error of indication values) 

The repeatability test is omitted and the results from the first calibration are taken into 

account. Also the eccentricity test was omitted and the results from the first calibration 

are taken into account. This can be done as only the sensitivity of the balance was 

adjusted and no influence to the repeatability and the eccentricity test can be estimated. 

Air density not calculated. 

Errors of indication 

Requirements given in 

Chapter 5.2, weights distributed 

fairly evenly 

Test loads each applied once; discontinuous loading only 

upwards, indication at no load reset to zero where 

necessary. 

Indications recorded: 

Load mref Indication I Error of indication E

0,000 0 g 0,000 0 g 0,000 0 g 

50,000 0 g 50,000 0 g 0,000 0 g 

99,999 9 g 99,999 8 g - 0,000 1 g 

149,999 9 g 149,999 9 g 0,000 0 g 

220,000 1 g 220,000 0 g - 0,000 1 g 

Option 2: Air density ρas during adjustment and air density ρacal during calibration are 

identical as an adjustment was carried out immediately before calibration. 

The air density is calculated according to the simplified CIPM formula (A1.1-1) 

 Measurement values used for calculation: 

Barometric pressure p: 990 hPa 

Relative humidity RH: 50 % 

Temperature t: 21 °C 

Density ρs and ρCal: (7950 ± 70) kg/m3 

Air density ρaCal: 1,173 kg/m3 

Calculated buoyancy correction δmB according to formula (4.2.4-4). 

Numerical value used for calculation: 

Density of the reference mass ρCal: (7950 + 70) kg/m3 

Buoyancy correction δmB: 2,138 x 10-8 mref

The calculated buoyancy correction δmB of mref of Load L following formula (4.2.4-4) is 

negligible as the relative resolution of the instrument is in the order of 10 -6 and thus 

much larger than the buoyancy correction. The above table is effectual. 

H1.3/B Errors and related uncertainties (budget of related uncertainties) 

Conditions: 

- The uncertainty for the zero position only results in the digitalisation d0 and 

repeatability s.

- The eccentric loading is taken into account for the calibration according to (7.1.1-

10). 

- The conventional mass of the test weights (class E2) is taken into account for the 

calibration results. Therefore U(mc) = U/k is calculated following formula 7.1.2-2. 
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- The drift of the weights was statistically monitored and the factor kD of formula 7.1.2-

11 was chosen as 1,25. 

- The degrees of freedom for the calculation of the coverage factor k are derived 

following appendix B3 and table G.2 of [1]. In the case of the example, the influence 

of the uncertainty of the repeatability test with 5 measurements is significant. 

- The information about the relative uncertainty U(E)rel = u(E)/mref is not mandatory, 

but helps to demonstrate the characteristics of the uncertainties. 

Uncertainty budget for option 1 (no buoyancy correction applied to the error of indication 

values) 

Additional condition: 

The balance is adjusted immediately before the calibration and no information about air 

density at the time of calibration is available. Therefore, formula (7.1.2-5c) is relevant. 

Quantity or Influence Load, indication and error in g 

Uncertainties in g Formula 

Loadmref /g 0,000 0 50,000 0 99,999 9 149,999 9 220,000 1 

Indication I / g 0,000 0 50,000 0 99,999 8 149,999 9 220,000 0 

Error of indication E /g 0,000 0 0,000 0 -0,000 1 0,000 0 -0,000 1 7.1-1

Repeatability u(δIrep)/g 0,000 114 7.1.1-5 

Resolution u(δIdig0) /g 0,000 029 7.1.1-2a 

Resolution u(δIdigL) /g 0,000 0 0,000 029 7.1.1-3a 

Eccentricity u(δIecc)/g 0,000 0 0,000 029 0,000 058 0,000 087 0,000 127 7.1.1-10 

Uncertainty of the 

indication u(I)/g 0,000 118 0,000 124 0,000 134 0,000 149 0,000 175 7.1.1-12 

Test loads mc /g 0,000 0 50,000 0 99,999 9 99,999 9

50,000 0

200,000 1 

20,000 0 

Conventional mass 

u(mc)/g 0,000 0 0,000 015 0,000 025 0,000 040 0,000 063 7.1.2-2 

Drift u(mD)/g 0,000 0 0,000 022 0,000 036 0,000 058 0,000 090 7.1.2-10 

Buoyancy u(mB)/g 0,000 000 0,000 014 0,000 022 0,000 036 0,000 055 

7.1.2-5c / 

Table 

E2.1 

Convection u(mconv)/g Not relevant in this case (weights are acclimatized) 7.1.2-13 

Uncertainty of the 

reference mass u(mref)/g 0,000 00 0,000 03 0,000 049 0,000 079 0,000 123 7.1.2-14 

Standard uncertainty of the 

error u(E) /g 0,000 118 0,000 128 0,000 143 0,000 169 0,000 214 7.1.3-1a 

eff(degrees of freedom) 4 6 9 19 49 B3-1 

k(95,45 %) 2,87 2,52 2,32 2,14 2,06 [1] 

U(E) = ku(E)/g 0,000 34 0,000 32 0,000 33 0,000 36 0,000 44 7.3-1 

Urel(E)/% ---- 0,000 64 0,000 33 0,000 24 0,000 20 

It would be acceptable to state in the certificate only the largest value of expanded 

uncertainty for all the reported errors: U(E)= 0,000 44 g, based on k = 2,06 accompanied 

by the statement that the coverage probability is at least 95 %.The certificate shall give 

the advice to the user that the expanded uncertainty stated in the certificate is only 

applicable, when the Error (E) is taken into account. 

 
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Uncertainty budget for option 2 (buoyancy correction applied to the error of indication 

values) 

Additional condition: 

The balance is adjusted immediately before calibration. The procedure according to 

option 2 is applied, taking into account the determination of the air density and buoyancy 

correction. Therefore, formula (7.1.2-5a) is applied for the uncertainty due to air 

buoyancy. 

As an adjustment has been carried out immediately before the calibration, the expected 

maximum values for pressure, temperature and humidity variations which can occur at 

the installation site of the instrument do not have to be taken into account in contrast to 

the scenario where the adjustment has been performed independent of the calibration. 

The only contributing factor to the standard uncertainty of the air density originates from 

the uncertainty of the measurement of the environmental parameters. 

The following numeric values are taken to calculate the relative uncertainty of the 

buoyancy correction, using formula (7.1.2-5a): 

Air density ρaCal: 1,173 kg/m3 

Density of the reference mass ρCal: (7950 ± 70) kg/m3

Furthermore, the following uncertainties for temperature, pressure and humidity 

measurement are taken for calculating the relative uncertainty of the air density 

according to (A3-1): 

u  T = 0,2 K 

u  p = 50 Pa 

u  RH = 1% 

This leads to 

a

a ( )



u  = 9,77 × 10-4, and ( )  a u = 0,00115 kg/m3. 

Formula (7.1.2-5a) leads to the relative uncertainty of the buoyancy correction of 

u(mB) = 3,014 × 10-8 

As an alternative the additional uncertainty of convection effects due to non-acclimatized 

weights u(mconv) for a temperature difference of 2 K is shown.

 
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Quantity or Influence Load, indication and error in g 

Uncertainties in g Formula 

Load mref /g 0,000 0 50,000 0 99,999 9 149,999 9 220,000 1 

Correction mB /g 0,000 0 0,000 001 0,000 002 0,000 003 0,000 005 4.2.4-4

Indication I /g 0,000 0 50,000 0 99,999 8 149,999 9 220,000 0 

Error of indication E /g 0,000 0 0,000 0 -0,000 1 0,000 0 -0,000 1 

Buoyancy u(mB)/g 0,000 0 0,000 001 5 0,000 003 0 0,000 004 5 0,000 006 6 7.1.2-5a 

Convection u(mconv)/g Not relevant in this case (weights are acclimatized).

Uncertainty of the 

reference mass u(mref) /g 0,000 000 0,000 026 0,000 044 0,000 066 0,000 110 7.1.2-14 

Standard uncertainty of 

the error u(E) /g 0,000 118 0,000 127 0,000 141 0,000 163 0,000 207 7.1.3-1a 

eff(degrees of freedom) 4 6 9 16 43 B3-1 

k(95,45 %) 2,87 2,52 2,32 2,17 2,06 [1] 

U(E) = ku(E)/g 0,000 34 0,000 32 0,000 33 0,000 35 0,000 43 7.3-1 

Urel(E)/ % ---- 0,000 64 0,000 33 0,000 23 0,000 20 

Alternative the additional uncertainty of convection effects due to non-acclimatized weights u(mconv) for 

a temperature difference of 2 K is shown.

Convection u(mconv)/g 0,000 000 0,000 029 0,000 046 0,000 075 0,000 092 7.1.2-13 

Uncertainty of the 

reference mass u(mref) /g 0,000 000 0,000 031 0,000 051 0,000 079 0,000 122 7.1.2-14 

Standard uncertainty of 

the error u(E) / g 0,000 118 0,000 128 0,000 144 0,000 168 0,000 214 7.1.3-1a 

eff (degrees of freedom) 4 6 10 19 49 B3-1 

k(95,45 %) 2,87 2,52 2,28 2,14 2,06 [1] 

U(E) = ku(E)/g 0,000 34 0,000 32 0,000 33 0,000 36 0,000 44 7.3-1 

Urel(E)/% ---- 0,000 64 0,000 33 0,000 24 0,000 20 

The expanded uncertainties of the error using option 1 and using the option 2 are almost 

identical as the uncertainty of the reference mass u(mref) is very small as compared to 

the uncertainty of the indication u(I). In this example, the determination of pressure and 

humidity on site to determine the buoyancy correction and to minimize the uncertainty 

contribution due to buoyancy does not significantly improve the results of the calibration. 

H1.4/B Uncertainty of a weighing result (for option 1) 

As stated in 7.4, the following information may be developed by the calibration laboratory 

or by the user of the instrument. The results must not be presented as part of the 

calibration certificate except for the approximated error of indication and the uncertainty 

of the approximated error which can form part of the certificate. Usually the information 

on the uncertainty of a weighing result is presented as an appendix to the calibration 

certificate or is otherwise shown if its contents are clearly separated from the calibration 

results. 

Normal conditions of use of the instrument, as assumed, or as specified by the user may 

include: 

- Built-in adjustment device available and activated (T ≥ 3 K) 

- Variation of room temperature T = 5 K 

- Tare balancing function operated 
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- Loads not always centred carefully 

The uncertainty of a weighing result is derived using a linear approximation of the error 

of indication according to (C2.2-16). 

The uncertainty of a weighing result is presented for option 1 only (no buoyancy 

correction applied to the error of indication values). The approximated error of indication 

per (C2.2-16) and the uncertainty of the approximated error of indication per (C2.2-16d) 

differ insignificantly between both options as the underlying weighting factors 

  p j u E j

2  1 differ in the order of a few per mil, and the errors of indication are the same 

for both options (buoyancy correction smaller than the resolution of the instrument). 

The designations R and W are introduced to differentiate from the weighing instrument 

indication I during calibration. 

R: Reading when weighing a load on the calibrated instrument obtained after the 

calibration 

W: Weighing result 

Note that within the following table the reading R and all results are in g. 

 
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Quantity or Influence Reading, weighing result and error in kg 

Uncertainties in g or as relative value Formula 

Error of Indication 

Eappr(R) for gross or net 

readings: Approximation 

by a straight line through 

zero 

E R R7

appr 3,895 10    C2.2-16 

Uncertainty of the approximated error of indication

Standard uncertainty of 

the error u(Eappr)     13 2 13 2

appr

2 u E 1,517 10 u R 4,015 10 R       11 C2.2-16d 

Standard uncertainty of 

the error, neglecting the 

offset 

uE  R7

appr 6,337 10  

Uncertainties from environmental influences

Temperature drift of 

sensitivity   6

rel temp 1 299 10 u R  ,  7.4.3-1 

Buoyancy   6

rel buoy 1,636 10  u R   7.4.3-4 

Change in characteristics 

due to drift 

Not relevant in this case (built-in adjustment activated and 

drift between calibrations negligible) 7.4.3-5 

Uncertainties from the operation of the instrument

Tare balancing operation   7

rel Tare 5 774 10 u R  ,  7.4.4-5 

Creep, hysteresis 

(loading time) Not relevant in this case (short loading time). 7.4.4-9a/b 

Eccentric loading   6

rel ecc 1154 10 u R  ,  7.4.4-10 

Uncertainty of a weighing result

Standard uncertainty, 

corrections to the 

readings u(Eappr) to be 

applied 

    8 2 12 2 u W 1,466 10 g 6,433 10 R      

7.4.5-1a 

7.4.5-1b 

Standard uncertainty, 

corrections to the 

readings u(Eappr) to be 

applied 

    8 2 12 2 U W 2 1,466 10 g 6,433 10 R       7.5.1-2b 

Simplified to first order UW  R 4 6 2,422 10 g 4,090 10       7.5.2-3d 

Global uncertainty of a weighing result without correction to the readings

Ugl     W  U W  Eappr R U W  R 4 6

gl 2,422 10 g 4,479 10       7.5.2-3a 

The condition regarding the observed chi-squared value following (C2.2-2a) was 

checked with positive result. The first linear regression taking into account the weighing 

factors p j

 , equation (C2.2-18b). 

Based on the global uncertainty, the minimum weight value for the instrument may be 

derived as per Appendix G. 

Example: 

Weighing tolerance requirement: 1% 

Safety factor: 3 

The minimum weight according to formula (G-9), using the above equation for the global 

 

11The first term is negligible as the uncertainty of the reading u(R) is in the order of some mg. Thus the first term is in the order of 

10-11 mg2

 while the second term represents values up to 10-7 mg2.
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uncertainty results in 0,0727 g; i.e. the user needs to weigh a net quantity of material 

that exceeds 0,0727 g in order to achieve a relative (global) measurement uncertainty 

for a relative weighing tolerance requirement of 1 % and a safety factor of 3 (equals a 

relative weighing tolerance of 0,33%). 

H2 Instrument of 60 kg capacity, multi-interval 

Preliminary note:

The calibration of a multi-interval balance with scale intervals 2 g / 5 g / 10 g is 

demonstrated. This example shows the complete standard procedure for the 

presentation of measurement results and the related uncertainties as executed by most 

laboratories. An alternative method for the consideration of air buoyancy effects is also 

presented as option 2 (in italic type). 

First situation: adjustment of sensitivity carried out independently of calibration 

H2.1/A Conditions specific for the calibration 

Instrument Electronic non-automatic weighing instrument, 

description and identification 

Upper limits of the 

intervals Maxi/Scale 

intervals di

12 000 g / 2 g 

30 000 g / 5 g 

60 000 g / 10 g 

Sensitivity of the 

instrument to 

temperature variation 

KT= 2×10-6/K (manufacturer’s manual); only necessary for 

calculation of the uncertainty of a weighing result. 

Built-in adjustment 

device 

Acts automatically after switching on the balance, and 

when T≥ 3 K; only necessary for calculation of 

uncertainty of a weighing result. Status: activated. 

Adjustment by 

calibrator Not adjusted immediately before calibration. 

Temperature during 

calibration 

21 °C at the beginning of calibration 

23 °C at the end of the calibration. 

Barometric pressure and 

humidity (optional) 990 hPa, 50 % RH. 

Room conditions 

Maximum temperature variation during use 10 K 

(laboratory room with windows). If used for the buoyancy 

uncertainty as per formula 7.1.2-5e, it must be presented 

in the calibration certificate. Not relevant for the 

uncertainty of a weighing result, when built-in adjustment 

device is activated (T≥ 3 K). In this case the maximum 

temperature variation for the estimation ofthe uncertainty 

of a weighing result is 3 K. 

Test loads / 

Acclimatization 

Standard weights, class F2, acclimatized to room 

temperature. 

 
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H2.2/A Tests and results 

Repeatability

Requirements given in 

Chapter 5.1

Indication at no load 

reset to zero where 

necessary 

Repeatability test carried 

out in interval 1 and 2 

Test load 10 000 g 

applied 5 times(standard 

deviation assumed 

constant over interval 1) 

Test load 25 000 g applied 

5 times(standard deviation 

assumed constant over 

interval 2 and 3)

9 998 g 24 995 g 

10 000 g 25 000 g 

9 998 g 24 995 g 

10 000 g 24 995 g 

10 000 g 25 000 g 

Standard deviation s = 1,095 g s = 2,739 g 

Eccentricity

Requirements given in 

Chapter 5.3

Indication set to zero prior to 

test; load put in centre first then 

moved to the other positions 

Position of the load Test load 20 000 g

Centre 19 995 g 

Front left 19 995 g 

Back left 19 995 g 

Back right 19 990 g 

Front right 19 990 g 

Maximum deviation max ecci I 5 g 

Errors of indication

General prerequisites: Requirements given in Chapter 5.2, weights distributed 

fairly evenly over the weighing range. 

Test loads each applied once; discontinuous loading only 

upwards, indication at no load reset to zero if necessary. 

Option 1: Air density unknown during adjustment and during calibration (i.e. no 

buoyancy correction applied to the error of indication values) 

Requirements given in 

chapter 5.2, weights 

distributed fairly evenly. 

Test loads each applied 

once; discontinuous 

loading only upwards; 

indication at no load reset 

to zero where necessary 

Load mref

(mN) Indication I Error of 

indication E

0 g 0 g 0 g 

10 000 g 10 000 g 0 g 

20 000 g 19 995 g -5 g 

40 000 g 39 990 g - 10 g 

60 000 g 59 990 g - 10 g 

 
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Option 2: Air density ρas during adjustment unknown and air density ρaCal during 

calibration calculated according to the simplified CIPM formula (A1.1-1) 

Measurement values used for calculation: 

Barometric pressure p: 990 hPa 

Relative humidity RH: 50 % 

Temperature t: 21 °C 

Air density ρaCal 1,173 kg/m3 

Calculated buoyancy correction δmB according to formula 4.2.4-4: 

Numerical value used for calculation 

Density of the reference mass ρCal : (7950 ± 70) kg/m3 

Buoyancy correction δmB: 2,138 × 10-8mN

The calculated correction δmB of the loads mN following formula 4.2.4-4 is negligible as 

the relative resolution of the instrument is in the order of 10-4 and thus much larger than 

the buoyancy correction. The above table is effectual. 

H2.3/A Errors and related uncertainties (budget of related uncertainties) 

Conditions common to both options: 

- The uncertainty of the error at zero only comprises the uncertainty of the no-load 

indication (scale interval d0 = d1= 2 g) and the repeatability s. The uncertainty of the 

indication at load is not taken into consideration at zero. 

- The eccentric loading is taken into account for the calibration according to (7.1.1-

10). 

- The error of indication is derived using the nominal weight value as reference value, 

therefore the maximum permissible errors of the test weights are taken into account 

for deriving the uncertainty contribution due to the reference mass: u(mc)is 

calculated as u(mc) = Tol/3 following formula (7.1.2-3). 

- The average drift of the weights monitored over 2 recalibrations in two-yearly 

intervals was Dmpe/2. Therefore the uncertainty contribution due to the drift of 

the weights was set to u(mD) = mpe/23. This corresponds to a kDfactor of 1,5 

(assuming the worst-case scenario of U = mpe / 3). 

- The weights are acclimatized with a residual temperature difference of 2 K to the 

ambient temperature. 

- The degrees of freedom for the calculation of the coverage factor k are derived 

following appendix B3 and table G.2 of [1]. In the case of the example, the influence 

of the uncertainty of the repeatability test with 5 measurements is significant. 

- The information about the relative uncertainty U(E)rel = U(E)/mref is not mandatory, 

but helps to demonstrate the characteristics of the uncertainties. 

Uncertainty budget for option 1 (no buoyancy correction applied to the error of indication 

values) 

Additional condition: 

The balance is not adjusted immediately before calibration. The procedure according to 

option 1 is applied, with no information about air density. Therefore formula (7.1.2-5d) is 

applied for the uncertainty due to air buoyancy. As an alternative in the table, formula 

(7.1.2-5e) was used, thereby assuming a temperature variation during use of 10 K. 
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Quantity or Influence Load, indication and error in g 

Uncertainties in g Formula 

Load mref(mN) /g 0 10 000 20 000 40 000 60 000 

Indication I /g 0 10 000 19 995 39 990 59 990 

Error of Indication E /g 0 0 - 5 - 10 - 10 7.1-1 

Repeatability u(Irep) /g 1,095 2,739 7.1.1-5 

Resolution u(Idig0) /g 0,577 7.1.1-2a 

Resolution u(IdigL) /g 0,000 0,577 1,443 2,887 2,887 7.1.1-3a 

Eccentricity u(Iecc) /g 0,000 0,722 1,443 2,887 4,330 7.1.1-10 

Uncertainty of the 

indicationu(I) /g 1,238 1,545 3,464 4,950 5,909 7.1.1-12 

Test loads mN /g 

0 10 000 20 000 20 000

20 000

20 000 

20 000 

20 000 

Weights u(mc) /g 0,000 0,092 0,173 0,346 0,554 7.1.2-3 

Drift u(mD) / g 0,000 0,046 0,087 0,173 0,277 7.1.2-11 

Buoyancy u(mB) /g 0,000 0,110 0,217 0,433 0,658 

7.1.2-5d 

/ Table 

E2.1 

Convection u(mconv) /g Not relevant in this case (only relevant for F1 and 

better). 7.1.2-13 

Uncertainty of the reference 

mass u(mref) /g 0,000 0,151 0,290 0,581 0,904 7.1.2-14 

Standard uncertainty of the 

error u(E) /g 1,238 1,552 3,476 4,984 5,978 7.1.3-1a 

eff (degrees of freedom) 6 16 10 43 90 B3-1 

k(95,45 %) 2,52 2,17 2,28 2,06 2,05 [1] 

U(E) = ku(E)/g 3,120 3,369 7,926 10,266 12,254 7.3-1 

Urel(E)/% ---- 0,0337 % 0,0396 % 0,0257 % 0,0204 % 

Alternative: Uncertainty due to buoyancy with formula (7.1.2-5e) instead of (7.1.2-5d), i.e. 

substituting the worst case approach with a value derived from the estimated room 

temperature variations of 10 K during use. 

Buoyancy u(mB) /g 0,000 0,046 0,089 0,178 0,276 7.1.2-5e 

Uncertainty of the reference 

mass u(mref) /g 0,000 0,113 0,213 0,462 0,678 7.1.2-14 

Standard uncertainty of the 

error u(E) /g 1,238 1,549 3,471 4,968 5,948 7.1.3-1a 

eff (degrees of freedom) 6 16 10 43 88 B3-1 

k(95,45 %) 2,52 2,17 2,28 2,06 2,05 [1] 

U(E) = ku(E)/g 3,120 3,362 7,913 10,234 12,193 7.3-1 

Urel(E)/% ---- 0,0336 0,0396 0,0256 0,0203 

It is seen in this example that the uncertainty of the reference mass is reduced 

significantly if an uncertainty contribution for buoyancy is taken into account that is 

based on the estimated room temperature changes during use rather than using the 

most conservative approach provided by (7.1.2-5d). However, as the uncertainty of the 

reference mass is very small compared to the uncertainty of the indication, the standard 

uncertainty of the error is almost not affected. 

It would be acceptable to state in the certificate only the largest value of the expanded 

uncertainty for all the reported errors: U(E)= 12,254 g, based on k = 2,05 accompanied 

by the statement that the coverage probability is at least 95 %. 

The certificate shall give the advice to the user that the expanded uncertainty stated in 
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the certificate is only applicable, when the Error (E) is taken into account. 

Uncertainty budget for option 2 (buoyancy correction applied to the error of indication 

values) 

Additional condition: 

The balance is not adjusted immediately before calibration. The procedure according to 

option 2 is applied, taking into account the determination of the air density and buoyancy 

correction. Therefore, formula (7.1.2-5a) is applied for the uncertainty due to air 

buoyancy. 

Note that the air density during adjustment (which occurred independent of the 

calibration) is unknown, so that the variation of air density over time is taken as an 

estimate for the uncertainty. Consequently, the uncertainty of the air density is derived 

based on assumptions for pressure, temperature and humidity variations which can 

occur at the installation site of the instrument. 

Appendix A3 provides advice to estimate the uncertainty of the air density. The example 

uses the approximation of the uncertainty based on (A3-2) instead of the general 

equation (A3-1), i.e. with temperature being the only free parameter. 

For a temperature variation of 10 K, the calculation with the approximation formula (A3-

2) leads to a relative uncertainty of u(a)/a = 1,55  10-2, which, for an air density at 

calibration of ρa = 1,173 kg/m3, leads to an uncertainty u(a) = 0,018 kg/m3. 

The following numeric values are taken to calculate the relative uncertainty of the 

buoyancy correction, using formula (7.1.2-5a): 

Air density ρaCal: (1,173 ± 0,018) kg/m3 

Density of the reference mass ρCal: (7950 ± 70) kg/m3

Formula (7.1.2-5a) leads to the relative uncertainty of the buoyancy correction of 

urel(mB) = 3,334 × 10 -8

The relative uncertainty of the buoyancy correction is negligible as compared to the 

other contributions to the uncertainty of the reference mass. 

This example has shown that the calculated correction of the error δmB and the 

calculated relative uncertainty of the buoyancy correction urel(mB) are both negligible. 

This leads to an updated measurement uncertainty budget: 

 
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Quantity or Influence Load, indication and error in g 

Uncertainties in g Formula 

Load mref(mN)/g 0 10 000 20 000 40 000 60 000 

Correction mB/g 0 0 0 0 0 4.2.4-4 

Indication I /g 0 10000 19 995 39 990 59 990 

Error of Indication E/g 0 0 - 5 - 10 - 10 7.1-1 

Repeatability u(Irep) /g 1,095 2,739 7.1.1-5 

Resolution u(Idig0) /g 0,577 7.1.1-2a 

Resolution u(IdigL) /g 0,000 0,577 1,443 2,887 2,887 7.1.1-3a 

Eccentricityu(Iecc) /g 0,000 0,722 1,443 2,887 4,330 7.1.1-10 

Uncertainty of the 

indicationu(I) /g 1,238 1,545 3,464 4,950 5,909 7.1.1-12 

Test loadsmN /g 

0 10 000 20 000 20 000

20 000

20 000 

20 000 

20 000 

Weightsu(mc) /g 0,000 0,092 0,173 0,346 0,554 7.1.2-3 

Driftu(mD)/ g 0,000 0,046 0,087 0,173 0,277 7.1.2-11 

Buoyancyu(mB) /g 0,000 0,000 0,001 0,001 0,002 7.1.2-5a 

Convectionu(mconv) /g Not relevant in this case (only relevant for F1 and 

better). 7.1.2-13 

Uncertainty of the reference 

mass u(mref)/g 0,000 0,103 0,194 0,387 0,620 7.1.2-14 

Standard uncertainty of the 

erroru(E) /g 1,238 1,549 3,470 4,965 5,941 7.1.3-1a 

eff (degrees of freedom) 6 15 10 43 88 B3-1 

k(95,45 %) 2,52 2,17 2,28 2,06 2,05 [1] 

U(E) = ku(E)/g 3,120 3,360 7,910 10,228 12,180 7.3-1 

Urel(E)/% ---- 0,0360 0,0396 0,0256 0,0203 

It can be seen from this example that the contribution of buoyancy to the standard 

uncertainty is insignificant. Furthermore, the standard uncertainties of the error using 

option 1 and option 2 are almost identical as the uncertainty of the reference mass 

u(mref) is very small as compared to the uncertainty of the indication u(I). The 

determination of pressure and humidity on site in addition to the temperature 

measurement to correct for buoyancy and to minimize the associated uncertainty 

contribution does not significantly improve the results of the calibration. 

H2.4/A Uncertainty of a weighing result (for option 1) 

As stated in 7.4, the following information may be developed by the calibration laboratory 

or by the user of the instrument. The results must not be presented as part of the 

calibration certificate except for the approximated error of indication and the uncertainty 

of the approximated error which can form part of the certificate. Usually the information 

on the uncertainty of a weighing result is presented as an appendix to the calibration 

certificate or is otherwise shown if its contents are clearly separated from the calibration 

results. 

Normal conditions of use of the instrument, as assumed, or as specified by the user may 

include: 

- Built-in adjustment device available and activated (T ≥ 3 K) 

- Variation of room temperature T = 10 K 

- Tare balancing function operated 
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- Loads not always centred carefully 

The uncertainty of a weighing result is derived using a linear approximation of the error 

of indication according to (C2.2-16). 

The uncertainty of a weighing result is presented for option 1 only (no buoyancy 

correction applied to the error of indication values). The approximated error of indication 

per (C2.2-16) and the uncertainty of the approximated error of indication per (C2.2-16d) 

differ insignificantly between both options as the underlying weighting factors 

  p j u E j

2  1 differ in the order of a few per mil, and the errors of indication are the same 

for both options (buoyancy correction smaller than the resolution of the instrument). 

Buoyancy according to chapter 7.4.3.2 is not taken into account as the estimation of the 

uncertainty at calibration has shown that this influence is negligible. 

The designations R and W are introduced to differentiate from the weighing instrument 

indication I during calibration. 

R: Reading when weighing a load on the calibrated instrument obtained after the 

calibration 

W: Weighing result 

Note that within the following table the reading R and all results are in g. 

 
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Quantity or Influence Reading, weighing result and error in g 

Uncertainties in g or as relative value Formula 

Error of Indication 

Eappr(R) for gross or net 

readings: Approximation 

by a straight line through 

zero 

R , R4 Eappr 1717 10    C2.2-16 

Uncertainty of the approximated error of indication

Standard uncertainty of 

the error u(Eappr)   8 2 9 2

appr

2 u E 2,950 10 u (R) 4,172 10 R       12 C2.2-16d 

Standard uncertainty of 

the error, neglecting the 

intercept 

uE  , R

5

appr 6 459 10  

Uncertainties from environmental influences

Temperature drift of 

sensitivity   6

rel temp 1 732 10 u R  ,  7.4.3-1 

Buoyancy Not relevant in this case. 7.4.3-2 

Change in characteristics 

due to drift 

Not relevant in this case (built-in adjustment activated and 

drift between calibrations negligible) 7.4.3-5 

Uncertainties from the operation of the instrument

Tare balancing operation   4

rel Tare 1 444 10 u R  ,  7.4.4-5 

Creep, hysteresis 

(loading time) Not relevant in this case (short loading time). 7.4.4-9a/b 

Eccentric loading   4

rel ecc 1443 10 u R  ,  7.4.4-10 

Uncertainty of a weighing result, for partial weighing intervals (PWI)

Standard uncertainty, 

corrections to the 

readings u(Eappr) to be 

applied 

PWI 1     2 8 2 u W 1,867g 4,589 10 R    

PWI 2     7.4.5-1b 2 8 2 u W 9,917g 4,589 10 R    

PWI 3     2 8 2 u W 16,167g 4,589 10 R    

Expanded uncertainty, 

corrections to the 

readings Eappr to be 

applied 

PWI 1     2 8 2 U W 2 1,867g 4,589 10 R    

PWI 2     7.5.1-2b 2 8 2 U W 2 9,917g 4,589 10 R    

PWI 3     2 8 2 U W 2 16,167g 4,589 10 R    

Simplified to first order 

PWI 1 UW  , , R4 2 733 g 2 574 10   

PWI 2 10190 g 3 434 10  12000 g 7.5.2-3f 4      U(W ) , , R

PWI 3   20 311g 3 923 10  30000 g 4      U W , , R

Global uncertainty of a weighing result without correction to the readings

Ugl     W  U W  Eappr R

PWI 1 U W  , , R4

gl 2 733g 4 291 10   

PWI 2   10190 g 5151 10  12000g 7.5.2-3a 4

gl      U W , , R

PWI 3   20 311g 5 641 10  30000 g 4

gl      U W , , R

The condition regarding the observed chi-squared value following (C2.2-2a) was 

checked with positive result. The linear regression was performed taking into account 

the weighing factors p j

 of equation (C2.2-18b). 

 

12The first term is negligible as the uncertainty of the reading u(R) is in the order of some g. Thus the first term is in the order of 10-7

g2

 while the second term represents values up to 15 g2.
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Based on the global uncertainty, the minimum weight value for the instrument may be 

derived as per Appendix G. 

Example: 

Weighing tolerance requirement: 1 % 

Safety factor: 2 

The minimum weight according to formula G-9, using the above equation for the global 

uncertainty in PWI 1 results in 598 g; i.e. the user needs to weigh a net quantity of 

material that exceeds 598 g in order to achieve a relative (global) measurement 

uncertainty for a relative weighing tolerance requirement of 1% and a safety factor of 2 

(equals a relative weighing tolerance of 0,5%). 

Second situation: adjustment of sensitivity carried out immediately before calibration

H2.1/B Conditions specific for the calibration 

Instrument Electronic non-automatic weighing instrument, 

description and identification 

Upper limits of the 

intervalsMaxi/Scale 

intervals di

12 000 g / 2 g 

30 000 g / 5 g 

60 000 g / 10 g 

Sensitivity of the 

instrument to 

temperature variation 

KT= 2×10-6/K (manufacturer’s manual); only necessary for 

calculation of uncertainty of a weighing result. 

Built-in adjustment 

device 

Acts automatically after switching on the balance, and 

when T≥ 3 K; only necessary for calculation of the 

uncertainty of a weighing result. Status: activated. 

Adjustment by 

calibrator 

Adjusted immediately before calibration (built-in 

adjustment weights). 

Temperature during 

calibration 

23 °C at the beginning of calibration 

24 °C at the end of the calibration 

Barometric pressure and 

humidity (optional) 990 hPa, 50 % RH. 

Room conditions 

Maximum temperature variation during use 10 K (room 

without windows).Not relevant, when built-in adjustment 

device is activated (T ≥ 3 K). In this case the maximum 

temperature variation for the estimation of uncertainty of a 

weighing result is 3 K. 

Test loads / 

acclimatization 

Standard weights, class F2, acclimatized to room 

temperature. 

 
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H2.2/B Tests and results 

Repeatability

Requirements given in 

Chapter 5.1

Indication at no load 

reset to zero where 

necessary 

Repeatability test carried 

out in interval 1 and 2 

Test load 10 000 g 

applied 5 times (standard 

deviation assumed 

constant over interval 1) 

Test load 25 000 g applied 

5 times (standard deviation 

assumed constant over 

interval 2 and 3)

10 000 g 25 000 g 

10 000 g 25 000 g 

9 998 g 25000g 

10 000 g 24995g 

10 000 g 25 000 g 

Standard deviation s = 0,894 g s = 2,236 g 

Eccentricity

Requirements given in 

Chapter 5.3

Indication set to zero prior to 

test; load put in centre first then 

moved to the other positions 

Position of the load Test load 20 000 g

Centre 20 000g 

Front left 20 000g 

Back left 20 000g 

Back right 20 000g 

Front right 19 995g 

Maximum deviation max ecci I 5 g 

Errors of indication

General prerequisites: 

Requirements given in Chapter 5.2, weights distributed fairly 

evenly over the weighing range. 

Test loads each applied once; discontinuous loading only 

upwards, indication at no load reset to zero if necessary. 

Option 1: Air density unknown during adjustment / calibration (i.e. no buoyancy 

correction applied to the error of indication values). 

Requirements given in 

chapter 5.2, weights 

distributed fairly evenly. 

Test loads each applied 

once; discontinuous 

loading only upwards; 

indication at no load reset 

to zero where necessary 

Load mref

(mN) 

Indication I Error of 

indication E

0 g 0 g 0 g 

10 000 g 10 000 g 0 g 

20 000 g 20 000 g 0 g 

40 000 g 40 000 g 0 g 

60 000 g 60 000 g 0 g 
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Option 2: Air density ρas during adjustment and air density ρaCal during calibration are 

identical as an adjustment was carried out immediately before calibration 

The air density is calculated according to the simplified CIPM formula (A1.1-1): 

Measurement values used for calculation: 

Barometric pressure p: 990 hPa 

Relative humidity RH: 50 % 

Temperature t: 23 °C 

Air density ρaCal: 1,165 kg/m3 

Calculated buoyancy correction mB according to formula (4.2.4-4): 

Numerical value used for calculation 

Density of the reference mass ρCal: (7950 ± 70) kg/m3 

Buoyancy correction mB: 2,762 × 10-8 mN

The calculated correction mB of the loads mN following formula (4.2.4-4) is negligible as 

the relative resolution of the instrument is in the order of 10-4 and thus much larger than 

the buoyancy correction. The above table is effectual. 

H2.3/B Errors and related uncertainties (budget of related uncertainties) 

Conditions common to both options: 

- The uncertainty of the error at zero only comprises the uncertainty of the no-load 

indication (scale interval d0 = d1= 2 g) and the repeatability s. The uncertainty of the 

indication at load is not taken into consideration at zero.

- The eccentric loading is taken into account for the calibration according to (7.1.1-

10). 

- The error of indication is derived using the nominal weight value as reference value, 

therefore the maximum permissible errors of the test weights are taken into account 

for deriving the uncertainty contribution due to the reference mass: u(mc)is 

calculated as u(mc) = Tol/3 following formula (7.1.2-3). 

- The average drift of the weights monitored over 2 recalibrations in two-yearly 

intervals was Dmpe/2. Therefore the uncertainty contribution due to the drift of 

the weights was set to u(mD) = mpe/23. This corresponds to a kDfactor of 1.5 

(assuming the worst-case scenario of U = mpe / 3). 

- The weights are acclimatized with a residual temperature difference of 2 K to the 

ambient temperature. 

- The degrees of freedom for the calculation of the coverage factor k are derived 

following appendix B3 and table G.2 of [1]. In the case of the example, the influence 

of the uncertainty of the repeatability test with 5 measurements is significant. 

- The information about the relative uncertainty U(E)rel = U(E)/mref is not mandatory, 

but helps to demonstrate the characteristics of the uncertainties. 

Uncertainty budget for option 1 (no buoyancy correction applied to the error of indication 

values) 

Additional condition: 

The balance is adjusted immediately before calibration. The procedure according to 

option 1 is applied, with no information about air density. Therefore, formula (7.1.2-5c) is 

applied for the uncertainty due to air buoyancy. 

 
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Quantity or Influence Load, indication and error in g 

Uncertainties in g Formula

Load mref(mN) /g 0 10 000 20 000 40 000 60 000 

Indication I /g 0 10 000 20 000 40 000 60 000 

Error of Indication E /g 0 0 0 0 0 7.1-1 

Repeatability u(Irep) /g 0,894 2,236 7.1.1-5 

Resolution u(Idig0) /g 0,577 7.1.1-2a 

Resolution u(IdigL) /g 0,000 0,577 1,443 2,887 2,887 7.1.1-3a 

Eccentricity u(Iecc) /g 0,000 0,722 1,443 2,887 4,330 7.1.1-10 

Uncertainty of the 

indicationu(I) /g 1,065 1,410 3,082 4,690 5,694 7.1.1-12 

Test loads mN /g 

0 10 000 20 000 20 000

20 000

20 000 

20 000 

20 000 

Weights u(mc) /g 0,000 0,092 0,173 0,346 0,554 7.1.2-3 

Drift u(mD) /g 0,000 0,046 0,087 0,173 0,277 7.1.2-11 

Buoyancy u(mB) /g 0,000 0,023 0,043 0,087 0,139 7.1.2-5c 

Convection u(mconv) /g Not relevant in this case (only relevant for F1 and 

better). 7.1.2-13 

Uncertainty of the reference 

mass u(mref) /g 0,000 0,106 0,198 0,397 0,635 7.1.2-14 

Standard uncertainty of the 

error u(E) /g 1,065 1,414 3,089 4,707 5,739 7.1.3-1a 

eff (degrees of freedom) 8 25 14 78 172 B3-1 

k(95,45 %) 2,37 2,11 2,20 2,05 2,025 [1], 

U(E) = ku(E)/g 2,523 2,983 6,795 9,650 11,601 7.3-1 

Urel(E)/% ---- 0,0298 0,0340 0,0241 0,0193 

It would be acceptable to state in the certificate only the largest value of the expanded 

uncertainty for all the reported errors: U(E) = 11,601 g, based on k = 2,025 accompanied 

by the statement that the coverage probability is at least 95%. 

The certificate shall give the advice to the user that the expanded uncertainty stated in 

the certificate is only applicable, when the Error (E) is taken into account. 

Uncertainty budget for option 2 (buoyancy correction applied to the error of indication 

values) 

Additional condition: 

The balance is adjusted immediately before calibration. The procedure according to 

option 2 is applied, taking into account the determination of the air density and buoyancy 

correction. Therefore formula (7.1.2-5a) is applied for the uncertainty due to air 

buoyancy. 

As an adjustment has been carried out immediately before the calibration, the expected 

maximum values for pressure, temperature and humidity variations which can occur at 

the installation site of the instrument do not have to be taken into account – in contrast to 

the scenario where the adjustment has been performed independent of the calibration. 

The only contributing factor to the standard uncertainty of the air density originates from 

the uncertainty of the measurement of the environmental parameters. 

The following numeric values are taken to calculate the relative uncertainty of the 

buoyancy correction, using formula (7.1.2-5a): 

Air density ρaCal: 1,165 kg/m3 
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Density of the reference mass ρCal: (7950 ± 70) kg/m3

Furthermore, the following uncertainties for temperature, pressure and humidity 

measurement are taken for calculating the relative uncertainty of the air density 

according to (A3-1): 

u  T = 0,2 K 

u  p = 50 Pa 

u  RH = 1% 

This leads to 

a

a ( )



u  = 9,77 × 10-4, and ( )  a u = 0,00114 kg/m3. 

Formula (7.1.2-5a) leads to the relative uncertainty of the buoyancy correction of 

urel(mB) = 3,892 × 10 -8.

The relative uncertainty of the buoyancy correction is negligible as compared to the 

other contributions to the uncertainty of the reference mass. 

This example has shown that the calculated correction of the error mBand the 

calculated relative uncertainty of the buoyancy correction urel(mB) are both negligible. 

This leads to an updated measurement uncertainty budget: 

 
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Quantity or Influence Load, indication and error in g 

Uncertainties in g Formula

Load mref(mN)/g 0 10 000 20 000 40 000 60 000 

Correction mB/g 0 0 0 0 0 4.2.4-4 

Indication I /g 0 10 000 20 000 40 000 60 000 

Error of Indication E/g 0 0 0 0 0 7.1-1 

Repeatability u(Irep)/g 0,894 2,236 7.1.1-5 

Resolution u(Idig0) /g 0,577 7.1.1-2a 

Resolution u(IdigL) /g 0,000 0,577 1,443 2,887 2,887 7.1.1-3a 

Eccentricityu(Iecc) /g 0,000 0,722 1,443 2,887 4,330 7.1.1-10 

Uncertainty of the 

indicationu(I) /g 

1,065 1,410 3,082 4,690 5,694 7.1.1-12 

Test loadsmN /g 0 10 000 20 000 20 000

20 000

20 000 

20 000 

20 000 

Weightsu(mc) /g 0,000 0,092 0,173 0,346 0,554 7.1.2-3 

Driftu(mD)/g 0,000 0,046 0,087 0,173 0,277 7.1.2-11 

Buoyancyu(mB) /g 0,000 0,000 0,001 0,001 0,002 7.1.2-5c 

Convectionu(mconv) /g Not relevant in this case (only relevant for F1 and 

better). 

7.1.2-13 

Uncertainty of the reference 

mass u(mref)/g 

0,000 0,103 0,194 0,387 0,620 7.1.2-14 

Standard uncertainty of the 

erroru(E) /g 

1,065 1,414 3,089 4,706 5,727 7.1.3-1a 

eff (degrees of freedom) 8 25 14 78 172 B3-1 

k(95,45 %) 2,37 2,11 2,20 2,05 2,025 [1] 

U(E) = ku(E)/g 2,523 2,983 6,794 9,648 11,598 7.3-1 

Urel(E)/% ---- 0,0301 0,0340 0,0241 0,0193 

The expanded uncertainties of the error using the standard procedure and using the 

option are almost identical as the uncertainty of the reference mass u(mref) is very small 

as compared to the uncertainty of the indication u(I). In this example, the determination 

of pressure and humidity on site to determine the buoyancy correction and to minimize 

the uncertainty contribution due to buoyancy does not significantly improve the results of 

the calibration. 

H2.4/B Uncertainty of a weighing result (for option 1) 

As stated in 7.4, the following information may be developed by the calibration laboratory 

or by the user of the instrument. The results must not be presented as part of the 

calibration certificate except for the approximated error of indication and the uncertainty 

of the approximated error which can form part of the certificate. Usually the information 

on the uncertainty of a weighing result is presented as an appendix to the calibration 

certificate or is otherwise shown if its contents are clearly separated from the calibration 

results. 

Normal conditions of use of the instrument, as assumed, or as specified by the user may 

include: 

- Built-in adjustment device available and activated (T ≥ 3 K) 

- Variation of room temperature T = 10 K 

- Tare balancing function operated 

- Loads not always centred carefully 

The uncertainty of a weighing result is derived using a linear approximation of the error 
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of indication according to (C2.2-16). 

The uncertainty of a weighing result is presented for option 1 only (no buoyancy 

correction applied to the error of indication values). The approximated error of indication 

per (C2.2-16) and the uncertainty of the approximated error of indication per (C2.2-16d) 

differ insignificantly between both options as the underlying weighting factors 

  p j u E j

2  1 differ in the order of a few per mil, and the errors of indication are the same 

for both options (buoyancy correction smaller than the resolution of the instrument). 

The designations R and W are introduced to differentiate from the weighing instrument 

indication I during calibration. 

R: Reading when weighing a load on the calibrated instrument obtained after the 

calibration 

W: Weighing result 

Note that within the following table the reading R and all results are in g. 

 
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Quantity or Influence Reading, weighing result and error in kg 

Uncertainties in g or as relative value Formula 

Error of Indication 

Eappr(R) for gross or net 

readings: Approximation 

by a straight line through 

zero 

Eappr R  0 C2.2-16 

Uncertainty of the approximated error of indication

Standard uncertainty of 

the error u(Eappr)   2 9 2

appr

2 u E 0 u (R) 3,651 10 R      C2.2-16d 

Standard uncertainty of 

the error, neglecting the 

offset 

uE  R5

appr 6,043 10  

Uncertainties from environmental influences

Temperature drift of 

sensitivity   6

rel temp 1 732 10 u R  ,  7.4.3-1 

Buoyancy Not relevant in this case. 7.4.3-2 

Change in adjustment 

due to drift 

Not relevant in this case (built-in adjustment activated and 

drift between calibrations negligible) 7.4.3-5 

Uncertainties from the operation of the instrument

Tare balancing operation urelRTare   0 7.4.4-5 

Creep, hysteresis 

(loading time) Not relevant in this case (short loading time). 7.4.4-9a/b 

Eccentric loading   4

rel ecc 1 443 10 u R  ,  7.4.4-10 

Uncertainty of a weighing result, for partial weighing intervals (PWI)

Standard uncertainty, 

corrections to the 

readings Eappr to be 

applied 

PWI 1     2 8 2 u W 1,467g 2,449 10 R    

PWI 2     7.4.5-1b 2 8 2 u W 7,417g 2,449 10 R    

PWI 3     2 8 2 u W 13,667g 2,449 10 R    

Expanded uncertainty, 

corrections to the 

readings Eapp to be 

applied 

PWI 1     2 8 2 U W 2 1,467g 2,449 10 R    

PWI 2     7.5.1-2b 2 8 2 U W 2 7,417g 2,449 10 R    

PWI 3     2 8 2 U W 2 13,667g 2,449 10 R    

Simplified to first order 

PWI 1 UW  , , R4 2 422 g 1706 10   

PWI 2   6 616 g 2 355 10  12000 g 7.5.2-3f 4      U W , , R

PWI 3   11951g 2 744 10  30000 g 4      U W , , R

Global uncertainty of a weighing result without correction to the readings 

U   W U W E R gl   appr

PWI 1 U   , , R4

gl W 2 422 g 1706 10   

PWI 2   6 616 g 2 355 10  12000g 7.5.2-3a 4

gl      U W , , R

PWI 3   11951g 2 744 10  30000 g 4

gl      U W , , R

The condition regarding the observed chi-squared value following (C2.2-2a) was 

checked with positive result. The linear regression was performed taking into account 

the weighing factors p j

 of equation (C2.2-18b). 

Based on the global uncertainty, the minimum weight value for the instrument may be 

derived as per Appendix G. 
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Example: 

Weighing tolerance requirement: 1 % 

Safety factor: 2 

The minimum weight according to formula G-9, using the above equation for the global 

uncertainty in PWI 1 results in 502 g; i.e. the user needs to weigh a net quantity of 

material that exceeds 502 g in order to achieve a relative (global) measurement 

uncertainty for a relative weighing tolerance requirement of 1% and a safety factor of 2 

(equals a relative weighing tolerance of 0,5%). 

H3 Instrument of 30 000 kg capacity, scale interval 10 kg 

Preliminary note:

The calibration of a weighbridge for road vehicles is demonstrated. This example shows 

the complete standard procedure for the presentation of measurement results and the 

related uncertainties as executed by most laboratories. 

Test loads should preferably consist only of standard weights that are traceable to the SI 

unit of mass. 

This example shows the use of standard weights and substitution loads. The instrument 

under calibration is used as comparator to adjust the substitution load so that it brings 

about approximately the same indication as the corresponding load made up of standard 

weights. 
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First situation: adjustment of sensitivity carried out independent of calibration 

(Instrument status: as it was found) 

H3.1/A Conditions specific for the calibration 

Instrument: Electronic non-automatic weighing instrument, description 

and identification, with OIML R76 certificate of conformity or 

EN 45501 type approval but not verified 

Maximum weighing capacity Max

/ scale interval d

30 000 kg / 10 kg 

Load receptor 3 m wide, 10 m long, 4 points of support 

Installation Outside, in plain air, under shadow 

Temperature coefficient KT = 2 × 10-6/K (manufacturer’s manual); only necessary for 

calculation of uncertainty of a weighing result. 

Built-in adjustment device Not provided. 

Adjustment by calibrator Not adjusted immediately before calibration. 

Scale interval for testing Higher resolution (service mode), dT= 1 kg 

Duration of tests From 9h to 13h (this information could be useful in relation with 

possible effects of creep and hysteresis) 

Temperature during calibration 17°C at the beginning of the calibration 

20°C at the end of the calibration

Barometric pressure and 

environmental conditions during 

calibration (optional) 

1 010 hPa ± 5 hPa ; no rain, no wind 

Test loads Standard weights: 

 10 parallelepiped standard weights, cast iron, 1 000 kg 

each, certified to class M1 tolerance of mpe = 50 g 

(OIML R111 [4]) 

Substitution loads made up of steel or cast iron: 

 2 steel containers filled with loose steel or cast iron, 

each weighing ≈ 2 000 kg; 

 2 steel containers filled with loose steel or cast iron, 

each weighing ≈ 3 000 kg; 

 trailer to support the standard weights or the steel 

containers, weight adjusted to ≈ 10 000 kg; 

 small metallic pieces, used to adjust the substitution 

loads. 

Lifting and manoeuvring means for standard weights and 

substitution loads: 

 forklift, weight ≈ 4500 kg, capacity 6 000 kg to move 

standard weights and substitution loads; 

 vehicle with trailer and crane, lifting capacity 10 000 kg, 

to transport and to move standard weights and 

substitution loads. 
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H3.2/A Tests and results 

Repeatability

Requirements given in 

Chapter 5.1

Indication at no load reset 

to zero where necessary 

After unloading, no-load 

indications were between 0 

and 2 kg 

Test load ≈ 10 420 kg: 

Fork lift with 2 steel 

containers, moved on 

alternating from either long 

end of load receptor, load 

centred by eyesight 

Test load ≈ 24 160 kg: 

Loaded vehicle moved on 

alternating from either long end 

of load receptor, load centred by 

eyesight (alternatively or 

additionally performed) 

10 405 kg 24 145 kg 

10 414 kg 24 160 kg 

10 418 kg 24 172 kg 

10 412 kg 24 152 kg 

10 418 kg 24 156 kg 

10 425 kg 24 159 kg 

Standard deviation s = 6,74 kg s = 9,03 kg 

Eccentricity

Requirements given in Chapter 5.3 

Indication set to zero prior to test; 

load put in centre first then moved to 

the other positions 

Position of the load Test load ≈ 10 420 kg: 

Fork lift with 2 steel 

containers 

Centre 10 420 kg 

Front left 10 407 kg 

Back left 10 435 kg 

Back right 10 433 kg 

Front right 10 413 kg 

Maximum difference between 

center indication and the offcenter indications (in the four 

corners)

max ecci I 15 kg 

Eccentricity (alternatively or 

additionally performed with rolling 

loads) 

Requirements given in Chapter 5.3 

Indication set to zero prior to test 

and prior the change of direction; 

Position of the load Test load ≈ 24 160 kg: 

heaviest and most 

concentrated available 

vehicle 

Left 24 160 kg 

Centre 24 157 kg 

Right 24 181 kg 

(change direction) 

Right 24 177 kg 

Centre 24 157 kg 

Left 24 162 kg 

Maximum difference between 

center indication and the two 

off-center indications (along the 

longitudinal axis)

max ecci I 24 kg 
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Errors of indication

Standard Procedure: Requirements given in chapter 5.2, weights distributed fairly 

evenly. 

Test loads built up by substitution, with 10 000 kg standard weights (10 weights × 1 000 

kg) and 2 substitution loads Lsub1 and Lsub2 of approximately 10 000 kg each (the trailer 

and the sum of 4 containers). Test loads applied once; continuous loading only upwards. 

This may include creep and hysteresis effects in the results, but reduces the amount of 

loads to be moved on and off the load receptor. 

Indications after removal of standard weights recorded but no correction applied; all 

loads arranged reasonably around centre of load receptor. 

Indications recorded: 

Air density ρas during adjustment is unknown and air density ρaCal is unknown. 

No buoyancy correction is applied to the error of indication values. Using standard 

weights class M1 the relative uncertainty for buoyancy effect is calculated according to 

(7.1.2-5d) is 1,6×10-5 (since the instrument is not adjusted immediately before 

calibration). The uncertainty is small enough, so a more elaborate calculation of this 

uncertainty component based on actual data for air density is superfluous (the 

uncertainty of buoyancy is smaller than the scale interval of the high resolution mode dT

and is negligible). 

The limit of density for class M1 standard weights is established to be ρ ≥ 4 400 kg m-3

[4]. This limit may be considered also for the substitution loads. In this case, the relative 

uncertainty estimated for the buoyancy effect of the substitution loads is the same as 

above (for standard weights) and is small enough; a more elaborate calculation of this 

uncertainty component based on actual data is superfluous. 

Note: In the estimation of density for substitution loads, it is necessary to take 

into account any internal cavities, which are not open to the atmosphere (for 

LOAD 

Standard weights 

mN

Substitution 

loads Lsub

Total test load 

LT = mN+Lsub

Indication I Error of 

indication E

0 kg 0 kg 0 kg 0 kg 0 kg 

5 000 kg 

1⁄2 mref

0 kg 5 000 kg 5 002 kg 

I(1⁄2 mref) 2 kg 

10 000 kg 

mref 0 kg 10 000 kg 10 010 kg 

I(mref) 10 kg 

0 kg 10 000 kg 

Lsub1

10 000 kg 10 010 kg 

I(Lsub1) 10 kg 

5 000 kg 

1⁄2 mref

10 000 kg 

Lsub1

15 000 kg 15 015 kg 

I(1⁄2 mref +Lsub1) 15 kg 

10 000 kg 

mref

10 000 kg 

Lsub1

20 000 kg 20 018 kg 

I(mref +Lsub1) 18 kg 

0 kg 20 010 kg 

Lsub1+Lsub2

20 010 kg 20 028 kg 

I(Lsub1+Lsub2) 18 kg 

5 000 kg 

1⁄2 mref

20 010 kg 

Lsub1+Lsub2

25 010 kg 25 035 kg 

I(1⁄2 mref +Lsub1+Lsub2) 25 kg 

10 000 kg 

mref

20 010 kg 

Lsub1+Lsub2

30 010 kg 30 040 kg 

I(mref +Lsub1+Lsub2) 30 kg 

0 kg 0 kg 0 kg 4 kg 4 kg 

E0
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example at tanks, reservoirs). It is necessary to estimate the density of such a load 

as a whole, not to suppose that it has the same density as the material from which 

it is built. 

H3.3/A Errors and related uncertainties (budget of related uncertainties) 

Conditions: 

- The uncertainty of the error at zero only comprises the uncertainty of the no-load 

indication (scale interval d = 1 kg) and the repeatability s. The uncertainty of the 

indication at load is not taken into consideration at zero 

- The eccentric loading is taken into account for the calibration according to (7.1.1-10) 

because it cannot be excluded during the error of indication test. If both eccentricity 

tests were performed, then the result with the largest relative value should be used. 

- The error of indication is derived using the nominal weight value as reference value, 

therefore the maximum permissible errors of the test weights are taken into account 

for deriving the uncertainty contribution due to the reference mass: u(mc) is 

calculated as u(mc) = mpe/3 following formula (7.1.2-3). For each standard weight 

of 1000 kg u(mc) =50/3  29 g. 

- In the absence of information on drift, the value of D is chosen D = mpe. For each 

standard weight of 1000 kg mpe = ± 50 gand u(mc) =50/3  29 g, following 

formula (7.1.2-11). 

- The instrument is not adjusted immediately before calibration. The standard 

procedure is applied, with no information about air density. Therefore, formula 

(7.1.2-5d) is applied for the uncertainty due to air buoyancy. 

- The load remains on the load receptor for a significant period of time during the 

calibration. Based on chapter 7.1.1 that states that additional uncertainty 

contributions might have to be taken into account, the creep and hysteresis effects 

in the results are calculated following formula (7.4.4-7) and included in the 

uncertainty of the indication. 

- The weights are acclimatized with a residual temperature difference of 5 K to the 

ambient temperature. The effects of convection are not relevant (usually they are 

only relevant for weights of class F1 or better). 

- The degrees of freedom for the calculation of the coverage factor k are derived 

following appendix B3 and table G.2 of [1]. In the case of the example, the influence 

of the uncertainty of the repeatability test with 6 measurements is significant. 

- The information about the relative uncertainty U(E)rel = U(E)/mref is not mandatory, 

but helps to demonstrate the characteristics of the uncertainties. 

 
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Quantity or Influence Load, indication, error and uncertainties in kg Formula

Total test load 

LT = mN+Lsubj/kg 

0 5 000 10 000

mref

10 000*)

Lsub1

15 000 

Indication I /kg 0 5 002 10 010

I (mref) 

10 010 

I (Lsub1) 

15 015 

Error of Indication E /kg 0 2 10 10 

I1 = 0 

15 7.1-1 

Repeatability u(Irep) /kg 6,74 7.1.1-5 

Resolution u(Idig0) /kg 0,29 7.1.1-2a 

Resolution u(IdigL) /kg 0,00 0,29 7.1.1-3a 

Eccentricity u(Iecc) /kg 0,00 2,08 4,16 4,16 6,24 7.1.1-10 

Creep / hysteresis urel(Itime)

/kg 

0,00 0,38 0,77 0,77 1,16 7.4.4-7 

Uncertainty of the indication

u(I)/kg

6,75 7,08 7,97 7,97 9,27 7.1.1-12 

Standard weights mN/kg 0 5 000 10 000 0 5 000 

Uncertainty u(mc) /kg 0,00 0,14 0,29 0,00 0,14 7.1.2-3 

Drift u(mD) /kg 0,00 0,14 0,29 0,00 0,14 7.1.2-11 

Buoyancy u(mB) /kg 0,00 0,08 0,16 0,00 0,08 7.1.2-5d 

Convection u(mconv) /kg Not relevant in this case 7.1.2-13 

Uncertainty of the reference 

mass u(mref) /kg 

0,00 0,22 0,44 0,00 0,22 7.1.2-14 

Substitution loads Lsubj /kg

000 10 000 

Lsub1=

mref+I1

10 000 

Lsub1

Uncertainty u(Lsubj)/kg 0,00 0,00 0,00 11,28 11,28 7.1.2-15b

Buoyancy u(mB) /kg 0,00 0,00 0,00 0,16 0,16 7.1.2-5d 

Convection u(mconv) /kg Not relevant in this case 

Uncertainty of substitution 

loads u(Lsubj) /kg 

0,00 0,00 0,00 11,28 11,28 7.1.2-15b

7.1.2-14 

Standard uncertainty of the 

error u(E) /kg 

6,75 7,08 7,98 -------- 14,60 7.1.3-1c 

eff (degrees of freedom) 5 6 9 -------- 109 B3-1 

k(95,45 %) 2,65 2,52 2,32 -------- 2,02 [1] 

U(E) = ku(E)/kg 18 18 19 -------- 29 7.3-1 

Urel(E) /% ---- 0,36 0,19 -------- 0,20 
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(continue) 

Quantity or Influence Load, indication, error and uncertainties in kg Formula 

Total test load 

LT = mN+Lsubj/kg 

20 000

mref2+Lsub2

20 010*) 25 010 30 010 

Indication I /kg 20 018

I(mref2+Lsub2)

20 028

I(Lsub1+Lsub2

) 

25 035 30 040 

Error of Indication E /kg 18 18

I2=10 

25 30 7.1-1 

Repeatability u(Irep) /kg 6,74 7.1.1-5 

Resolution u(Idig0) /kg 0,29 7.1.1-2a 

Resolution u(IdigL) /kg 0,29 7.1.1-3a 

Eccentricity u(Iecc) /kg 8,32 8,32 10,40 12,48 7.1.1-10 

Creep / hysteresis urel(Itime)

/kg 

1,54 1,54 1,93 2,31 7.4.4-7 

Uncertainty of the indication

u(I)/kg

10,82 10,82 12,54 14,38 7.1.1-12 

Standard weights mN/kg 10 000 0 5 000 10 000 

Uncertainty u(mc) /kg 0,29 0,00 0,14 0,29 7.1.2-3 

Drift u(mD) /kg 0,29 0,00 0,14 0,29 7.1.2-11 

Buoyancy u(mB) /kg 0,16 0,00 0,08 0,16 7.1.2-5d 

Convection u(mconv) /kg Not relevant in this case 7.1.2-13 

Uncertainty of the reference 

mass u(mref) /kg 

0,44 0,00 0,22 0,44 7.1.2-14 

Substitution loads Lsubj /kg 10 000

Lsub1

20 010

Lsub1+Lsub2=

2mref1+I2

20 010

Lsub1+Lsub2

20 010 

Lsub1+Lsub2

Uncertainty u(Lsubj)/kg 11,28 19,02 19,02 19,02 7.1.2-15b 

Buoyancy u(mB) /kg 0,16 0,32 0,32 0,32 7.1.2-5d 

Convection u(mconv) /kg Not relevant in this case 

Uncertainty of substitution 

loads u(Lsubj) /kg 

11,28 19,02 19,02 19,02 7.1.2-15b 

7.1.2-14 

Standard uncertainty of the 

error u(E) /kg 

15,64 -------- 22,79 23,85 7.1.3-1c 

eff (degrees of freedom) 144 -------- 653 783 B3-1 

k(95,45 %) 2,02 -------- 2,00 2,00 [1] 

U(E) = ku(E)/kg 32 -------- 46 48 7.3-1 

Urel(E) /% 0,16 -------- 0,18 0,16 

*) The values written in this column (for the same total load value as in previous column, after 

substitution of standard weights with substitution loads) are not reported in the calibration 

certificate, but are used in next columns. In order to remember this, the bold font is not used in 

this column and the final 5 cells are empty. 

It would be acceptable to state in the certificate only the largest value of the expanded 

uncertainty for all the reported errors: U(E) = 48 kg, based on k = 2 accompanied by the 

statement that the coverage probability is at least 95 %. 

The certificate shall give the advice to the user that the expanded uncertainty stated in 

the certificate is only applicable, when the Error (E) is taken into account. 
EURAMET Calibration Guide No. 18 

Version 4.0 (11/2015) - 98 - 

H3.4/A Uncertainty of a weighing result 

As stated in 7.4, the following information may be developed by the calibration laboratory 

or by the user of the instrument. The results must not be presented as part of the 

calibration certificate, except for the approximated error of indication and the uncertainty 

of the approximated error which can form part of the certificate. Usually the information 

on the uncertainty of a weighing result is presented as an appendix to the calibration 

certificate or is otherwise shown if its contents are clearly separated from the calibration 

results. 

The normal conditions of use of the instrument, as assumed, or as specified by the user 

may include: 

Variation of temperature T =40 K 

Loads not always centred carefully 

Tare balancing function operated 

Loading times: normal, that is shorter than at calibration 

Readings in normal resolution, d = 10 kg 

The error of indication at 30 000 kg is 30 kg, and this value is taken for the change in 

adjustment due to drift. 

The designations R and W are introduced to differentiate from the weighing instrument 

indication I during calibration. 

R: Reading when weighing a load on the calibrated instrument obtained after the 

calibration 

W: Weighing result 

Note that within the following table the reading R and all results are in kg. 

 
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Quantity or Influence Reading, weighing result and error in kg 

Uncertainties in g or as relative value Formula 

Error of Indication 

Eappr   R for gross or net 

readings: Approximation 

by a straight line through 

zero 

E   R R4

appr 9.379 10   

C2.2-16 

Uncertainty of the approximated error of indication 

Standard uncertainty of 

the error   u Eappr

  7 2 7 2 u Eappr 8,797 10 u (R) 1,316 10 R       C2.2-16d 

Standard uncertainty of 

the error, neglecting the 

offset 

uE  R

4

appr 3,627 10  

Uncertainties from environmental influences 

Temperature drift of 

sensitivity   5

6

rel temp 2,309 10

12

2 10 40 



    u R  7.4.3-1 

Buoyancy Not relevant in this case. 7.4.3-3 

Change in adjustment 

due to drift (change of 

E(Max) over 1 year = 

30 kg) 

    4

rel adj 30 30000 3 5,774 10 u R   

7.4.3-6 

Uncertainties from the operation of the instrument 

Tare balancing operation   4

rel Tare 3,457 10 u R   7.4.4-5 

Creep, hysteresis 

(loading time) Not relevant in this case (short loading time). 7.4.4-7 

Eccentric loading   4

rel ecc 8,311 10 u R   7.4.4-10 

Uncertainty of a weighing result

Standard uncertainty, 

corrections to the 

readings Eappr to be 

applied 

    2 6 2 u W 62,133 kg 1,276 10 R    

7.4.5-1a 

7.4.5-1b 

Expanded uncertainty, 

corrections to the 

readings Eapprto be 

applied 

    2 6 2 U W 2 62,133 kg 1,276 10 R     7.5.1-2b 

Simplified to first order UW  R3 16 kg 1,79 10    7.5.2-3d 

Global uncertainty of a weighing result without correction to the readings

Ugl     W  U W  Eappr R U W R3

gl 16 kg 2,73 10    7.5.2-3a 

The condition regarding the observed chi-squared value following (C2.2-2a) was 

checked with positive result. The first linear regression taking into account the weighing 

factors p j

 , equation (C2.2-18b). 

Based on the global uncertainty, the minimum weight value for the instrument may be 

derived as per Appendix G. 

Example: 

Weighing tolerance requirement: 1 % 

Safety factor: 1 
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Version 4.0 (11/2015) - 100 - 

The minimum weight according to formula (G-9), using the above equation for the global 

uncertainty results in 2 169 kg; i.e. the user needs to weigh a net quantity of material that 

exceeds 2 169 kg in order to achieve a relative (global) measurement uncertainty for a 

relative weighing tolerance requirement of 1 % and a safety factor of 1. 

If a safety factor is included, it might be chosen to be 2. Because of the large global 

uncertainty, a higher safety factor might not be able to be realised. 

The minimum weight according to formula (G-9), using the above equation for the global 

uncertainty results in 6 950 kg; i.e. the user needs to weigh a net quantity of material that 

exceeds 6 950 kg in order to achieve a relative (global) measurement uncertainty for a 

relative weighing tolerance requirement of 1 % and a safety factor of 2 (equals a relative 

weighing tolerance of 0,50 %). 
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Second situation:adjustment of sensitivity carried out immediately before calibration 

(Previously, repair and maintenance operations were performed on the instrument) 

H3.1/B Conditions specific for the calibration 

Instrument: Electronic non-automatic weighing instrument, description 

and identification, with OIML R76 certificate of conformity or 

EN 45501 type approval but not verified 

Maximum weighing capacity Max / 

scale interval d

30 000 kg / 10 kg 

Load receptor 3 m wide, 10 m long, 4 points of support 

Installation Outside, in plain air, under shadow 

Temperature coefficient 

KT = 2 × 10-6/K (manufacturer’s manual); only necessary for 

calculation of uncertainty of a weighing result. 

Built-in adjustment device Not provided. 

Adjustment by calibrator Adjusted immediately before calibration. 

Scale interval for testing Higher resolution (service mode), T d = 1 kg 

Duration of tests From 14h to 18h 

Temperature during calibration 

22°C at the beginning of the calibration 

18°C at the end of the calibration 

Barometric pressure during calibration 

1 010 hPa ± 5 hPa; no rain, no wind 

Test loads Standards weights: 

 10 parallelepiped standard weights, cast iron, 1 000 kg 

each, certified to class M1 tolerance of mpe = 50 g (OIML 

R111 [4]) 

Substitution loads made up of steel or cast iron: 

 2 steel containers filled with loose steel or cast iron, 

each weighing ≈ 2 000 kg; 

 2 steel containers filled with loose steel or cast iron, 

each weighing ≈ 3 000 kg; 

 trailer to support the standard weights or the steel 

containers, weight adjusted to ≈ 10 000 kg; 

 small metallic pieces, used to adjust the substitution 

loads. 

Lifting and manoeuvring means for standard weights and 

substitution loads: 

 forklift, weight ≈ 4500 kg, capacity 6 000 kg to move 

standard weights and substitution loads; 

 vehicle with trailer and crane, lifting capacity 10 000 kg, 

to transport and to move standard weights and 

substitution loads. 
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H3.2/B Tests and results 

Repeatability

Requirements given in 

Chapter 5.1

Indication at no load reset 

to zero where necessary 

After unloading, no-load 

indications were between 0 

and 2 kg 

Test load ≈ 10 420 kg: 

Fork lift with 2 steel 

containers (or the empty 

trailer), moved on alternating 

from either long end of load 

receptor, load centred by 

eyesight 

Test load ≈ 24 160 kg: 

Loaded vehicle (or loaded 

trailer), moved on alternating 

from either long end of load 

receptor, load centred by 

eyesight (alternatively or 

additionally performed) 

10 415 kg 24 155 kg 

10 418 kg 24 160 kg 

10 422 kg 24 162 kg 

10 416 kg 24 152 kg 

10 422 kg 24 156 kg 

10 419 kg 24 159 kg 

Standard deviation s = 2,94 kg s = 3,67 kg 

Eccentricity

Requirements given in 

Chapter 5.3 

Indication set to zero prior to test; 

load put in centre first then moved to 

the other positions 

Position of the load Test load ≈ 10 420 kg: 

Fork lift with 2 steel 

containers 

Centre 10 420 kg 

Front left 10 417 kg 

Back left 10 423 kg 

Back right 10 425 kg 

Front right 10 425 kg 

Maximum difference between 

center indication and the offcenter indications (in the four 

corners)

max ecci I 5 kg 

Eccentricity (alternatively or 

additionally performed with rolling 

loads) 

Requirements given in Chapter 5.3 

Indication set to zero prior to test 

and prior the change of direction; 

Position of the load Test load ≈ 24 160 kg: 

heaviest and most 

concentrated available 

vehicle 

Left 24 151 kg 

Centre 24 160 kg 

Right 24 169 kg 

(change direction) 

Right 24 167 kg 

Centre 24 160 kg 

Left 24 150 kg 

Maximum difference between 

center indication and the two 

off-center indications (along the 

longitudinal axis)

max ecci I 10 kg 
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Errors of indication

Standard Procedure: Requirements given in chapter 5.2, weights distributed fairly 

evenly. 

Test loads built up by substitution, with 10 000 kg standard weights (10 weights × 1 000 

kg) and 2 substitution loads Lsub1 and Lsub2 of approximately 10 000 kg each (the trailer 

and the sum of 4 containers). Test loads applied once; continuous loading only upwards. 

This may include creep and hysteresis effects in the results, but reduces the amount of 

loads to be moved on and off the load receptor. 

Indications after removal of standard weights recorded but no correction applied; all 

loads arranged reasonably around centre of load receptor. 

Indications recorded: 

Air density ρas during adjustment is unknown and air density ρaCal is unknown. 

No buoyancy correction is applied to the error of indication values. Using standard 

weights class M1, the relative uncertainty for buoyancy effect is calculated according to 

(7.1.2-5c) and it is 7,2×10-6 (since the instrument is adjusted immediately before 

calibration), The uncertainty is small enough; a more elaborate calculation of this 

uncertainty component based on actual data for air density is superfluous (the 

uncertainty of buoyancy is smaller than the scale interval of the high resolution mode dT

and is negligible). 

The limit of density for class M1 standard weights is established to be ρ ≥ 4 400 kg m-3

[4]. This limit may be considered also for the substitution loads. In this case, the relative 

uncertainty estimated for the buoyancy effect of the substitution loads is the same as 

above (for standard weights) and is small enough; a more elaborate calculation of this 

uncertainty component based on actual data is superfluous. 

Note: In the estimation of density for substitution loads, it is necessary to take 

into account any internal cavities, which are not open to the atmosphere (for 

LOAD 

Standard weights 

mN

Substitution 

loads Lsub

Total test load 

LT=mN+Lsub

Indication I Error of 

indication E

0 kg 0 kg 0 kg 0 kg 0 kg 

5 000 kg 

1⁄2 mref

0 kg 5 000 kg 5 002 kg 

I(1⁄2 mref) 2 kg 

10 000 kg 

mref

0 kg 10 000 kg 10 005 kg 

I (mref) 5 kg 

0 kg 10 000 kg 

Lsub1

10 000 kg 10 005 kg 

I(Lsub1) 5 kg 

5 000 kg 

1⁄2 mref

10 000 kg 

Lsub1

15 000 kg 15 007 kg 

I(1⁄2 mref+Lsub1) 7 kg 

10 000 kg 

mref

10 000 kg 

Lsub1

20 000 kg 20 008 kg 

I(mref+Lsub1) 8 kg 

0 kg 20 010 kg 

Lsub1+ Lsub2 20 010 kg 20 018 kg 

I(Lsub1+Lsub2) 8 kg 

5 000 kg 

1⁄2 mref

20 010 kg 

Lsub1+ Lsub2

25 010 kg 25 020 kg 

I(1⁄2 mref Lsub1+Lsub2) 10 kg 

10 000 kg 

mref

20 010 kg 

Lsub1+ Lsub2

30 010 kg 30 022 kg 

I(mref+Lsub1+Lsub2) 12 kg 

0 kg 0 kg 0 kg 4 kg 4 kg 

E0
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example at tanks, reservoirs). It is necessary to estimate the density of such a load 

as a whole, not to suppose that it has the same density as the material from which 

it is built. 

H3.3/B Errors and related uncertainties (budget of related uncertainties) 

Conditions: 

- The uncertainty of the error at zero only comprises the uncertainty of the no-load 

indication (scale interval d = 1 kg) and the repeatability s. The uncertainty of the 

indication at load is not taken into consideration at zero. 

- The eccentric loading is taken into account for the calibration according to (7.1.1-10) 

because it cannot be excluded during the error of indication test. If both eccentricity 

tests were performed, then the result with the largest relative value should be used. 

- The error of indication is derived using the nominal weight value as reference value, 

therefore the maximum permissible errors of the test weights are taken into account 

for deriving the uncertainty contribution due to the reference mass: u(mc) is 

calculated as u(mc) = mpe/3 following formula (7.1.2-3). For each standard weight 

of 1000 kg u(mc) =50/3  29 g. 

- In the absence of information on drift, the value of D is chosen D = mpe. For each 

standard weight of 1000 kg mpe = ± 50 gand u(mc) =50/3  29 g, following 

formula (7.1.2-11). 

- The instrument is adjusted immediately before calibration. The standard procedure 

is applied, with no information about air density. Therefore formula (7.1.2-5c) is 

applied for the uncertainty due to air buoyancy. 

- The load remains on the load receptor for a significant period of time during the 

calibration. Based on chapter 7.1.1 that states that additional uncertainty 

contributions might have to be taken into account, the creep and hysteresis effects 

in the results are calculated following formula (7.4.4-7) and included in the 

uncertainty of the indication. 

- The weights are acclimatized with a residual temperature difference of 5 K to the 

ambient temperature. The effects of convection are not relevant (usually they are 

only relevant for weights of class F1 or better). 

- The degrees of freedom for the calculation of the coverage factor k are derived 

following appendix B3 and table G.2 of [1]. In the case of the example, the influence 

of the uncertainty of the repeatability test with 6 measurements is significant. 

- The information about the relative uncertainty U(E)rel = U(E)/mref is not mandatory, 

but helps to demonstrate the characteristics of the uncertainties. 

 
EURAMET Calibration Guide No. 18 

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Quantity or Influence Load, indication, error and uncertainties in kg Formula 

Total test load 

LT = mN+Lsubj/kg 

0 5 000 10 000

mref

10 000*)

Lsub1

15 000 

Indication I /kg 0 5 002 10 005

I (mref) 

10 005

I (Lsub1) 

15 007 

Error of Indication E /kg 025 5

I1=0

7 7.1-1 

Repeatability u(Irep) /kg 2,94 7.1.1-5 

Resolution u(Idig0) /kg 0,29 7.1.1-2a 

Resolution u(IdigL) /kg 0,00 0,29 7.1.1-3a 

Eccentricity u(Iecc) /kg 0,00 0,69 1,39 1,39 2,08 7.1.1-10 

Creep / hysteresis 

urel(Itime) /kg 

0,00 0,39 0,77 0,77 1,16 7.4.4-7 

Uncertainty of the indication

u(I)/kg

2,96 3,08 3,37 3,37 3,81 7.1.1-12 

Standard weights mN/kg 0 5 000 10 000 0 5 000 

Uncertainty u(mc) /kg 0,00 0,14 0,29 0,00 0,14 7.1.2-3 

Drift u(mD) /kg 0,00 0,14 0,29 0,00 0,14 7.1.2-11 

Buoyancy u(mB) /kg 0,00 0,04 0,07 0,00 0,04 7.1.2-5c 

Convection u(mconv) /kg Not relevant in this case 7.1.2-13 

Uncertainty of the reference 

mass u(mref) /kg 

0,00 0,21 0,42 0,00 0,21 7.1.2-14 

Substitution loads Lsubj 

/kg

000 10 000 

Lsub1=

mref+I1

10 000 

Lsub1

Uncertainty u(Lsubj)/kg 0,00 0,00 0,00 4,78 4,78 7.1.2-15b

Buoyancy u(mB) /kg 0,00 0,00 0,00 0,07 0,07 7.1.2-4 

Convection u(mconv) /kg Not relevant in this case 

Uncertainty of substitution 

loads u(Lsubj) /kg 

0,00 0,00 0,00 4,78 4,78 7.1.2-15b

7.1.2.-4 

Standard uncertainty of the 

error u(E) /kg 

2,96 3,08 3,39 -------- 6,12 7.1.3-1c 

eff (degrees of freedom) 5 6 8 -------- 93 B3-1 

k(95,45 %) 2,65 2,52 2,32 -------- 2,03 [1] 

U(E) = ku(E)/kg 888 -------- 12 7.3-1 

Urel(E) /% ---- 0,16 0,08 -------- 0,08 
EURAMET Calibration Guide No. 18 

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(continue) 

Quantity or Influence Load, indication, error and uncertainties in kg Formula 

Total test load 

LT = mN+Lsubj/kg 

20 000

mref2+Lsub2

20 010*) 25 010 30 010 

Indication I /kg 20 008

I(mref2+Lsub2)

20 018

I(Lsub1+Lsub2)

25 020 30 022 

Error of Indication E /kg 8 8

I2=10 

10 12 7.1-1 

Repeatability u(Irep) /kg 2,94 7.1.1-5 

Resolution u(Idig0) /kg 0,29 7.1.1-2a 

Resolution u(IdigL) /kg 0,29 7.1.1-3a 

Eccentricity u(Iecc) /kg 2,77 2,77 3,47 4,16 7.1.1-10 

Creep / hysteresis urel(Itime)

/kg 

1,54 1,54 1,93 2,31 7.4.4-7 

Uncertainty of the indication

u(I)/kg

4,34 4,34 4,95 5,61 7.1.1-12 

Standard weights mN/kg 10 000 0 5 000 10 000 

Uncertainty u(mc) /kg 0,29 0,00 0,14 0,29 7.1.2-3 

Drift u(mD) /kg 0,29 0,00 0,14 0,29 7.1.2-11 

Buoyancy u(mB) /kg 0,07 0,00 0,04 0,07 7.1.2-5c 

Convection u(mconv) /kg Not relevant in this case 7.1.2-13 

Uncertainty of the reference 

mass u(mref) /kg 

0,42 0,00 0,21 0,42 7.1.2-14 

Substitution loads Lsubj /kg 10 000

Lsub1

20 010

Lsub1+Lsub2=2

mref1+I2

20 010

Lsub1+Lsub2

20 010 

Lsub1+Lsub2

Uncertainty u(Lsubj)/kg 4,78 7,80 7,80 7,80 7.1.2-15a 

Buoyancy u(mB) /kg 0,07 0,14 0,14 0,14 7.1.2-5c 

Convection u(mconv) /kg Not relevant in this case 7.1.2-13 

Uncertainty of substitution 

loads u(Lsubj) /kg 

4,78 7,80 7,80 7,80 7.1.2-15a 

7.1.2-4 

Standard uncertainty of the 

error u(E) /kg 

6,47 -------- 9,24 9,62 7.1.3-1a 

eff (degrees of freedom) 117 -------- 486 569 B3-1 

k(95,45 %) 2,02 -------- 2,01 2,00 [1] 

U(E) = ku(E)/kg 13 -------- 19 19 7.3-1 

Urel(E) /% 0,06 -------- 0,07 0,06 

*) The values written in this column (for the same total load value as in previous column, after 

substitution of standard weights with substitution loads) are not reported in the calibration 

certificate, but are used in next columns. In order to remember this, the bold font is not used in 

this column and the final 5 cells are empty. 

It would be acceptable to state in the certificate only the largest value of the expanded 

uncertainty for all the reported errors: U(E) = 19 kg, based on k = 2 accompanied by the 

statement that the coverage probability is at least 95 %. 

The certificate shall give the advice to the user that the expanded uncertainty stated in 

the certificate is only applicable, when the Error (E) is taken into account. 
EURAMET Calibration Guide No. 18 

Version 4.0 (11/2015) - 107 - 

H3.4/B Uncertainty of a weighing result 

As stated in 7.4, the following information may be developed by the calibration laboratory 

or by the user of the instrument. The results must not be presented as part of the 

calibration certificate, except for the approximated error of indication and the uncertainty 

of the approximated error which can form part of the certificate. Usually the information 

on the uncertainty of a weighing result is presented as an appendix to the calibration 

certificate or is otherwise shown if its contents are clearly separated from the calibration 

results. 

The normal conditions of use of the instrument, as assumed, or as specified by the user 

may include: 

Variation of temperature T = 40 K 

Loads not always centred carefully 

Tare balancing function operated 

Loading times: normal, that is shorter than at calibration 

Readings in normal resolution, d = 10 kg 

For the change in adjustment due to drift, the error of indication at 30 000 kg is assumed 

to be 15 kg. This is the mpe at initial verification, and taken as the instrument is in good 

condition after maintenance and repair. 

The designations R and W are introduced to differentiate from the weighing instrument 

indication I during calibration. 

R: Reading when weighing a load on the calibrated instrument obtained after the 

calibration 

W: Weighing result 

Note that within the following table the reading R and all results are in kg. 

 
EURAMET Calibration Guide No. 18 

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Quantity or Influence Reading, weighing result and error in kg 

Uncertainties in g or as relative value Formula 

Error of Indication 

Eappr   R for gross or net 

readings: Approximation 

by a straight line through 

zero 

E R R4

appr 4,280 10  

C2.2-16 

Uncertainty of the approximated error of indication 

Standard uncertainty of 

the error   u Eappr

  7 2 8 2

appr

2 u E 1,832 10 u (R) 2,204 10 R      

C2.2-16d 

Standard uncertainty of 

the error, neglecting the 

offset 

uE  R4

appr 1,485 10  

Uncertainties from environmental influences 

Temperature drift of 

sensitivity   5

6

rel temp 2,309 10

12

2 10 40 



    u R  7.4.3-1 

Buoyancy Not relevant in this case. 7.4.3-2 

Change in adjustment 

due to drift (change of 

E(Max) over 1 year = 15 

kg) 

    4

rel adj 15 30000 3 2 887 10  u R   , 

7.4.3-6 

Uncertainties from the operation of the instrument 

Tare balancing operation   4

rel Tare δR 1,154 10 u   7.4.4-5 

Creep, hysteresis 

(loading time) 

Not relevant in this case (short loading time). 7.4.4-7 

Eccentric loading   4

rel ecc 2,770 10 u R   7.4.4-10 

Uncertainty of a weighing result

Standard uncertainty, 

corrections to the 

readings Eapprto be 

applied 

    2 7 2 u W 25,333 kg 1,960 10 R    

7.4.5-1a 

7.4.5-1b 

Expanded uncertainty, 

corrections to the 

readingsEapprto be 

applied 

    2 7 2 U W 2 25,333 kg 1,960 10 R     7.5.1-2b 

Simplified to first order U W  R4 10,067 kg 6,113 10     7.5.2-3d 

Global uncertainty of a weighing result without correction to the readings

U   W U W E R gl   appr U W R

3

gl 10 kg 1,04 10    7.5.2-3a 

The condition regarding the observed chi-squared value following (C2.2-2a) was 

checked with positive result. The first linear regression taking into account the weighing 

factors p j

 , equation (C2.2-18b). 

Based on the global uncertainty, the minimum weight value for the instrument may be 

derived as per Appendix G. 
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Version 4.0 (11/2015) - 109 - 

Example: 

Weighing tolerance requirement: 1 % 

Safety factor: 1 

The minimum weight according to formula (G-9), using the above equation for the global 

uncertainty results in 1 123 kg; i.e. the user needs to weigh a net quantity of material that 

exceeds 1 123 kg in order to achieve a relative (global) measurement uncertainty for a 

relative weighing tolerance requirement of 1 % and a safety factor of 1. 

If a safety factor is included, it might be chosen to be 2. Because of the large global 

uncertainty, a higher safety factor might not be able to be realised. 

The minimum weight according to formula (G-9), using the above equation for the global 

uncertainty results in 2 542 kg; i.e. the user needs to weigh a net quantity of material that 

exceeds 2 542 kg in order to achieve a relative (global) measurement uncertainty for a 

relative weighing tolerance requirement of 1 % and a safety factor of 2 (equals a relative 

weighing tolerance of 0,50 %). 

H3.5 Further information to the example: Details of the substitution procedure (4.3.3) 

It is highly recommended to let the substitution load indicate – as far as possible – 

the same value as the standard load (as demonstrated for the indication of 10 005 kg for 

the second situation). 

For this purpose, the substitution load can be adjusted by adding or removing small 

metallic parts until you get the same indication value (10 005 kg). The value of mass 

assigned to first substitution load is Lsub1 = mN = 10 000 kg. 

Note: Both mN and mref can be used (mref = mN). 

In the same table, the situation when it was not possible to adjust the substitution load to 

achieve the indication value 20 008 kg is presented. The value of mass assigned to 

second substitution load is Lsub2 = mN + I(Lsub2) − I(mN) = 10 000 kg + 20 018 kg −20 008 

kg = 10 010 kg, and the total substitution load Lsub is Lsub = Lsub1 + Lsub2 = 20 010 kg. 

H4 Determination of the error approximation function 

Preliminary note:

In this example the main procedure for the determination of the coefficients of the 

calibration function and the evaluation of the related uncertainties as described in 

Appendix C is shown. 

H4.1 Conditions specific for the calibration 

Instrument Electronic weighing instrument 

Maximum Capacity 

Max/ 

Scale interval d 

400 g / 0,000 1 g 

Adjustment by 

calibrator 

Adjusted immediately before calibration (built-in 

adjustment weights). 

Room conditions Temperature 23 °C 

Air density ρaCal=1,090 kg/m3, u(ρaCal)=0,004 kg/m3

Test loads / 

acclimatization 

Standard weights, Class E2, acclimatized to room 

temperature: mconv =0; u(mconv)=0. 
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H4.2 Tests and calibration results

Repeatability test 

performed at 200 g s(I) = 0,052 mg 7.1.1-5 

Eccentricity test 

performed at 200 g 

∆Ieccimax= 0,10 mg 

urel(Iecc)= 0,000 144 7.1.1-11 

Calibration method 

The test loads applied 

increasing by steps with 

unloading between the 

separate steps. 

Number of test points n = 9. 

Number of cycles N = 3. 

Uncertainty due to the 

repeatability uIrep  sI j N = 0,030 mg 7.1.1-6 

H4.3 Errors and related uncertainties (budget of related uncertainties) 

Conditions: 

- The uncertainty of the error at zero comprises the uncertainty of the no-load 

indication and the repeatability.

- The eccentric loading is taken into account for the calibration according to (7.1.1-10) 

- The error of indication is derived using the calibration value as reference value, the 

uncertainty contribution due to the reference mass is given by the Calibration 

Certificate u(mc) = U/2.

- In addition, also the air density at the time of calibration ρa1 is known.

- The drift of the weights is estimated by subsequent recalibrations.

The results are: 

mN mc/g U (mc) / 

mg 

u(mD) / mg 

50 g 50,000 006 0,030 0,005 

100 g 99,999 987 0,050 0,010 

200 g 200,000 013 0,090 0,015 

200 g* 199,999 997 0,090 0,015 

ρCal=8000 kg/m3, u(ρCal)=60 kg/m3

Calibration carried out at an air density ρa1=1,045 kg/m3. 

From equation (4.2.4-4) mB = 0, therefore mref= mc. 

- The weights are acclimatized to the ambient temperature, the temperature variation 

during the balance calibration is negligible. 

- The balance is adjusted immediately before calibration and air density at the 

calibration time is determined. 

- The air buoyancy uncertainty is determined by 

    B

2

rel

2

B N

2 u m  m u m

by (7.1.2-5b). 

Note that in this example this contribution is negative, for this reason, the variance 

contribution instead of the uncertainty is given. 
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Loads from 0 g to 200 g 

Quantity or Influence Load and indication in g 

Error and uncertainties in mg Formula 

Load mref/g 0 50,000 006 99,999 987 149,999 993 200,000 013

Indication I /g 

(mean value) 0,000 000 50,000 067 100,000 100 150,000 233 200,000 267 

Error of Indication E

/mg 0,000 0,061 0,113 0,240 0,254 7.1-1 

Repeatability s /mg 0,030 7.1.1-6 

Resolution u(Idig0)

/mg 0,029 7.1.1-2a 

Resolution u(IdigL)/mg 0,000 0,029 7.1.1-3a 

Eccentricity u(Iecc) 

/mg 

0,000 0,007 0,014 0,022 0,029 7.1.1-10 

Uncertainty of the 

indicationu(I) /mg 0,042 0,051 0,053 0,055 0,058 7.1.1-12 

Test loads mN/g 0 50 100 100 

50 

200 

Weights u(mc) /mg 0,000 0,015 0,025 0,040 0,045 7.1.2-3 

Drift u(mD) /mg 0,000 0,005 0,010 0,015 0,015 7.1.2-11 

Buoyancyu2(mB)/mg2 0,000 -4,8310-5 -1,9310-4 -4,3510-4 -7,7310-4 7.1.2-5b 

Convection 

u(mconv)/mg Not relevant in this case. 7.1.2-13 

Uncertainty of the 

reference mass 

u(mref)/ mg 

0,000 0,014 0,023 0,037 0,038 7.1.2-14 

Standard uncertainty of 

the error u(E) / mg 0,042 0,053 0,058 0,067 0,070 7.1.3-1a 

 
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Loads from 250 g to 400 g 

Quantity or Influence Load and indication in g 

Error and uncertainties in mg Formula

Load mref (mN) /g 250,000 019 300,000 000 350,000 006 400,000 010 

Indication I/g 250,000 100 300,000 200 350,000 267 400,000 400 

Error of Indication E

/mg 0,081 0,200 0,261 0,390 7.1-1 

Repeatability s /mg 0,030 7.1.1-6 

Resolution u(Idig0) /mg 0,029 7.1.1-2a 

Resolution u(IdigL)/mg 0,000 0,029 7.1.1-3a 

Eccentricity u(Iecc) /mg 0,036 0,043 0,051 0,058 7.1.1-10 

Uncertainty of the 

indicationu(I) /mg 0,062 0,067 0,072 0,077 7.1.1-12 

Test loadsmN/g 50

200

100

200

50 

100 

200 

200 

200 * 

Weights u(mc) /mg 0,060 0,070 0,085 0,090 7.1.2-3 

Drift u(mc)/mg 0,020 0,025 0,030 0,030 7.1.2-11 

Buoyancyu2(mB)/mg2 -1,2110-3 -1,7410-3 -2,3710-3 -3,0910-3 7.1.2-5b 

Convection u(mconv) /mg Not relevant in this case. 7.1.2-13 

Uncertainty of the 

reference massu(mref)

/mg 

0,053 0,062 0,076 0,077 7.1.2-14 

Standard uncertainty of 

the error u(E) /mg 0,082 0,091 0,104 0,109 7.1.3-1a 

From the calibration results the calibration function E  f I is determined. 

As an example the linear regression model E a I  1 is considered. 

The coefficientsa1 is determined by equation C2.2-6. 

Table H4.1 shows the matrix X and the vector e. The relevant covariance matrix U(e) is 

given in Table H4.4, which is determined by (C2.2-3a). 

Table H4.2 shows the covariance matrices U(mref), which is determined by (C2.2-3b), 

where the column vector mref s is given by the uncertainties of the reference mass u(mref) . 

Table H4.3 shows the covariance matrix U(Ical) which is a diagonal matrix having on the 

diagonal the square values of U(Ical). 

At the first step no contribution is considered for U(mod) (sm = 0). 

As the number of test points is n = 9 and the number of parameters is npar = 1, the 

degrees of freedom are  = n - npar = 8. 

 
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Table H4.1: Matrix X and vector e 

X/g e/mg

0 0,000 

50,000 067 0,061 

100,000 100 0,213 

150,000 233 0,274 

200,000 267 0,254 

250,000 100 0,181 

300,000 200 0,200 

350,000 267 0,261 

400,000 400 0,390 

Table H4.2: Covariance matrix U(mref) 

0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 

0,000 2,01710-4 3,27410-4 5,29410-4 5,45710-4 7,50310-4 8,73610-4 1,07710-3 1,09110-3

0,000 3,27410-4 5,31610-4 8,59610-4 8,86010-4 1,21810-3 1,41810-3 1,74910-3 1,77210-3

0,000 5,29410-4 8,59610-4 1,39010-3 1,43310-3 1,97010-3 2,29410-3 2,82910-3 2,86510-3

0,000 5,45710-4 8,86010-4 1,43310-3 1,47710-3 2,03010-3 2,36410-3 2,91510-3 2,95310-3

0,000 7,50310-4 1,21810-3 1,97010-3 2,03010-3 2,79210-3 3,25010-3 4,00910-3 4,06010-3

0,000 8,73610-4 1,41810-3 2,29410-3 2,36410-3 3,25010-3 3,78510-3 4,66810-3 4,72810-3

0,000 1,07710-4 1,74910-3 2,82910-3 2,91510-3 4,00910-3 4,66810-3 5,75610-3 5,83110-3

0,000 1,09110-4 1,77210-3 2,86510-3 2,95310-3 4,06010-3 4,72810-3 5,83110-3 5,90610-3

Table H4.3: Covariance matrix U(ICal) 

1,73510-3 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 

0,000 2,62010-3 0,000 0,000 0,000 0,000 0,000 0,000 0,000 

0,000 0,000 2,77610-3 0,000 0,000 0,000 0,000 0,000 0,000 

0,000 0,000 0,000 3,03710-3 0,000 0,000 0,000 0,000 0,000 

0,000 0,000 0,000 0,000 3,40110-3 0,000 0,000 0,000 0,000 

0,000 0,000 0,000 0,000 0,000 3,87010-3 0,000 0,000 0,000 

0,000 0,000 0,000 0,000 0,000 0,000 4,44310-3 0,000 0,000 

0,000 0,000 0,000 0,000 0,000 0,000 0,000 5,12010-3 0,000 

0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 5,90110-3
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Table H4.4: Covariance matrix U(e) with sm = 0 

1,73510-3 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 

0,000 2,82210-3 3,27410-4 5,29410-4 5,45710-4 7,50310-4 8,73610-4 1,07710-3 1,09110-3

0,000 3,27410-4 3,30810-3 8,59610-4 8,86010-4 1,21810-3 1,41810-3 1,74910-3 1,77210-3

0,000 5,29410-4 8,59610-4 4,42710-3 1,43310-3 1,97010-3 2,29410-3 2,82910-3 2,86510-3

0,000 5,45710-4 8,86010-4 1,43310-3 4,87810-3 2,03010-3 2,36410-3 2,91510-3 2,95310-3

0,000 7,50310-4 1,21810-3 1,97010-3 2,03010-3 6,66210-3 3,25010-3 4,00910-3 4,06010-3

0,000 8,73610-4 1,41810-3 2,29410-3 2,36410-3 3,25010-3 8,22810-3 4,66810-3 4,72810-3

0,000 1,07710-3 1,74910-3 2,82910-3 2,91510-3 4,00910-3 4,66810-3 1,08810-2 5,83110-3

0,000 1,09110-3 1,77210-3 2,86510-3 2,95310-3 4,06010-3 4,72810-3 5,83110-3 1,18110-2

H4.4 Results 

Applying (C2.2-6) and (C2.2-9), the results are 

 a1 = 0,00083 mg/g 

The covariance matrix U  aˆ is 

5,10910-8 (mg/g)2

from which 

 u(a1)= 0,00023 mg/g 

From (C2.2-8) 

2  obs =12,5 

As in this case the 2test (C2.2-2a) fails, an uncertainty contribution sm is added. 

Considering sm = 0,05 mg, the correspondent covariance matrix U(mod) is given by a 

diagonal matrix 9x9 having 2

ms =0,052 on the diagonal. Table H4.5 shows the 

correspondent covariance matrix U(e). 

Table H4.5: Covariance matrix U(e) evaluated with ms =0,05 mg 

4,23510-3 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 

0,000 5,32210-33 3,27410-4 5,29410-4 5,45710-4 7,50310-4 8,73610-4 1,07710-3 1,09110-3

0,000 3,27410-4 5,808E-03 8,59610-4 8,86010-4 1,21810-3 1,41810-3 1,74910-3 1,77210-3

0,000 5,29410-4 8,59610-4 6,92710-3 1,43310-3 1,97010-3 2,29410-3 2,82910-3 2,86510-3

0,000 5,45710-4 8,86010-4 1,43310-3 7,37810-3 2,03010-3 2,36410-3 2,91510-3 2,95310-3

0,000 7,50310-4 1,21810-3 1,97010-3 2,03010-3 9,16210-3 3,25010-3 4,00910-3 4,06010-3

0,000 8,73610-4 1,41810-3 2,29410-3 2,36410-3 3,25010-3 1,07310-2 4,66810-3 4,72810-3

0,000 1,07710-3 1,74910-3 2,82910-3 2,91510-3 4,00910-3 4,66810-3 1,33810-2 5,83110-3

0,000 1,09110-3 1,77210-3 2,86510-3 2,95310-3 4,06010-3 4,72810-3 5,83110-3 1,43110-2
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Version 4.0 (11/2015) - 115 - 

The new results are 

 a1= 0,00084 mg/g 

The covariance matrix is 

5,63710-8 (mg/g)2

from which 

 u(a1)= 0,00024 mg/g 

and 

2  obs =7,3 

In this case, the 2 test (C2.2-2a) passes. The plot of the result is shown in Figure H4-1. 

Figure H4-1: Measured errors of indication E and the linear fitting function with the 

associated uncertainty bands 

The residuals and the uncertainties associated with the calibration points are calculated 

by (C2.2-7) and (C2.2-11) respectively, and are shown in Table H4.6. 
EURAMET Calibration Guide No. 18 

Version 4.0 (11/2015) - 116 - 

Table H4.6: Calculated error, residuals and uncertainties associated to the calibration 

points 

I /g E /mg Eappr /mg Residual 

 /mg 

u(Eappr) 

/mg 

U(Eappr) 

/mg 

Residual Test 

(C2.2-2b) 

0 0,000 0,000 0,000 0,000 0,000 YES 

50,000 067 0,061 0,042 -0,019 0,012 0,024 YES 

100,000 200 0,113 0,084 -0,029 0,024 0,047 YES 

150,000 267 0,240 0,126 -0,114 0,036 0,071 NO 

200,000 267 0,254 0,168 -0,086 0,047 0,095 YES 

250,000 200 0,081 0,210 0,129 0,059 0,119 NO 

300,000 200 0,200 0,252 0,052 0,071 0,142 YES 

350,000 267 0,261 0,293 0,032 0,083 0,166 YES 

400,000 400 0,390 0,335 -0,055 0,095 0,190 YES 

If the alternative method given with (C2.2-2b) is followed, which is much more restrictive, 

the residual test fails in two points according to Table H4.6. 

In order to obtain the goodness of the fit according to the condition (C2.2-2b), it is 

necessary to consider a contribution sm = 0,25 mg and therefore a new matrix U(e) is 

calculated, which is given in Table H4.7. 

Table H4.7: Covariance matrix U(e) evaluated with sm =0,25 mg 

6,42310-2 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,000 

0,000 6,53210-2 3,27410-4 5,29410-4 5,45710-4 7,50310-4 8,73610-4 1,07710-3 1,09110-3

0,000 3,27410-4 6,58110-2 8,59610-4 8,86010-4 1,21810-3 1,41810-3 1,74910-3 1,77210-3

0,000 5,29410-4 8,59610-4 6,69310-2 1,43310-3 1,97010-3 2,29410-3 2,82910-3 2,86510-3

0,000 5,45710-4 8,86010-4 1,43310-3 6,73810-2 2,03010-3 2,36410-3 2,91510-3 2,95310-3

0,000 7,50310-4 1,21810-3 1,97010-3 2,03010-3 6,91610-2 3,25010-3 4,00910-3 4,06010-3

0,000 8,73610-4 1,41810-3 2,29410-3 2,36410-3 3,25010-3 7,07310-2 4,66810-3 4,72810-3

0,000 1,07710-3 1,74910-3 2,82910-3 2,91510-3 4,00910-3 4,66810-3 7,33810-2 5,83110-3

0,000 1,09110-3 1,77210-3 2,86510-3 2,95310-3 4,06010-3 4,72810-3 5,83110-3 7,43110-2

With this approach the result is 

 a1= 0,00084 mg/g 

The covariance matrixis 

1,745  10-7 (mg/g)2

Therefore 

 u(a1)= 0,00042 mg/g 

The plot of the results is shown in Figure H4-2. The calculated residuals and the 

uncertainties associated to the calibration points are shown in Table H4.8. 
EURAMET Calibration Guide No. 18 

Version 4.0 (11/2015) - 117 - 

Figure H4-2: Measured errors of indication E and the linear fitting function with the 

associated uncertainty bands 

Table H4.8: Calculated error, residuals and uncertainties associated to the 

calibration points 

I 

/g 

E 

/mg 

Eappr

/mg 

Residual 

 /mg 

u(Eappr) 

/mg 

U(Eappr) 

/mg 

Residual 

Test 

0 0,000 0,000 0,000 0,000 0,000 YES 

50,000 067 0,061 0,042 -0,019 0,021 0,042 YES 

100,000 200 0,213 0,084 -0,029 0,042 0,084 YES 

150,000 267 0,274 0,126 -0,114 0,063 0,125 YES 

200,000 267 0,254 0,168 -0,086 0,084 0,167 YES 

250,000 200 0,181 0,210 0,129 0,104 0,209 YES 

300,000 200 0,200 0,252 0,052 0,125 0,251 YES 

350,000 267 0,261 0,294 0,033 0,146 0,292 YES 

400,000 400 0,390 0,336 -0,054 0,167 0,334 YES 
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