uncertainty & specifications · pdf fileestimating weighing uncertainty from balance data...

Estimating Weighing Uncertainty FromBalance Data Sheet Specifications

Sources Of Measurement Deviations And UncertaintiesDetermination Of The Combined Measurement BiasEstimation Of The Combined Measurement Uncertainty

© METTLER TOLEDO, Arthur Reichmuth March 2000

Uncertainty & Specs 1.1

AbstractTo a lesser or greater extent, the performance of any scale orbalance is limited. These limitations are given in the specifica-tion sheet.It is common practice to accompany the measurement resultwith its uncertainty. This paper shows how the uncertainty ofa weighing or the minimum allowable weight can beestimated from the specifications of a balance given in thedata sheet. The model and assumptions used for thisdeduction, together with its limitations and neglections, arediscussed.

© Mettler Toledo, A. Reichmuth Uncertainty & Specs 1.1 Prtd.: October 27, 2000

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Sources Of Measurement Deviations And UncertaintiesSources of measurement deviations and uncertainties withlaboratory balances are (including, but not limited to):

Readability• rounding of the measurement value to the last digit inher-

ently introduces quantization noiseRepeatability

• noise of the electronic circuits (especially by the A/Dconverter’s reference, predominantly 1/f and burst noise)

• wind draft at the site of the balance (especially with resolu-tions of 1mg and below)

• vibrations• pressure fluctuations

Non-Linearity• kinematic non-linearities of the weighing cell, especially of

the parallel guiding mechanism and the lever (where pre-sent)

• load dependent deformations of the weighing cell• the electrodynamic transducer’s inherent non-linearity

between current and force• non-linear A/D conversion

Sensitivity Accuracy And Sensitivity TemperatureCoefficient• Adjustment tolerance, or determination accuracy, of the

calibration weightWithout re-calibration and adjustment:• deviations induced by both temperature and spontaneous

drift of the lever's mechanical advantage, theelectrodynamic transducer’s magnetic flux, and the A/Dconverter's reference


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Determination Of The Combined Measurement BiasSystematic deviations (bias) of the balance's transfercharacteristic from weighing load to reading—provided, theyare of systematic origin and invariable—are eliminated eitherthrough adjustment after assembly, or measured and storedin the balance, such that these deviations can be com-pensated on-line by means of signal processing algorithms.These include:• non-linearity correction• correction of temperature influence (with on-line measure-

ment of the temperature)• correction of the calibration weight’s adjustment deviation• on-site adjustments for sensitivity and sometimes non-line-

arity with many balancesThe remaining deviations after adjustment or compensation…• are—provided, they are of systematic origin—too small by

definition to be compensated (had they been largeenough, they would have been compensated);

• are time dependent in an unknown manner (unknownsystematic deviations);

• are caused by unknown ambient conditions (such as tem-perature or humidity);

• are of entirely unknown origin—neither their source oramount, nor their course over time are known—andtherefore are by definition not identifiable as systematicdeviations.

Hence, these influences must be regarded as randomcontributions, and are treated here as such.

Estimation Of The Combined Measurement UncertaintyThe basis for the following strategy is derived from probabili-ty theory. From error analysis, it is known that…• the variances of multiple random influences on a mea-

surement—provided they are mutually independent, or atleast uncorrelated—may be added;

• this resulting sum of variances may be used as variance ofall influences combined.


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We now apply this method to a balance, while consideringthe following influences:• readability• repeatability• non-linearity• sensitivity• temperature coefficient (of sensitivity)However, we do not consider here influences caused by• eccentric loading

ReadabilityThe internal measurement value is generally rounded halfwaybetween the readability steps d of the balance (4-5rounding). The variance introduced by this process can becalculated as follows

sRD2 = 1

12 d 2 . unit: [g2]Generally, with laboratory balances the display step issmaller compared to repeatability. In this case, not only theuncertainty introduced by rounding may be neglected infavor of repeatability, but there is also no bias introduced.What is more, for practical reasons repeatability can not bedetermined isolated from the contribution of readability, sinceboth their contributions will be measured at the same time 1).Without further notice, it is understood here that theuncertainty contribution from readability is included in themeasurement or specification of the repeatability.Therefore, the readability’s contribution as such need not beconsidered any further and therefore it will be dismissedhere. 2)

RepeatabilityRepeatability of the balance is specified by the standard de-viation. It is valid for one weighing and can usually be foundin the data sheet. For some balance models there may be

1) Unless a smaller readability (smaller step size) is available whendetermining repeatability. In this case, the influence of readabilitycan be eliminated.

2) Particularly the ”±1 count” specification, often seen in this context,is inappropriate.


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multiple repeatability specifications given, ranked accordingto weighing load. This would reflect the fact that repeatabilitydepends on total load (sum of tare and weighing sample),usually increasing with weighing load.The repeatability’s variance that accompanies the weighingresult equals the square of the repeatability specification

sRP2 = SPC RP

2 3). unit: [g2]As a rule, the repeatability specification describes the corre-sponding property of the balance, not the one of the weigh-ing object. To determine the repeatability specification,uncritical test loads are used, usually weight standards orother compact metal weights.If the weighing object possesses a large surface, or otherproperties detrimental to the weighing process, it may de-grade the repeatability of the weighing process. In suchcases, or when no figure for repeatability is available, it maybe sensible to determine the repeatability on-site, preferablywith the weighing object in question. With laboratorybalances, a measurement series of ten weighings is usuallycarried out, which is evaluated as follows:

sRP

2 = 1n–1 xi– x 2Σ

i = 1

n

., where unit: [g2]

xi is a single measurement value (a weighing), obtainedas the difference of the reading when the tare alone isplaced on the weighing pan, and the reading when thetare and sample weight together are placed on theweighing pan (pair of readings, making up the weighingof an object 4);

3) ”SPC” stands here, and in all following instances, for the value of aproperty's specification as given in the data sheet.

4) i) If there is no tare weight (such as a beaker, boat, or othercontainer), the reading is taken with empty pan instead.ii) If the balance is re-zeroed at any load, then the first reading iszero, by definition. Consequently, the reading with the sample isthen equal to the second reading (of the tare and sample weight),and the difference need not be calculated by the operator, as thebalance ”took” the first reading and has already subtracted it fromthe second reading.


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x= 1

n xiΣi = 1

n is the mean of this measurement series (or

weighings, i.e., differences of pairs of readings); n is the number of measurements (or weighings, i.e.,

differences of pairs of readings) 5).

Non-LinearityThe non-linearity of a balance can be read from the datasheet, too. This specification describes the largest deviationbetween the actual and the ideal, i.e., linear characteristiccurve 6)

SPC NL max yNL .As the characteristic curve of an individual balance, althougha systematic deviation, is generally unknown to the user, wehave to treat the actual deviation for any given load as a ran-dom contribution.Because the non-linearity specification only gives the limitswithin which the linearity deviation lies, we have to make anassumption about its random distribution to determine itsvariance. For lack of further knowledge, we assume here auniform distribution of the non-linearity within the specifiedlimits:

pNL x = 12SPC NL

within –SPC NL x SPC NL

With this assumption we are able to evaluate an equivalentvariance of the non-linearity:

s NL

2 = x2 pNL x dx–SPC NL

SPC NL

= x2 12SPC NL

dx–SPC NL

SPC NL

=

5) Be aware, that even with as many as 10 weighings, the standarddeviation derived from such a series may vary considerably, as itsoutcome is itself a random process, hence subject to stray. Thisapplies even more with fewer weighings.

6) The characteristic curve of a balance is the relationship betweendisplayed value and load. To get hold of a balance’s characteristiccurve, one has to load the balance from zero load to its full capaci-ty, in small (enough) load steps, and record all correspondingreadings.


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= 12SPC NL

13 x3

–SPC NL

SPC NL = 16 SPC NL

x3–SPC NL

SPC NL =

= 16SPC NL

SPC NL3– –SPC NL

3 = 13 SPC NL

2 [g2]

Sensitivity (Deviation)With the assumption that the balance’s sensitivity was ad-justed with an internal or external calibration weight, we firsthave to deal with the weight’s tolerance. Usually its deviationis given in the data sheet as a tolerance band

SPCCAL max mCAL ,and we find ourselves in the same situation as we were whenderiving a variance for the non-linearity. With the same rea-soning we can write for the variance of the calibration weightdeviation

sCAL2 = 1

3 SPCCAL2 . unit: [g2]

Furthermore, as calibration can only take place through aweighing of the calibration weight, strictly speaking, wewould have to consider the repeatability of this calibrationweighing. However, calibration is a special case insofar as aspecial signal processing is applied to it, i.e., usually astronger filtering and a longer measurement interval, therebyimproving the repeatability of the calibration weighing. Forpractical reasons, this contribution may therefore be ne-glected, which we will do here.A further complication occurs if the calibration weight doesnot amount to the balance’s full weighing capacity. When thecalibration weight is smaller than the weighing capacity, theobtained calibration measurement is extrapolated to thecorresponding calibration value at full capacity.Unfortunately, this calculation increases a potential linearitydeviation occurring at the load of the calibration weight bythe same factor. Particularly with precision balances,possessing weighing capacities of several kilograms, it is forpractical reasons not always possible to build in a calibrationweight equal to its full capacity. With analytical balances, onthe other hand, the built-in calibration weight usuallyembraces the full weighing capacity of the balance.


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We do not further pursue here the consequences of this con-tribution.Remark:As we will show later, it is sometimes more convenient to usethe relative calibration deviation, instead of the absolute one.The relative deviation is the absolute deviation normalized tothe mass of the calibration weight. The corresponding ex-pressions are

SPCCAL,rel =

SPCCAL

mCALmax

mCAL

mCALunit: [1]

sCAL ,rel

2 =sCAL

2

mCAL2 = 1

3SPC CAL

2

mCAL2 unit: [1]

Temperature CoefficientThe temperature coefficient of sensitivity may also be takenfrom the data sheet (if this item is specified). This specificationdescribes the largest static sensitivity deviation caused by achange in ambient temperature.

SPCTCS max TCS .Again, we use the same procedure to obtain the variancefrom the band limits (see derivation under the linearity devia-tion)

sTCS2 = 1

3SPCTCS2 . unit: [1/K2]

About the properties of the temperature excursion we canonly speculate here. Unless there is additional knowledgeavailable about the course of ambient temperature, thefollowing assumptions seem reasonable:• If the temperature at the location of the weighing is con-

stant, we can drop the influence of the temperature (coeffi-cient) altogether.

• The temperature excursion at the location of the weighingstays within a band of ±d t degrees.

• If an automatically induced calibration (for example”FACT”) is active, then it is realistic to assume that amaximum temperature change of ±2°C may occur, beforethe balance gets adjusted.


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In the latter two cases, the assumption d t max t , with d t 2°C

is justifiable, if we assume a temperature band of ±2°C (or±2K).For the variance of the ambient temperature we get—usingthe well known assumption of uniform distribution—

st2 = 1

3d t2 . unit: [K2]

We obtain the change of the balance’s sensitivity as productof temperature coefficient and temperature change

dTS = dTCSd t . unit: [1]As a last step, we need to determine the variance of this de-viation. Because we have a product of two individual contri-butions, its derivation is not trivial. It can be shown, however,that the product of the variances is a reasonable approxima-tion, which we will use here

sTS2 = sTCS

2 st2 .

Finally, we get sTS2 = 1

3SPC TCS21

3d t2 = 1

9 SPCTCSd t2 unit: [1]


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Combination Of The Variances 7)The combined variance of all deviations considered, can nowbe obtained by adding all single variances of the individualdeviations.

sTOT

2 = si2

i = 1

n .

A required condition to justify this operation is statistical in-dependence, or at least uncorrelatedness, between the singlecontributions. It can be shown that the individual causes forthe balance’s deviation from its ideal performance are inde-pendent from each other.All contributions to a single measurement—the difference of atare-weighing and a sample weighing (i.e., tare and weigh-ing object)—now produce the following result:

RepeatabilityRepeatability is an absolute deviation and by definition wasdetermined from the difference of pairs of readings.Therefore, its contribution to one weighing (i.e., difference oftwo readings) is the simple variance

sRP2 . unit: [g2]

Non-LinearityNon-linearity is an absolute deviation. It occurs when weigh-ing the tare, as well as the sample (tare and weighing ob-

7) We do not consider corner load deviations that may occur, if theweighing object is not placed in the center of the weighing pan. (Ifthe weighing object is placed in the center of the platform, thisdeviation vanishes.)Neither do we consider any other influences on the weighing pro-cess, besides those explicitly stated in the text.Particularly, we do not consider influences such as (including, butnot limited to): ambient climate (rapid temperature change, humidi-ty change), air draft, pressure fluctuation, heat radiation, mechani-cal influence (leveling, vibration), electromagnetic influence (elec-trostatic or magnetic), air buoyancy.In case of such influences, the effects have to be dealt with sepa-rately.


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ject) 8). Since the two loadings are statisticallyindependent 9), their contribution is twice the variance

2s NL2 . unit: [g2]

Tolerance Of Calibration WeightThe normalized calibration weight tolerance is a relative de-viation. The deviation of a weighing (i.e., difference of pairof readings) is proportional to the sample weight m. Itscontribution is the simple variance times the square of thesample weight

sCAL ,rel2m2 . unit: [g2]

Sensitivity DriftSensitivity temperature drift is a relative deviation. The devia-tion of a weighing (i.e., difference of pair of readings) isproportional to the sample weight m. Its contribution is thesimple variance times the square of the sample weight

sTS2 m2 . unit: [g2]

Total VarianceUnder these assumptions, we get for the combined varianceof a weighing (i.e., difference of pair of readings)

s2 = sRP2 +2s NL

2 +sCAL,rel2m2+sTS

2 m2 = = sRP

2 +2s NL2 +m2 sCAL ,rel

2+sTS2 .

We now substitute the variances with their previously deter-mined expressions and obtain

s2 = SPC RP2+2

3SPC NL2+1

3m2 SPCCAL,rel2+1

3 SPC TCSd t2

unit: [g2]Total Variance, Normalized To Sample Weight

Most often we are interested in the normalized variance, i.e.,the quotient of absolute variance and sample weight. We find

8) With the exception of weighings that include either zero load, fullcapacity, or weighings with zero sample weight.All three cases are but of academic interest, therefore they are notconsidered here.

9) There may be a dependence of the readings for small samples. Ifthere is knowledge about such a correlation, it can be used; here,we do not consider it for the sake of simplicity.


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this value by dividing the former expression by the square ofthe sample mass

srel

2 = s2

m2 =

= 1m2 SPC RP

2+23 SPC NL

2 +13 SPCCAL,rel

2+13 SPC TCSd t

2

unit: [1]From this variance, we can derive the normalized standarddeviation for one weighing (i.e., difference of pair ofreadings). This yields srel =

= 1

m2 SPC RP2+2

3 SPC NL2 +1

3 SPCCAL,rel2+1

3 SPC TCSd t2

10) unit: [1]

Measurement UncertaintyIt is reasonable to assume that a balance’s combined mea-surement deviation resembles a normal distribution. As oneof its justifications we mention the fact that some contributionsthemselves are normally distributed already (e.g., repeatabili-ty). A second reason is that there are multiple contributingsources of independent deviations which favors a normaldistribution of their combined deviation.From the combined standard deviation we can determine anuncertainty interval from the laws of normal distribution,provided a confidence level is given. We first derive from theconfidence level the expansion (or coverage) factor k, i.e.,the quotient relating uncertainty to the standard deviation

k P =

us

,

10) This combined standard deviation considers the influences of re-peatability, non-linearity, calibration weight adjustment and tem-perature coefficient, under the assumption of a temperature band.Not considered are, among others, eccentric load and deviationdue to calibration weights not comprising the weighing capacity.


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which is a function of the confidence level P 11). Multiplyingthe standard deviation with this factor yields the (single sided)uncertainty interval

u = k P ·s .Hence, with a probability of P , the true value can be as-sumed to lie between the limits

R–u m R+uwhere R is the weighing result (i.e., the difference of tworeadings), and m the true sample weight 12).With a probability of

Q = 1–Pthe true value will lie outside these limits.

Example: Determining The Weighing Uncertainty On AnAnalytical Balance

Balance Type: AT201: 200g/0.01mgUsing this balance, a sample of 1g shall be weighed in a190g container. What is the resulting uncertainty of thisweighing, conforming to a 95% confidence level?

11) Expansion Factor Confidence Level Expected Missing(Single Sided) (Expectation Prob.) Probability

k P Q=1–P——————— ——————— ———————

1 68.27% 31.73%1.645 90% 10%1.960 95% 5%2 95.45% 4.55%2.576 99% 1%3 99.73% 0.27%4 99.994% 0.006%5 99.99994% 0.00006%

12) To keep things simple, we have consequently refrained from deter-mining, or correcting for, the degree of freedom. Of course, no-thing stands against the notion of correcting for the degree of free-dom, if it is known of all individual contributions. An instruction forhow to determine the correction factor can be found in ”Guide ToThe Expression Of Uncertainty In Measurement”, first edition[1995], ISBN 92-67-10188-9.


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Specifications from data sheet Spec (SPC) SPC2

• Readability 0.01mg 1x10–10g2

• Repeatability up to 50g 0.015mg 2.3x10–10g2

50-200g 0.04mg 1.6x10–9g2

• Non-Linearity within 10g 0.03mg 9x10–10g2

within 200g 0.12mg 1.4x10–8g2

• Calibration Weight Tolerance 1.5ppm 2.3x10–12

• Temperature Coefficient 1.5ppm/K 2.3x10–12K–2

Environment (Assumption) Spec (SPC) SPC2

• Ambient Temp. Excursion 2 K 4 K2

The formula valid for the combined normalized standard de-viation for a single sample weighing is

srel = 1

m2SPC RP

2+23 SPC NL

2 +13 SPCCAL ,rel

2+13 SPCTCSd t

2

As the repeatability specification at 191g is unavailable, weuse the 200g specification instead. Thus, we obtain asstandard deviation for a 1g sample:

srel = 11g 2 1.6 10–9g 2+2

39 10–10g 2 +1

32.3 10–12+1

32.3 10–12K–24K 2

=

= 2.2 10–9+13

2.3 10–12+3.1 10–12 = 2.2 10–9+1.8 10–12 47 10–6

Conclusion:The mass of a 1g sample, weighed in a 200g container, canbe determined on this balance with a relative standard de-viation of approximately

srel < 50 10–6 .Based on a confidence level of 95%, the corresponding un-certainty amounts to twice the standard deviation, namely

urel = 2srel= 1 10–4 .

Remark:It can be seen from this example that with small sampleweights 13) the contribution to uncertainty originating fromthe balance’s sensitivity (calibration weight tolerance and un-

13) small compared to the balance’s weighing capacity


— 14 —


— 15 —

certainty due to drifting temperature) are minor, compared tothose stemming from repeatability and non-linearity. Thisapplies to most balance types. We will use this property laterwhen determining the minimum sample weight.The diagram on the opposite page shows the relative uncer-tainty versus sample weight and (total) load, respectively.

Minimum Sample WeightThe minimum sample weight to be weighed conformally on abalance (a.k.a. ”minimum (sample) weight”) can beestimated, provided the specifications of the balance, as wellas the uncertainty and confidence level to be met, are given.To this end, we need once more the expansion factor k, thistime as quotient of relative uncertainty and relative standarddeviation

k P =

urel

srel=

u /ms/m

=us

,

which is clearly the same function of the confidence level Pas introduced above. From the required properties of thesample weighing, namely the relative uncertainty and theconfidence level, from which the expansion factor was deter-mined, we derive the standard deviation

srel =

urel

k( P) .

The formula used in the previous chapter for the relativestandard deviation we now solve for the sample weight m .We obtain

m =SPC RP

2 + 23SPC NL

2

srel2 – 1

3 SPCCAL,rel2+1

3 SPCTCSd t2

12

mMIN .

Substituting the expression for the relative standard deviationyields for the minimum sample weight

mMIN

SPC RP2 + 2

3SPC NL2

urel

k( P)

2– 1

3 SPCCAL,rel2+1

3 SPC TCSd t2

12

.


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ApproximationsRevisiting the figures of the previous example, we recognizethat with a sample weight of 1/200 of the weighing capacity,the variance of the sensitivity deviation, calibration weighttolerance and temperature coefficient, equals 1.8 10–12, anamount negligible compared to the variance due torepeatability and non-linearity ( 2.2 10–9). It may be presumed,that this be the case for all small sample weights. We willshow that this is true.Taking the radicand of the relative uncertainty 1

m2 SPC RP2+2

3 SPC NL2 +1

3 SPC CAL,rel2+1

3 SPC TCSd t2

and expanding it by m2, we get SPC RP

2+23 SPC NL

2 +m2

3SPCCAL,rel

2+13 SPC TCSd t

2

According to the assumption, that m is small (minimumsample weight!), a fact which is even more true for its square( m2), we may drop the second term in favor of the first, andwe get as an approximation

srel

1m2 SPC RP

2+23 SPC NL

2 =

= 1m SPC RP

2+23 SPC NL

2

(valid for small samples).Respecting the previous requirements, we determine from thisformula the approximate sample weight to be

mMIN

1srel

SPC RP2+2

3 SPC NL2 =

= k

u relSPC RP

2+23 SPC NL

2

(valid for small samples).From this we conclude that the minimum sample weight is es-sentially determined by the two specifications of repeatabilityand non-linearity; calibration weight tolerance and tempera-ture coefficient do not occur in the formula. Traditionally,sensitivity adjustment is given too much attention whendealing with small sample weights: With the exception of


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weighing heavy samples 14), sensitivity plays but an inferiorrole.If, for any reason, additional information is available aboutthe linearity deviation of a balance which shows that non-linearity is inferior compared to repeatability, the contributionof non-linearity may be neglected, too. Such informationcould be gained by on-site measurements, or could stem fromother sources.If the non-linearity specification (SPCNL) is 1/2 of therepeatability spec (SPCRP), its contribution reduces to 14% ofthe combined uncertainty, if non-linearity amounts to 1/3 ofrepeatability, its contribution is 7%. Regarding these figures,one may decide to drop this term altogether. In this case theformula for minimum weight would reduce to

mMIN = k

urelSPC RP

2 = kurel

SPC RP

(valid for small non-linearity).

Example: Determining The Minimum Sample Weight On AnAnalytical Balance

Balance Type: AT201: 200g/0.01mgWhat is the minimum sample weight required using a 190gcontainer, observing a relative uncertainty of 0.1% at a con-fidence level of 95% (corresponding to k≈2)?

Specifications from data sheet Spec (SPC) SPC2

• Readability 0.01mg 1x10–10g2

• Repeatability up to 50g 0.015mg 2.3x10–10g2

50-200g 0.04mg 1.6x10–9g2

• Non-Linearity within 10g 0.03mg 9x10–10g2

within 200g 0.12mg 1.4x10–8g2

• Calibration Weight Tolerance 1.5ppm 2.3x10–12

• Temperature Coefficient 1.5ppm/K 2.3x10–12K–2

Environment (Assumption) Spec (SPC) SPC2

• Ambient Temp. Excursion 2 K 4 K2

14) usually larger than 1/10 to 1/4 of the balance’s weighing capacity,yet independent of total load


— 18 —

Since we are dealing here with a minimum sample weight,we may use the approximation formula

m MIN

kurel

SPC RP2+2

3 SPC NL2 .

As the repeatability specification at 191g is unavailable, weuse the 200g specification instead. Thus, we obtain as aminimum sample mass

m MIN2

10–3 1.6 10–9g2+239 10–10g2 = 2000 1.6+0.6 10–9g 2 =

= 2000 2.2 10–9g 2 = 2000 47 10–6g = 94 mg .

If the sample amounts to about 100mg or more, then we canbe assured that the given requirements, namely the mass de-termination with 0.1% uncertainty at 95% confidence, can beachieved on this balance.If we had additional information about this balance, such thatits linearity deviation is smaller than 0.02mg, this figurewould amount to 1/2 of repeatability (0.04mg). In this case,its contribution is small, as can be seen

m MIN

210–3 1.6 10–9g 2+2

3 20 10–6g 2 =

= 2000 1.6 10–9g 2+0.27 10–9g2 = 2000 1.9 10–9g2 = = 2000 43 10–6g = 86 mg ,

and we may decide to neglect it after all. We then have as aminimal weight estimation

m MIN2

10–3 0.04 mg = 2000 0.04 mg = 80 mg .


— 19 —

ConclusionIn many instances, the weighing result needs to be qualified.To this end, the measurement uncertainty accompanying theweighing process is required, but usually not readily avail-able, not least because it is dependent on the application athand. At other times, the operator needs to know the mini-mum amount of mass he/she is able to conformally weigh toa required relative uncertainty and confidence level(minimum weight).This paper explains how uncertainty and minimum weightcan be estimated from the data sheet specifications of abalance. The assumptions and restrictions, under which thisdeduction is valid, as well as when, and under whichconditions, neglections can be made, are discussed. Two ex-amples with actual data from analytical balances are givenas illustrations.The theory and examples provided enables the user toestimate the appropriate figures of uncertainty or minimumsample weight for his/her balance application.


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