1 Estimating Weighing Uncertainty From Balance Data Sheet Specifications Sources Of Measurement Deviations And Uncertainties Determination Of The Combined Measurement Bias Estimation Of The Combined Measurement Uncertainty METTLER TOLEDO, Arthur Reichmuth March 000 Uncertainty & Specs 1.1
2 Abstract To a lesser or greater extent, the performance of any scale or balance is limited. These limitations are given in the specification sheet. It is common practice to accompany the measurement result with its uncertainty. This paper shows how the uncertainty of a weighing or the minimum allowable weight can be estimated from the specifications of a balance given in the data sheet. The model and assumptions used for this deduction, together with its limitations and neglections, are discussed.
3 1 Sources Of Measurement Deviations And Uncertainties Sources of measurement deviations and uncertainties with laboratory balances are (including, but not limited to): Readability rounding of the measurement value to the last digit inherently introduces quantization noise Repeatability noise of the electronic circuits (especially by the A/D converter s reference, predominantly 1/f and burst noise) wind draft at the site of the balance (especially with resolutions 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 present) 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 Temperature Coefficient Adjustment tolerance, or determination accuracy, of the calibration weight Without re-calibration and adjustment: deviations induced by both temperature and spontaneous drift of the lever's mechanical advantage, the electrodynamic transducer s magnetic flux, and the A/D converter's reference
4 Determination Of The Combined Measurement Bias Systematic deviations (bias) of the balance's transfer characteristic from weighing load to reading provided, they are of systematic origin and invariable are eliminated either through adjustment after assembly, or measured and stored in the balance, such that these deviations can be compensated on-line by means of signal processing algorithms. These include: non-linearity correction correction of temperature influence (with on-line measurement of the temperature) correction of the calibration weight s adjustment deviation on-site adjustments for sensitivity and sometimes non-linearity with many balances The remaining deviations after adjustment or compensation are provided, they are of systematic origin too small by definition to be compensated (had they been large enough, they would have been compensated); are time dependent in an unknown manner (unknown systematic deviations); are caused by unknown ambient conditions (such as temperature or humidity); are of entirely unknown origin neither their source or amount, nor their course over time are known and therefore are by definition not identifiable as systematic deviations. Hence, these influences must be regarded as random contributions, and are treated here as such. Estimation Of The Combined Measurement Uncertainty The basis for the following strategy is derived from probability theory. From error analysis, it is known that the variances of multiple random influences on a measurement provided they are mutually independent, or at least uncorrelated may be added; this resulting sum of variances may be used as variance of all influences combined.
5 3 We now apply this method to a balance, while considering the following influences: readability repeatability non-linearity sensitivity temperature coefficient (of sensitivity) However, we do not consider here influences caused by eccentric loading Readability The internal measurement value is generally rounded halfway between the readability steps d of the balance (4-5 rounding). The variance introduced by this process can be calculated as follows s RD = 1 1 d. unit: [g ] Generally, with laboratory balances the display step is smaller compared to repeatability. In this case, not only the uncertainty introduced by rounding may be neglected in favor of repeatability, but there is also no bias introduced. What is more, for practical reasons repeatability can not be determined isolated from the contribution of readability, since both their contributions will be measured at the same time 1 ). Without further notice, it is understood here that the uncertainty contribution from readability is included in the measurement or specification of the repeatability. Therefore, the readability s contribution as such need not be considered any further and therefore it will be dismissed here. ) Repeatability Repeatability of the balance is specified by the standard deviation. It is valid for one weighing and can usually be found in the data sheet. For some balance models there may be 1) Unless a smaller readability (smaller step size) is available when determining repeatability. In this case, the influence of readability can be eliminated. ) Particularly the ±1 count specification, often seen in this context, is inappropriate.
6 4 multiple repeatability specifications given, ranked according to weighing load. This would reflect the fact that repeatability depends on total load (sum of tare and weighing sample), usually increasing with weighing load. The repeatability s variance that accompanies the weighing result equals the square of the repeatability specification s RP = SPC 3 RP ). unit: [g ] As a rule, the repeatability specification describes the corresponding property of the balance, not the one of the weighing object. To determine the repeatability specification, uncritical test loads are used, usually weight standards or other compact metal weights. If the weighing object possesses a large surface, or other properties detrimental to the weighing process, it may degrade the repeatability of the weighing process. In such cases, or when no figure for repeatability is available, it may be sensible to determine the repeatability on-site, preferably with the weighing object in question. With laboratory balances, a measurement series of ten weighings is usually carried out, which is evaluated as follows: s RP = n 1 1 n Σ i = 1 x i x., where unit: [g ] x i is a single measurement value (a weighing), obtained as the difference of the reading when the tare alone is placed on the weighing pan, and the reading when the tare and sample weight together are placed on the weighing pan (pair of readings, making up the weighing of an object 4 ); 3) SPC stands here, and in all following instances, for the value of a property's specification as given in the data sheet. 4) i) If there is no tare weight (such as a beaker, boat, or other container), the reading is taken with empty pan instead. ii) If the balance is re-zeroed at any load, then the first reading is zero, by definition. Consequently, the reading with the sample is then equal to the second reading (of the tare and sample weight), and the difference need not be calculated by the operator, as the balance took the first reading and has already subtracted it from the second reading.
7 n Σ 5 x= n 1 x i is the mean of this measurement series (or i = 1 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-Linearity The non-linearity of a balance can be read from the data sheet, too. This specification describes the largest deviation between the actual and the ideal, i.e., linear characteristic curve 6 ) SPC NL max y NL. As the characteristic curve of an individual balance, although a systematic deviation, is generally unknown to the user, we have to treat the actual deviation for any given load as a random contribution. Because the non-linearity specification only gives the limits within which the linearity deviation lies, we have to make an assumption about its random distribution to determine its variance. For lack of further knowledge, we assume here a uniform distribution of the non-linearity within the specified limits: p NL x = 1 SPC NL within SPC NL x SPC NL With this assumption we are able to evaluate an equivalent variance of the non-linearity: SPC NL = x p NL x dx = x 1 dx SPC NL s NL SPC NL SPC NL = SPC NL 5) Be aware, that even with as many as 10 weighings, the standard deviation derived from such a series may vary considerably, as its outcome is itself a random process, hence subject to stray. This applies even more with fewer weighings. 6) The characteristic curve of a balance is the relationship between displayed value and load. To get hold of a balance s characteristic curve, one has to load the balance from zero load to its full capacity, in small (enough) load steps, and record all corresponding readings.
8 6 = 1 SPC NL 1 3 x 3 SPC NL = 1 x SPC NL 6 SPC 3 SPC NL SPC NL = NL = 1 6SPC NL SPC NL 3 SPC NL 3 = 1 3 SPC NL [g ] Sensitivity (Deviation) With the assumption that the balance s sensitivity was adjusted with an internal or external calibration weight, we first have to deal with the weight s tolerance. Usually its deviation is given in the data sheet as a tolerance band SPC CAL max m CAL, and we find ourselves in the same situation as we were when deriving a variance for the non-linearity. With the same reasoning we can write for the variance of the calibration weight deviation s CAL = 1 3 SPC CAL. unit: [g ] Furthermore, as calibration can only take place through a weighing of the calibration weight, strictly speaking, we would have to consider the repeatability of this calibration weighing. However, calibration is a special case insofar as a special signal processing is applied to it, i.e., usually a stronger filtering and a longer measurement interval, thereby improving the repeatability of the calibration weighing. For practical reasons, this contribution may therefore be neglected, which we will do here. A further complication occurs if the calibration weight does not amount to the balance s full weighing capacity. When the calibration weight is smaller than the weighing capacity, the obtained calibration measurement is extrapolated to the corresponding calibration value at full capacity. Unfortunately, this calculation increases a potential linearity deviation occurring at the load of the calibration weight by the same factor. Particularly with precision balances, possessing weighing capacities of several kilograms, it is for practical reasons not always possible to build in a calibration weight equal to its full capacity. With analytical balances, on the other hand, the built-in calibration weight usually embraces the full weighing capacity of the balance.
9 7 We do not further pursue here the consequences of this contribution. Remark: As we will show later, it is sometimes more convenient to use the relative calibration deviation, instead of the absolute one. The relative deviation is the absolute deviation normalized to the mass of the calibration weight. The corresponding expressions are SPC CAL,rel = SPC CAL m CAL s CAL,rel = s CAL m = 1 3 CAL max SPC CAL m CAL m CAL m CAL unit:  unit:  Temperature Coefficient The temperature coefficient of sensitivity may also be taken from the data sheet (if this item is specified). This specification describes the largest static sensitivity deviation caused by a change in ambient temperature. SPC TCS max TC S. Again, we use the same procedure to obtain the variance from the band limits (see derivation under the linearity deviation) s TCS = 1 3 SPC TCS. unit: [1/K ] About the properties of the temperature excursion we can only speculate here. Unless there is additional knowledge available about the course of ambient temperature, the following assumptions seem reasonable: If the temperature at the location of the weighing is constant, we can drop the influence of the temperature (coefficient) altogether. The temperature excursion at the location of the weighing stays within a band of ±d t degrees. If an automatically induced calibration (for example FACT ) is active, then it is realistic to assume that a maximum temperature change of ± C may occur, before the balance gets adjusted.
10 8 In the latter two cases, the assumption d t max t, with d t C is justifiable, if we assume a temperature band of ± C (or ±K). For the variance of the ambient temperature we get using the well known assumption of uniform distribution s t = 1 3 d t. unit: [K ] We obtain the change of the balance s sensitivity as product of temperature coefficient and temperature change d TS = d TCS d t. unit:  As a last step, we need to determine the variance of this deviation. Because we have a product of two individual contributions, its derivation is not trivial. It can be shown, however, that the product of the variances is a reasonable approximation, which we will use here s TS = s TCS s t. Finally, we get s TS = 1 3 SPC 1 TCS 3 d t = 1 9 SPC TCSd t unit: 
11 9 Combination Of The Variances 7) The combined variance of all deviations considered, can now be obtained by adding all single variances of the individual deviations. s TOT n = s i i = 1. A required condition to justify this operation is statistical independence, or at least uncorrelatedness, between the single contributions. It can be shown that the individual causes for the balance s deviation from its ideal performance are independent from each other. All contributions to a single measurement the difference of a tare-weighing and a sample weighing (i.e., tare and weighing object) now produce the following result: Repeatability Repeatability is an absolute deviation and by definition was determined from the difference of pairs of readings. Therefore, its contribution to one weighing (i.e., difference of two readings) is the simple variance s RP. unit: [g ] Non-Linearity Non-linearity is an absolute deviation. It occurs when weighing the tare, as well as the sample (tare and weighing ob- 7) We do not consider corner load deviations that may occur, if the weighing object is not placed in the center of the weighing pan. (If the weighing object is placed in the center of the platform, this deviation vanishes.) Neither do we consider any other influences on the weighing process, besides those explicitly stated in the text. Particularly, we do not consider influences such as (including, but not limited to): ambient climate (rapid temperature change, humidity change), air draft, pressure fluctuation, heat radiation, mechanical influence (leveling, vibration), electromagnetic influence (electrostatic or magnetic), air buoyancy. In case of such influences, the effects have to be dealt with separately.
12 10 ject) 8 ). Since the two loadings are statistically independent 9 ), their contribution is twice the variance s NL. unit: [g ] Tolerance Of Calibration Weight The normalized calibration weight tolerance is a relative deviation. The deviation of a weighing (i.e., difference of pair of readings) is proportional to the sample weight m. Its contribution is the simple variance times the square of the sample weight s CAL,rel m. unit: [g ] Sensitivity Drift Sensitivity temperature drift is a relative deviation. The deviation of a weighing (i.e., difference of pair of readings) is proportional to the sample weight m. Its contribution is the simple variance times the square of the sample weight s TS m. unit: [g ] Total Variance Under these assumptions, we get for the combined variance of a weighing (i.e., difference of pair of readings) s = s RP +s NL +s CAL,rel m +s TS m = = s RP +s NL +m s CAL,rel +s TS. We now substitute the variances with their previously determined expressions and obtain s = SPC RP + 3 SPC NL m SPC CAL,rel SPC TCSd t unit: [g ] 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, full capacity, or weighings with zero sample weight. All three cases are but of academic interest, therefore they are not considered here. 9) There may be a dependence of the readings for small samples. If there is knowledge about such a correlation, it can be used; here, we do not consider it for the sake of simplicity.
13 11 this value by dividing the former expression by the square of the sample mass s rel = s m = = 1 m SPC RP + 3 SPC NL SPC CAL,rel SPC TCSd t unit:  From this variance, we can derive the normalized standard deviation for one weighing (i.e., difference of pair of readings). This yields s rel = = 1 m SPC RP + 3 SPC NL SPC CAL,rel SPC TCSd t 10) unit:  Measurement Uncertainty It is reasonable to assume that a balance s combined measurement deviation resembles a normal distribution. As one of its justifications we mention the fact that some contributions themselves are normally distributed already (e.g., repeatability). A second reason is that there are multiple contributing sources of independent deviations which favors a normal distribution of their combined deviation. From the combined standard deviation we can determine an uncertainty interval from the laws of normal distribution, provided a confidence level is given. We first derive from the confidence level the expansion (or coverage) factor k, i.e., the quotient relating uncertainty to the standard deviation k P = u s, 10) This combined standard deviation considers the influences of repeatability, non-linearity, calibration weight adjustment and temperature coefficient, under the assumption of a temperature band. Not considered are, among others, eccentric load and deviation due to calibration weights not comprising the weighing capacity.
14 1 which is a function of the confidence level P 11 ). Multiplying the 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 assumed to lie between the limits R u m R+u where R is the weighing result (i.e., the difference of two readings), and m the true sample weight 1 ). With a probability of Q = 1 P the true value will lie outside these limits. Example: Determining The Weighing Uncertainty On An Analytical Balance Balance Type: AT01: 00g/0.01mg Using this balance, a sample of 1g shall be weighed in a 190g container. What is the resulting uncertainty of this weighing, conforming to a 95% confidence level? 11) Expansion Factor Confidence Level Expected Missing (Single Sided) (Expectation Prob.) Probability k P Q=1 P % 31.73% % 10% % 5% 95.45% 4.55% % 1% % 0.7% % 0.006% % % 1) To keep things simple, we have consequently refrained from determining, or correcting for, the degree of freedom. Of course, nothing stands against the notion of correcting for the degree of freedom, if it is known of all individual contributions. An instruction for how to determine the correction factor can be found in Guide To The Expression Of Uncertainty In Measurement, first edition , ISBN
15 13 Specifications from data sheet Spec (SPC) SPC Readability 0.01mg 1x10 10 g Repeatability up to 50g 0.015mg.3x10 10 g 50-00g 0.04mg 1.6x10 9 g Non-Linearity within 10g 0.03mg 9x10 10 g within 00g 0.1mg 1.4x10 8 g Calibration Weight Tolerance 1.5ppm.3x10 1 Temperature Coefficient 1.5ppm/K.3x10 1 K Environment (Assumption) Spec (SPC) SPC Ambient Temp. Excursion K 4 K The formula valid for the combined normalized standard deviation for a single sample weighing is s rel = 1 m SPC RP + 3 SPC NL SPC CAL,rel SPC TCS d t As the repeatability specification at 191g is unavailable, we use the 00g specification instead. Thus, we obtain as standard deviation for a 1g sample: s rel = 1 1g g g K 4K = = = Conclusion: The mass of a 1g sample, weighed in a 00g container, can be determined on this balance with a relative standard deviation of approximately s rel < Based on a confidence level of 95%, the corresponding uncertainty amounts to twice the standard deviation, namely u rel = s rel = Remark: It can be seen from this example that with small sample weights 13 ) the contribution to uncertainty originating from the balance s sensitivity (calibration weight tolerance and un- 13) small compared to the balance s weighing capacity
17 15 certainty due to drifting temperature) are minor, compared to those stemming from repeatability and non-linearity. This applies to most balance types. We will use this property later when determining the minimum sample weight. The diagram on the opposite page shows the relative uncertainty versus sample weight and (total) load, respectively. Minimum Sample Weight The minimum sample weight to be weighed conformally on a balance (a.k.a. minimum (sample) weight ) can be estimated, provided the specifications of the balance, as well as the uncertainty and confidence level to be met, are given. To this end, we need once more the expansion factor k, this time as quotient of relative uncertainty and relative standard deviation k P = u rel = u /m s rel s/m = u, s which is clearly the same function of the confidence level P as introduced above. From the required properties of the sample weighing, namely the relative uncertainty and the confidence level, from which the expansion factor was determined, we derive the standard deviation s rel = u rel. k( P) The formula used in the previous chapter for the relative standard deviation we now solve for the sample weight m. We obtain m = SPC RP + 3 SPC NL s rel 1 3 SPC CAL,rel SPC TCSd t 1 m MIN. Substituting the expression for the relative standard deviation yields for the minimum sample weight m MIN u rel k( P) SPC RP + 3 SPC NL 1 3 SPC CAL,rel SPC TCSd t 1.
18 16 Approximations Revisiting the figures of the previous example, we recognize that with a sample weight of 1 /00 of the weighing capacity, the variance of the sensitivity deviation, calibration weight tolerance and temperature coefficient, equals , an amount negligible compared to the variance due to repeatability and non-linearity ( ). It may be presumed, that this be the case for all small sample weights. We will show that this is true. Taking the radicand of the relative uncertainty 1 m SPC RP + 3 SPC NL SPC CAL,rel SPC TCSd t and expanding it by m, we get SPC RP + 3 SPC NL + m 3 SPC CAL,rel SPC TCSd t According to the assumption, that m is small (minimum sample weight!), a fact which is even more true for its square ( m ), we may drop the second term in favor of the first, and we get as an approximation s 1 rel m SPC RP + 3 SPC NL = = m 1 SPC RP + 3 SPC NL (valid for small samples). Respecting the previous requirements, we determine from this formula the approximate sample weight to be m 1 MIN SPC s RP + 3 SPC NL = rel = u k SPC RP + rel 3 SPC NL (valid for small samples). From this we conclude that the minimum sample weight is essentially determined by the two specifications of repeatability and non-linearity; calibration weight tolerance and temperature coefficient do not occur in the formula. Traditionally, sensitivity adjustment is given too much attention when dealing with small sample weights: With the exception of
19 17 weighing heavy samples 14 ), sensitivity plays but an inferior role. If, for any reason, additional information is available about the linearity deviation of a balance which shows that nonlinearity is inferior compared to repeatability, the contribution of non-linearity may be neglected, too. Such information could be gained by on-site measurements, or could stem from other sources. If the non-linearity specification (SPC NL ) is 1 / of the repeatability spec (SPC RP ), its contribution reduces to 14% of the combined uncertainty, if non-linearity amounts to 1 /3 of repeatability, its contribution is 7%. Regarding these figures, one may decide to drop this term altogether. In this case the formula for minimum weight would reduce to m MIN = k u rel SPC RP = k u rel SPC RP (valid for small non-linearity). Example: Determining The Minimum Sample Weight On An Analytical Balance Balance Type: AT01: 00g/0.01mg What is the minimum sample weight required using a 190g container, observing a relative uncertainty of 0.1% at a confidence level of 95% (corresponding to k )? Specifications from data sheet Spec (SPC) SPC Readability 0.01mg 1x10 10 g Repeatability up to 50g 0.015mg.3x10 10 g 50-00g 0.04mg 1.6x10 9 g Non-Linearity within 10g 0.03mg 9x10 10 g within 00g 0.1mg 1.4x10 8 g Calibration Weight Tolerance 1.5ppm.3x10 1 Temperature Coefficient 1.5ppm/K.3x10 1 K Environment (Assumption) Spec (SPC) SPC Ambient Temp. Excursion K 4 K 14) usually larger than 1 /10 to 1 /4 of the balance s weighing capacity, yet independent of total load
20 18 Since we are dealing here with a minimum sample weight, we may use the approximation formula m k MIN SPC u RP + 3 SPC NL. rel As the repeatability specification at 191g is unavailable, we use the 00g specification instead. Thus, we obtain as a minimum sample mass m MIN g g = g = = g = g = 94 mg. If the sample amounts to about 100mg or more, then we can be assured that the given requirements, namely the mass determination with 0.1% uncertainty at 95% confidence, can be achieved on this balance. If we had additional information about this balance, such that its linearity deviation is smaller than 0.0mg, this figure would amount to 1 / of repeatability (0.04mg). In this case, its contribution is small, as can be seen m MIN g g = = g g = g = = g = 86 mg, and we may decide to neglect it after all. We then have as a minimal weight estimation m MIN mg = mg = 80 mg.
21 19 Conclusion In many instances, the weighing result needs to be qualified. To this end, the measurement uncertainty accompanying the weighing process is required, but usually not readily available, not least because it is dependent on the application at hand. At other times, the operator needs to know the minimum amount of mass he/she is able to conformally weigh to a required relative uncertainty and confidence level (minimum weight). This paper explains how uncertainty and minimum weight can be estimated from the data sheet specifications of a balance. The assumptions and restrictions, under which this deduction is valid, as well as when, and under which conditions, neglections can be made, are discussed. Two examples with actual data from analytical balances are given as illustrations. The theory and examples provided enables the user to estimate the appropriate figures of uncertainty or minimum sample weight for his/her balance application.
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