Calculating Measurement Uncertainty of the Conventional value of the result of weighing in air

Transcription

1 Calculating Measurement Uncertainty of the Conventional value of the result of weighing in air Technical note for submission to NCSLI Measure Celia J. Flicker* Hy D. Tran Abstract The conventional value of the result of weighing in air is frequently used in commercial calibrations of balances. The guidance in OIML D-028 for reporting uncertainty of the conventional value is too terse. When calibrating mass standards at low measurement uncertainties, it is necessary to perform a buoyancy correction before reporting the result. When calculating the conventional result after calibrating true mass, the uncertainty due to calculating the conventional result is correlated with the buoyancy correction. We show through Monte Carlo simulations that the measurement uncertainty of the conventional result is less than the measurement uncertainty when reporting true mass. The Monte Carlo simulation tool is available in the online version of this article. 1.1 Introduction Mass is the physical property of matter that indicates the amount of matter. This property has been given the unit of kilogram (kg) in the SI unit system. The kilogram is one of the seven base units in the SI from which all other measurement units may be derived. Mass was one of the earliest recorded measurements, used in commerce. The mass unit of the carat is based on the weight of the carob seed: 0.2 grams [1],[2]. More recently, the SI defined the kilogram as the mass of the international prototype kilogram. Mass is fundamental in commerce because it provides the amount of material being used or traded. Produce in grocery stores is sold by mass, prescription drugs are sold by mass, and chemical composition and purity is determined by the mass of the chemical and impurities. Mass is also fundamental in a number of sciences. For example, force is derived from mass and acceleration through Newton s second law; pressure is derived from force and area; torque is derived from force and distance. As technology has progressed, the uncertainty of mass calibrations tracing to the international prototype kilogram is limiting the uncertainties of derived SI units such as the volt [3]. A number of proposals have been made to fix a fundamental physical constant, and derive the kilogram from the value of the physical constant. Planck s constant is being proposed as one of the fixed values, with the realization of the kilogram through the use of a watt-balance [4]. Even with the redefinition of the realization of the kilogram, it will be necessary to compare mass standards through comparison weighings [5].

2 Currently, in the United States, a calibration laboratory s reference kilogram might be compared to an accredited laboratory s master kilogram standard, which in turn is compared to a National Institute of Standards and Technology (NIST) working kilogram standard, which is compared to the NIST K20 standard, and the K20 is compared to the international prototype kilogram. Even with the redefinition of the realization of the kilogram, most National Metrological Institutes (NMI) would still perform mass calibrations to a working mass standard, commonly called weight. 1.2 Conventional Result of the Value of Weighing in Air In commerce, weights are commonly used to calibrate balances. When the weights are calibrated, they are compared to another weight with a lower overall measurement uncertainty. Most mass measurements are actually force measurements, measuring the normal force due to gravity of a weight on an electronic balance. The problem is that the actual force of the weight is the force of the weight minus the buoyancy force. In commercial operations, the conventional result of weighing in air is often used as the reference mass value [6]. Nowadays, most weight standards are made of ferrous alloys, with densities ranging from about 7400 kg/m 3 to 8700 kg/m 3, depending on the type of alloy. Stainless steels tend to be closer to 8000 kg/m 3. Conventional result provides a mass value that is corrected for buoyancy of air at sea level, and for a reference weight with a density of 8000 kg/m 3, as shown in the definition from the International Organization of Legal Metrology, International Document 28 (OIML D-028): The conventional mass value of a body is equal to the mass m c of a standard that balances this body under conventionally chosen conditions. The unit of the quantity conventional mass is the kilogram... The conventionally chosen conditions are: t ref = 20 C 0 = 1.2 kg m -3 c = 8000 kg m -3 Where t ref is the reference temperature, 0 is the density of the atmosphere at conventional conditions, and c is the density of the standard. Figure 1 illustrates this definition for conventional mass value. [Figure 1 Here] When the unknown and the reference are perfectly balanced, the net force on each object is equal. Figure 2 shows the net force on a weight. This is the gravitational force on the weight, minus the force of buoyancy on the weight. The instrument measures F net. [Figure 2 Here] Mass comparisons use a single-pan electronic balance rather than the dual-pan balance as illustrated in Figure 1, so the balance measures force rather than mass. Weight calibrations are performed by substitution weighings [7],[8]. Substitution weighing compares the force applied by the unknown weight to the force applied by the reference weight. In essence, with an unknown weight X, and a reference weight S, the mass of the unknown is the difference from the comparison plus the mass of the reference, along with a correction for differences in buoyancy force between the two weights. The process for making buoyancy corrections is well described in a number of documentary standards [9].

3 The mass of the unknown, m X, is calculated from: m m (1) X Where m S is the mass of the standard, is a comparison correction for the difference in masses and the effects of buoyancy: = O + B. When comparing two different weights, the observed difference would be: S ( m g V g) ( m g V g) (2) O B X X a S S a Where V X is the volume of the unknown, a is the density of air where the weighing is being made, V S is the volume of the reference, and g is the local gravitational acceleration. In addition to the published NIST Standard Operating Procedures (SOPs), a number of references, such as Reference [1], provide detailed derivations that support the published equations. The conventional result of the value of weighing in air is called conventional result or conventional mass for short. In this paper, true mass is used to further differentiate mass value from the conventional result. The use of conventional result when calibrating commercial balances avoids many complexities associated with buoyancy corrections on commercial balances [1]. Equation (3) is commonly used to calculate the conventional result m cx when the true mass m X is known, and X is the density of the object (in kg/m 3 ). 91 m cx m X X (3) Note that although the term 1 uses the density of the unknown weight, the correction for X buoyancy physically originates from the volume of the unknown weight s displacement of air. 1.3 Uncertainty of the Conventional Result of the Value of Weighing in Air OIML D-028 [6] provides a recommendation for estimating the uncertainty of the conventional value of the result of weighing in air; Equation 13 in the D-028 document is reproduced here as Equation (4) u( m ) u u ( m ) u u (4) cx W cs b ba Here, u(m cx ) is the standard uncertainty of the conventional result, u W is the standard uncertainty of the weighing process, u(m cs ) is the standard uncertainty of the conventional result of the reference weight, u b is the standard uncertainty of the buoyancy correction, and u ba is the standard uncertainty of the balance.

4 Many labs report mass value (true mass) and not the conventional result. To obtain the conventional result, these laboratories apply Equation (3) to the calibrated mass result. Equation (3) does not provide guidance for reporting uncertainty. Equation (4) ignores the correlation between the buoyancy correction in determining mass and the buoyancy correction in calculating the conventional result. This problem was observed in 2013 [10]; Gläser [11], and later Bich and Malengo [12], stated that the buoyancy corrections were correlated. The remainder of this paper illustrates some unintuitive implications of the conventional result s uncertainty. The uncertainty of the measurement for the conventional value of the result is estimated using the Guide to the expression of uncertainty in measurement, commonly referred to as the GUM [13]. NIST Technical Note TN1297 [14] provides more detailed guidance than the GUM. The estimate of measurement uncertainty must incorporate all influence factors into the evaluation of the uncertainty; some of the evaluations are statistical, and others are non-statistical. The estimate then combines both statistical and non-statistical evaluations. In general, having a measurement equation is a necessary starting point for estimating measurement uncertainty. In the case of mass calibration, the measurement equation for substitution weighing is Equation (1). If reporting the conventional result, the measurement equation would be Equation (1) combined with Equation (3). At first glance, this would appear to be: m x ( m ) cx S O B Section of the GUM suggests combining the standard uncertainty for the measurement equation, Equation (5), by calculating: (5) N 2 f 2 c i1 xi u ( y) u ( xi) (6) Where u c is the combined standard uncertainty, y is the measurand, f is the measurement equation, and each x i is the individual influence factor in the measurement equation. The GUM points out that this approach is valid only if the individual influence factors are uncorrelated. The GUM does provide guidance for correlated uncertainties, based on adding covariance terms to Equation (6); however, this can become mathematically tedious. References [11] and [12] have shown derivations of the analytical solution. The GUM supplement [15] suggests using Monte Carlo methods to propagate uncertainties. When using a Monte Carlo simulation, it is not necessary to calculate individual weighting factors, because a properly performed simulation automatically assigns the correct weightings Calculating Uncertainties by Monte Carlo Simulation The influence factors identified for the uncertainties of measuring true mass and the conventional result of weighing in air are:

5 Uncertainty of the reference mass Uncertainty of the reference weight s density Uncertainty of the unknown weight s density Uncertainty in measurement of air density Uncertainty due to instrument readability Uncertainty due to weighing process These are summarized in Table 1. [Insert Table 1 Here] A Monte Carlo simulation is a virtual experiment: see for example Reference [16]. Monte Carlo methods were initially developed to solve difficult physics problems where closed-form solutions were difficult to obtain, and have become popular as computers have become more powerful. They are especially useful in showing the effects of changing different variables on all of the results of a complex calculation, without the cost of running an actual experiment multiple times. One has to be careful in running a Monte Carlo simulation because the simulation relies on the quality of random numbers generated by the computer (more properly, pseudorandom numbers) [17],[18]. The more important property of a pseudorandom number generator for a Monte Carlo simulation is the distribution of the numbers [19]. Many computer science courses demonstrate the Monte Carlo method by estimating the value of π. This is illustrated in Figure 3 [20]. A good Monte Carlo simulation for estimating measurement uncertainty not only requires the correct measurement equation, but also a good pseudorandom number generator. For the purposes of this paper, Excel 2010 is used for Monte Carlo simulations. The simulation tool is available in the online appendix. The Excel tool that the authors of this paper developed is illustrated in Figure 4. [Insert Figures 3 and 4 Here] The statistical procedures and distributions of earlier versions of Excel are known to have inaccuracies [21],[22],[23]. An evaluation of Excel 2010 s random number generator (RAND()) was more satisfactory than those of previous versions [24]. While the evaluation was not comprehensive, due to the need to test 2 35 pseudorandom numbers, they found that the 2010 pseudorandom number generator passed the distribution tests. This evaluation concluded that the 2010 random number generator performs similarly to a Mersenne Twister pseudorandom generator, which is considered adequate for Monte Carlo simulations. International Recommendation 111 (OIML R111) provides descriptions of weight classes suitable for different purposes, ranging from reference weights used at national metrological institutes to field weights used to calibrate balances in various commercial settings. There are other weight classes used; American Society for Testing and Materials (ASTM) E is one such common alternative to the OIML weights [25]. In both R111 and in ASTM E617, the standards specify acceptable limits for the properties of the weight (properties primarily being allowable deviations from nominal mass, density, shape, and

6 surface finish). This paper will focus on the uncertainties for high-quality weights: This would be OIML E2 class or better. Monte Carlo simulations were used to estimate the measurement uncertainty for calibrating 1 kg mass standards with E2 characteristics, per OIML R111. It was assumed that the mechanical and magnetic properties were acceptable, and a range of acceptable densities (7810 kg/m 3 to 8210 kg/m 3 ) were used, along with a range of acceptable uncertainties of density per OIML R111 (0.08% to 1.8 % of actual density, of an 8000 kg/m 3 weight) and a range of air densities from near sea level (1.2 kg/m 3 ) to the highest capital city in the world (La Paz, Bolivia at an altitude of approximately 3600 m with air density of approximately 0.77 kg/m 3 ). The range 0.08% to 1.8% of the actual density is chosen based on the density estimation methods allowable for an E2 class of weight per OIML R111-1 Tables B.8 and Section B The uncertainty of the measurement result in a Monte Carlo simulation is the standard deviation of all experimentally determined calculations of that result. The supplement to the GUM suggests using on the order of 10 6 simulations to get 2 significant figures in the uncertainty at a coverage factor k=2, or doing a convergence study. For this calculation, 500,000 ( ) simulations for each individual case were run. While we did not perform a formal convergence study, at 500,000 simulations, the results from one simulation to the next changed by between 0.04% and 0.1%, with the change usually a difference of 1 or 2 in the third or fourth significant digit, and not in the first or second significant digit. Table 2 shows a sample of ten simulation results at 500,000 trials. [Insert Table 2 Here] Based on the equipment and standards in our lab, and based on possible E2 class unknown weights, the values of the uncertainties simulated are detailed in Table 1. We simulated the uncertainties when performing double substitution per NIST SOP4. SOP4 provides detailed guidance for the actual weighing observations and calculations of buoyancy observations. In double substitution, four mass measurement observations are made. In the option chosen for our laboratory, the first and last measurements are of the reference weight, and the second and third are of the unknown weight. The average of the difference between the unknown and reference weights is taken to correct for comparator drift. This comparison difference and the known mass of the reference, together with corrections for the effects of buoyancy on both the reference weight and unknown weight, are used to calculate the mass of the unknown. The Monte Carlo simulations used the NIST SOP4 double substitution Equation , with correction for buoyancy for true mass and omitting sensitivity weights. This is shown in Equation (7): 201 m X O O O O a ms 1 S 2 a 1 X (7)

7 Where m X is the measurement result, m S is the mass of the reference, a is the density of air where the calibration is performed, S is the density of the reference, X is the density of the unknown, and O 1 through O 4 are the individual observations of the mass comparator. The simulation then applies a correction to obtain the conventional value of the result of weighing in air, per Equation (3). Because of the definition of the conventional result of weighing in air, the actual density of the unknown weight in implementing equation (3) was used, but the uncertainty of the density of the unknown is used in the calculation of true mass. For each simulated measurement result (trial), the Monte Carlo method calculates the result using a slightly different mass and density for the reference weight, a slightly different air density, and a slightly different density of the unknown weight. As long as the different masses, densities, and instrument and process effects are incorporated in each Monte Carlo trial, a large number of simulations will reflect the actual statistical behavior of the measurement without actually performing the physical measurements in the case of this paper, 500,000 trials per set of conditions (such as air density, or density of the unknown, as listed in Table 1). In addition to the factors associated with the environment and the weights, a value of 1 digit of resolution is added to each observation to simulate instrument readability, and the historical process standard deviation of the laboratory is added to the overall measurement result. The type of distribution for the random values is given in Table 1. In Excel, when adding a variation u RECT due to a rectangular distribution, an equation of the form is used: 220 urect width (2RAND() 1) (8) Where width is the half-width of the uncertainty, e.g. for a ±1 kg/m 3 rectangular uncertainty, width=1 kg/m 3 is used. When adding a variation due to a standard uncertainty, u GAUSS, an equation of the form is used: 224 ugauss NORM.INV(RAND(), x, ) (9) where x is the mean and is the standard deviation for the uncertainty. RAND() is a built-in Excel function that returns a uniformly distributed pseudorandom number between 0 and 1, and NORM.INV is a built-in Excel function that returns a pseudorandom number that matches a normal (Gaussian) probability density function with the chosen mean and standard deviation. The built-in Excel functions AVERAGE and STDEV.S were used to calculate the mean and the standard deviation of the simulation results. Since 500,000 simulations are performed, the difference between STDEV.S and STDEV.P (estimate of standard deviation of the sample and estimate of the standard deviation of the population) is insignificant: 233 STDEV.S: ,999 1 STDEV.P: = 500,000 (10)

8 234 STDEV.S was used on the results of 500,000 simulations, to calculate a standard uncertainty Results of the Monte Carlo Simulation The calibration of a 1 kg unknown weight is simulated, where the true mass result is reported. Equation (3) is then applied to obtain the conventional result. Note that the actual density of the unknown is used in calculating both the true mass and the conventional result. Because 500,000 simulations were run, the standard deviation of the results of all of the simulations represent the standard uncertainty of the calibration. The simulation results are presented in Table 3. The calibration uncertainties are standard uncertainties. For any given set of master density, uncertainty of master density, uncertainty of unknown density, and air density, the standard uncertainty when calculating the conventional result is always smaller than the standard uncertainty of the true mass. This appears paradoxical, because one would expect that applying one more equation (Equation (3)) to a result would add one more factor to the measurement uncertainty. However, Equation (3) is correlated with Equation (7); Equation (6) is only applicable if the uncertainty components are uncorrelated. [Insert Table 3 Here May need to be separate pages due to size of table] To validate the results of the Monte Carlo simulation with published analytical uncertainty analysis, the conditions of Example 2 from [12] were set as conditions for the simulation. Malengo and Bich found uncertainties of 0.92 mg for the true mass and mg for the conventional result, while the simulation gave uncertainties of 0.91 mg and mg. Some of the uncertainties the simulation requires, such as the uncertainty of air density, or the uncertainty due to instrument readability, were not included explicitly in the example, which might account for the small difference in uncertainties of true mass Discussion There is another way of considering the uncertainty when calculating uncertainty of the conventional result. This method uses the volume of the reference, and the volume of the unknown, rather than their densities. This may be a more intuitive approach. To estimate the uncertainty in conventional result, consider the measurement model of Equation (1), repeated here: mx ms O B (11) Where m X is the measurement result, m S is the mass of the standard, O is the observed difference, and B is the correction for buoyancy differences. Whether the calculation is for conventional result or true mass, this equation applies. C is the specific correction to the conventional result, but is otherwise the same as B.

9 If the local air density were 1.2 kg/m 3, and the reference S had a density of 8000 kg/m 3, C 0, because of the definition of conventional mass. The uncertainty components for Equation (11) still include the uncertainty mass and volume of S, the uncertainty in volume of X (because the environment is not at 1.2 kg/m 3 ), the uncertainty in density of air, and the process standard deviation. However, the uncertainties associated with the volume of the reference, the volume of the unknown, and the uncertainty of the density of air should be smaller for the conventional result. When calculating the conventional result for weighing in air, the effect of the volume of X (V X ) is referred to the nominal volume at 8000 kg/m 3 ; similarly for V S. In addition, the effect of local air density is relative to 1.2 kg/m 3 rather than correcting to vacuum. Consider a comparison of X with S in air at 1.2 kg/m 3. For the moment, assume X has a density of 8000 kg/m 3. The conventional result would be: 279 m m cx S O C ( m V ) ( m V ) O X X 1.2 S S 1.2 (12) where 1.2 is 1.2 kg/m 3 and the observed difference includes buoyancy terms. For true mass, it would be necessary to perform a complete set of buoyancy corrections. For conventional result, one only corrects if the density of S is not 8000 kg/m 3. Assume that V S > V 8000 (i.e. the density S < 8000 kg/m 3 ). The buoyant force on S is larger than it is in standard conventional conditions. S does not balance as much weight as it should in conventional conditions, so a correction to m cx is needed to increase the weight of m cx from the observed difference: 286 C ( V V ) (13) 8000 S Now assume that the air density a is not 1.2 kg/m 3 ; a < 1.2. The buoyant force on S is smaller than it is in standard conditions, so S balances a larger weight than it should. A negative correction to C is needed: ( V V )( ) (14) C 8000 S a 1.2 Finally, assume V S =V 8000, but V X > V 8000 ( X < 8000 kg/m 3 ), and again, the air density is not 1.2 kg/m 3 ; a < 1.2. In this case, the observation of X has a smaller buoyant force, add a positive correction to C is needed: Combining Equations (13), (14), and (15): ( V V )( ) (15) C X 8000 a ( V V ) ( V V )( ) ( V V )( ) C 8000 S S a 1.2 X 8000 a 1.2 ( V V ) ( V V )( ) C 8000 S 1.2 X S a 1.2 (16)

10 The first term in Equation (16), (V V S ) is the correction for m S to m cs to obtain the conventional result for the standard S. If the conventional value for the reference S is used in weighing X, the correction is simply: ( V V )( ) (17) C X S a The uncertainties associated with the buoyancy correction are uncertainties in V S, V X, and a, the uncertainty of the reference S, and the process standard deviation. The sensitivity of C with respect to V S is now easy to calculate. By inspection, C V S a 1.2 (18) 305 Similarly, the sensitivity with respect to V X is 306 C V X a 1.2 (19) 307 And the sensitivity with respect to a is 308 C a V X V S (20) 309 The combined uncertainty due to the buoyancy correction for the conventional result is: u u u u u V C 2 C 2 C VS VX a ms Process S VX a Note that the first three terms in Equation (21) are the uncertainties associated with buoyancy corrections for the conventional result, and the magnitude of these terms are smaller than when calculating true mass. (21) In summary, when calibrating true mass, a dominant term in the uncertainty is the uncertainty of the volume of the unknown. When calibrating conventional result, the effect of the volume uncertainty of the unknown is greatly reduced. In addition, the uncertainty of the volume of the reference is reduced to the difference from the volume to the 8000 kg/m 3 reference density Conclusions When calibrating mass for low measurement uncertainties, it is generally necessary to correct for volume differences between the reference and the unknown weights. When reporting the conventional result, because the definition of conventional result explicitly states reference density and standard

11 conditions, the uncertainty of the buoyancy correction for the conventional result is smaller than the uncertainty of the buoyancy correction for true mass. Because these uncertainties are correlated, they cannot be added as independent terms. The covariance terms must be accounted for, and the correlation is negative. The use of conventional results can lead to an uncertainty evaluation that appears abnormally low; however, proper application of the GUM requires this low uncertainty because of the correlation of the volume of the unknown to both the calibration of true mass and the following conversion to conventional result. This might lead to erroneous conclusions in interlaboratory comparisons, if the requested measurand is a conventional value. This simulation tool has not been used to simulate an interlaboratory comparison with the worst-case condition for assumed density at high altitude, or a reference lab at sea level, but anyone can use this simulation tool to explore the possible Type B uncertainties in calibrating mass and conventional results Acknowledgements Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL Certain commercial equipment, instruments, or materials are identified in this paper in order to adequately describe the experimental procedure. Such identification does not imply recommendation or endorsement by the authors or by Sandia National Laboratories, nor does it imply that the materials or equipment identified are the only or best available for the purpose. The authors acknowledge the support of the Sandia STAR Fellowship Program from the Community Involvement department at Sandia National Laboratories, and discussions with Rick Mertes, Eric Forrest, and Sam Ramsdale References [1] F. E. Jones and R. M. Schoonover, Handbook of Mass Measurement. Boca Raton, FL: CRC Press, [2] H. A. Klein, The Science of Measurement: A Historical Survey. New York: Dover Publications, [3] I. M. Mills, P. J. Mohr, T. J. Quinn, B. N. Taylor, and E. R. Williams, "Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1 (CI- 2005)," Metrologia, vol. 43, pp , [4] J. R. Pratt, "How to Weigh Everything from Atoms to Apples Using the Revised SI," NCSLI Measure: The Journal of Measurement Science, vol. 9, p. 12, [5] N. M. Zimmerman, J. R. Pratt, M. R. Moldover, D. B. Newell, and G. F. Strouse, "The Redefinition of the SI: Impact on Calibration Services at NIST," NCSLI Measure: The Journal of Measurement Science, vol. 10, p. 5, [6] OIML, "Conventional value of the result of weighing in air," ed, 2004.

12 [7] OIML, "Weights of classes E1, E2, F1, F2, M1, M1 2, M2, M2 3 and M3," ed, [8] NIST, "Recommended Standard Operations Procedure for Weighing by Double Substitution Using a Single-Pan Mechanical Balance, a Full Electronic Balance, or a Balance with Digital Indications and Built-In Weights," ed: National Institute of Standards and Technology, [9] NIST, "Recommended Standard Operating Procedure for Applying Air Buoyancy Corrections," ed, [10] H. D. Tran, "Calculating uncertainties for conventional mass," unpublished memo. [11] M. Gläser, "Covariances in the determination of conventional mass," Metrologia, vol. 37, p. 249, [12] A. Malengo and W. Bich, "Buoyancy contribution to uncertainty of mass, conventional mass and force," Metrologia, vol. 53, pp , [13] JCGM, "Evaluation of measurement data Guide to the expression of uncertainty in measurement.," ed, [14] B. N. Taylor and C. E. Kuyatt, "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results," NIST1994. [15] JCGM, "Evaluation of measurement data Supplement 1 to the Guide to the expression of uncertainty in measurement Propagation of distributions using a Monte Carlo method.," ed, [16] M. Angeles Herrador and A. G. Gonzalez, "Evaluation of measurement uncertainty in analytical assays by means of Monte-Carlo simulation," Talanta, vol. 64, pp , Oct [17] S. K. Zaremba, "The Mathematical Basis of Monte Carlo and Quasi-Monte Carlo Methods," SIAM Review, vol. 10, pp , [18] F. James, "A review of pseudorandom number generators," Computer Physics Communications, vol. 60, pp , [19] A. B. Owen, "Uniform Random Numbers," in Monte Carlo theory, methods and examples, ed [20] CaitlinJo. (2011, April 7, 2016). Pi 30K. Available: [21] B. D. McCullough and D. A. Heiser, "On the accuracy of statistical procedures in Microsoft Excel 2007," Computational Statistics & Data Analysis, vol. 52, pp , [22] B. D. McCullough and B. Wilson, "On the accuracy of statistical procedures in Microsoft Excel 2003," Computational Statistics & Data Analysis, vol. 49, pp , [23] A. T. Yalta, "The accuracy of statistical distributions in Microsoft Excel 2007," Computational Statistics & Data Analysis, vol. 52, pp , [24] G. Mélard, "On the accuracy of statistical procedures in Microsoft Excel 2010," Computational Statistics, vol. 29, pp , [25] ASTM, "Standard Specification for Laboratory Weights and Precision Mass Standards," ed: ASTM International,

13 Appendix 1. Excel Simulation Tool The Excel simulation tool is available in the online version of this article. The printed appendix shows the variables used and the formulas used in the simulation. The computer used for these simulations was a Dell OptiPlex 980, with Windows 7 Enterprise 64 bit, 4 GB RAM, and an Intel core i7 870 processor at 2.93 GHz. A simulation of 500,000 runs has a file size of approximately 60 MB. The version provided for online access only has 22 runs, and is 16 KB. While the simulation only takes 2 seconds to run, opening the file takes approximately 17 seconds, and saving the file takes approximately 9 seconds. When debugging the simulation, it is advantageous to have a much smaller number of runs. Increasing the number of runs is simple by using copy/paste and using Excel s functions to go to a specific cell. Table 4 shows the inputs into the simulation. For each simulation, a reference mass is used, m R, that is varied with the uncertainty u R of one of the reference references, using a rectangular distribution to reflect the expected variation over the calibration interval. The density R provided by the calibration certificate is used with a standard uncertainty u R because densities are not expected to vary the density of the weight is not known perfectly, but the density will not change over time. T is the density of the unknown, which is within the acceptable range for OIML E2 weights. The uncertainty u T of the density of the unknown is based on the acceptable methods for obtaining density of an E2 weight. a is the density of air in the laboratory where the calibration is performed; u a is the uncertainty of the measurement of air density, based on the instruments used for measuring air pressure, temperature, and relative humidity. u instr is the uncertainty due to instrument readability, and u W is the uncertainty due to the process (from the standard deviation control chart). O1 through O4 are simulated observations. Table 5 shows five outputs of the simulation. [Insert Tables 4 and 5 Here] The actual reference mass m R used is calculated with: =mr+ur*(rand()*2-1). The actual density of the reference R is calculated with: =(NORM.INV(RAND(),ϱR,uϱR)). The actual density of the unknown T is calculated with=(norm.inv(rand(),ϱt,uϱt)). The actual density of air a is calculated with: =NORM.INV(RAND(),ϱa,uϱa). Calibrated mass is calculated with: =NORM.INV(RAND(),0,uW)+(F3*(1- I3/G3)+((((O2_+uinstR*(RAND()*2-1))-(O1_+uinstR*(RAND()*2-1)))+((O3_+uinstR*(RAND()*2-1))- (O4_+uinstR*(RAND()*2-1))))/2))/(1-I3/H3), where F3, I3, G3, and H3 are the values of m R, a, R, and T, respectively, for this particular simulation. The row number, 3 in this case, changes depending on which simulation is run. The conventional result is calculated with: =J3*(1-ϱ0/H3)/(1-ϱ0/8000), where J3 is the calibrated mass calculation. Note that O1 in the Excel formula has a label O1_, to differentiate from the cell address O1, as do the other observations. Once all the calculations in the spreadsheet are validated, copy the calculations from the first row into a number of rows equal to the desired number of simulations to be run. Automatic Recalculation should be turned off. Manual recalculation can be used and the statistics of the result can be evaluated. With 500,000 simulations, the standard deviation changes in the fourth significant digit, which is consistent with performing a convergence study per the GUM Supplement recommendation. MathType

14 Professional (the upgrade to Microsoft Word s equation editor) can be used to check the nesting of parentheses in Excel formulas.

15 All Figures and Tables Figures Reference Weight Unknown Weight Figure 1: The conventional value of the result of weighing in air is the weight of the reference mass that exactly balances the unknown when the density of the reference is exactly 8000 kg/m 3 and atmospheric air density is exactly 1.2 kg/m 3.

16 Weight A F b = V a a g F g = m A g F net = F g F b Figure 2: Free Body Diagram showing forces acting upon a weight in a balance or mass comparator. The net force measured by the instrument (F net) is the gravitational force on the weight (F g) minus the buoyancy force (F b).

17 Figure 3: A Monte Carlo simulation of throwing a dart at a wall with a square target. If the dot lands inside the section of the square that is inside the quarter circle, the dot is red. If the dart-thrower is truly random, the probability of a dot being red is π/4. With a sufficiently large number of dart throws, one should be able to estimate π to a large number of significant digits. 446

18 Figure 4. A screenshot of the simulation tool in use. Cell colors are used to help identify the inputs to the simulation, intermediate results, and the simulation results. 449

19 450 Tables Table 1: Uncertainty components used in simulation. Where multiple values are given, they would represent our best reference versus our most commonly used reference, and the best, typical, and worst commercially available E2 densities. Uncertainty Component Value Distribution Reference mass mg Rectangular Density of reference (0.0675, 0.7) kg/m 3 Standard (Gaussian) Density of unknown (5, 10, 140, 170, 200) kg/m 3 Rectangular Measurement of air density kg/m 3 Standard Instrument readability mg Rectangular Weighing process mg Standard

20 Table 2: Informal convergence verification for 500,000 trials Standard Deviation of mt (mg) Standard Deviation of mct (mg) Sim Sim Sim Sim Sim Sim Sim Sim Sim Sim

21 Table 3: Summary of the results of the simulations. Each group of three results is at sea level, 1600 m, and 3600 m. The results are then grouped by uncertainty of the density of the unknown, then by density of the unknown itself, then by density and uncertainty of density of two of our lab s reference standards. Master Density (kg/m^3) Master Density Uncertainty (kg/m^3) (rectangular) Master Density Uncertainty as percentage (%) Density of UUT (kg/m^3) Uncertainty in Density of UUT, as percentage of density (%) (k=1) Uncertainty of UUT Density (kg/m^3) (k=1) Density of Air (kg/m^3) True mass uncertainty (standard) (mg) Conventional mass uncertainty (standard) (mg) Reduction in conventional mass uncertainty as a percentage (%) % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

22 % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

23 Table 4. The input variables for the simulation. The highlighted variables are used in the SOP4 calculations. The aqua and purple cells are calculated results. The orange cells are inputs into the simulation. Name Description Value Unit Distrib. m R true mass of ref. weight 1.00 kg u R unc. of true mass of ref. weight kg rectangular m CR conv. mass of ref. weight kg um cr unc. of conv. mass of ref. weight E-08 kg calculated ϱ R density of reference weight kg/m^3 u ϱr unc. of density of ref. weight 0.70 kg/m^3 gaussian (k= m T true mass of unknown weight kg m CT kg ϱ T density of unknown weight kg/m^3 u ϱt unc. of density of unknown weight kg/m^3 rectangular ϱ a density of air 0.77 kg/m^3 u ϱa unc. of density of air 2.30E-04 kg/m^3 gaussian std ϱ 0 air density in conv. conditions 1.20 kg/m^3 u instr unc. due to instrument readability 1.00E-09 kg rectangular u W unc. due to weighing process 1.50E-09 kg gaussian (sta O1 (R1) first observation of ref. weight kg O2 (T1) first observation of unknown weight kg O3 (T2) second observation of unknown weight kg O4 (R2) second observation of ref. weight kg P a g e

24 Table 5. The simulation results are stored in rows. The header and first five rows of a simulation are shown. The first four columns show small differences due to slightly different conditions in a Monte Carlo simulation. The last two columns are calculated results for mass and conventional result. Mean and standard deviation can be calculated from the last two columns. The number of rows is the number of simulated measurements performed m R ϱ R ϱ T ϱ a m T m CT E E E E E E E E E E E E E E E P a g e