Calculating Measurement Uncertainty of the “Conventional value of the 1

result of weighing in air” 2

Technical note for submission to NCSLI Measure 3

4

Celia J. Flicker* 5

Hy D. Tran† 6

Abstract 7

The conventional value of the result of weighing in air is frequently used in commercial calibrations of 8

balances. The guidance in OIML D-028 for reporting uncertainty of the conventional value is too terse. 9

When calibrating mass standards at low measurement uncertainties, it is necessary to perform a 10

buoyancy correction before reporting the result. When calculating the conventional result after 11

calibrating true mass, the uncertainty due to calculating the conventional result is correlated with the 12

buoyancy correction. We show through Monte Carlo simulations that the measurement uncertainty of 13

the conventional result is less than the measurement uncertainty when reporting true mass. The Monte 14

Carlo simulation tool is available in the online version of this article. 15

1.1 Introduction 16

Mass is the physical property of matter that indicates the amount of matter. This property has been 17

given the unit of “kilogram” (kg) in the SI unit system. The kilogram is one of the seven base units in the 18

SI from which all other measurement units may be derived. Mass was one of the earliest recorded 19

measurements, used in commerce. The mass unit of the carat is based on the weight of the carob seed: 20

0.2 grams [1],[2]. More recently, the SI defined the kilogram as the mass of the international prototype 21

kilogram. Mass is fundamental in commerce because it provides the amount of material being used or 22

traded. Produce in grocery stores is sold by mass, prescription drugs are sold by mass, and chemical 23

composition and purity is determined by the mass of the chemical and impurities. Mass is also 24

fundamental in a number of sciences. For example, force is derived from mass and acceleration through 25

Newton’s second law; pressure is derived from force and area; torque is derived from force and 26

distance. 27

As technology has progressed, the uncertainty of mass calibrations tracing to the international 28

prototype kilogram is limiting the uncertainties of derived SI units such as the volt [3]. A number of 29

proposals have been made to fix a fundamental physical constant, and derive the kilogram from the 30

value of the physical constant. Planck’s constant is being proposed as one of the fixed values, with the 31

realization of the kilogram through the use of a watt-balance [4]. Even with the redefinition of the 32

realization of the kilogram, it will be necessary to compare mass standards through comparison 33

weighings [5]. 34

Currently, in the United States, a calibration laboratory’s reference kilogram might be compared to an 35

accredited laboratory’s master kilogram standard, which in turn is compared to a National Institute of 36

Standards and Technology (NIST) working kilogram standard, which is compared to the NIST K20 37

standard, and the K20 is compared to the international prototype kilogram. Even with the redefinition of 38

the realization of the kilogram, most National Metrological Institutes (NMI) would still perform mass 39

calibrations to a working mass standard, commonly called “weight.” 40

1.2 Conventional Result of the Value of Weighing in Air 41

In commerce, weights are commonly used to calibrate balances. When the weights are calibrated, they 42 are compared to another weight with a lower overall measurement uncertainty. Most mass 43 measurements are actually force measurements, measuring the normal force due to gravity of a weight 44 on an electronic balance. The problem is that the actual force of the weight is the force of the weight 45 minus the buoyancy force. In commercial operations, the conventional result of weighing in air is often 46 used as the reference mass value [6]. 47 Nowadays, most weight standards are made of ferrous alloys, with densities ranging from about 7400 48 kg/m3 to 8700 kg/m3, depending on the type of alloy. Stainless steels tend to be closer to 8000 kg/m3. 49 Conventional result provides a mass value that is corrected for buoyancy of air at sea level, and for a 50 reference weight with a density of 8000 kg/m3, as shown in the definition from the International 51 Organization of Legal Metrology, International Document 28 (OIML D-028): 52 53

The conventional mass value of a body is equal to the mass mc of a standard 54 that balances this body under conventionally chosen conditions. The unit of 55 the quantity “conventional mass” is the kilogram... 56 57 The conventionally chosen conditions are: tref = 20 C 0 = 1.2 kg m-3

c = 58 8000 kg m-3 59

60

Where tref is the reference temperature, 0 is the density of the atmosphere at conventional conditions, 61

and c is the density of the standard. Figure 1 illustrates this definition for conventional mass value. 62

[Figure 1 Here] 63

When the unknown and the reference are perfectly balanced, the net force on each object is equal. 64

Figure 2 shows the net force on a weight. This is the gravitational force on the weight, minus the force of 65

buoyancy on the weight. The instrument measures Fnet. 66

[Figure 2 Here] 67

Mass comparisons use a single-pan electronic balance rather than the dual-pan balance as illustrated in 68

Figure 1, so the balance measures force rather than mass. Weight calibrations are performed by 69

substitution weighings [7],[8]. Substitution weighing compares the force applied by the unknown weight 70

to the force applied by the reference weight. In essence, with an unknown weight X, and a reference 71

weight S, the mass of the unknown is the difference from the comparison plus the mass of the 72

reference, along with a correction for differences in buoyancy force between the two weights. The 73

process for making buoyancy corrections is well described in a number of documentary standards [9]. 74

The mass of the unknown, mX, is calculated from: 75

X Sm m (1) 76

Where mS is the mass of the standard, is a comparison correction for the difference in masses and the 77

effects of buoyancy: = O + B. When comparing two different weights, the observed difference 78

would be: 79

( ) ( )O B X X a S S am g V g m g V g (2) 80

Where VX is the volume of the unknown, a is the density of air where the weighing is being made, VS is 81

the volume of the reference, and g is the local gravitational acceleration. In addition to the published 82

NIST Standard Operating Procedures (SOPs), a number of references, such as Reference [1], provide 83

detailed derivations that support the published equations. 84

The conventional result of the value of weighing in air is called “conventional result” or “conventional 85

mass” for short. In this paper, “true mass” is used to further differentiate mass value from the 86

conventional result. The use of conventional result when calibrating commercial balances avoids many 87

complexities associated with buoyancy corrections on commercial balances [1]. Equation (3) is 88

commonly used to calculate the conventional result mcX when the true mass mX is known, and X is the 89

density of the object (in kg/m3). 90

1.21

1.21

8000

XcX Xm m

(3) 91

Note that although the term 1.2

1X

uses the density of the unknown weight, the correction for 92

buoyancy physically originates from the volume of the unknown weight’s displacement of air. 93

1.3 Uncertainty of the Conventional Result of the Value of Weighing in Air 94

OIML D-028 [6] provides a recommendation for estimating the uncertainty of the conventional value of 95

the result of weighing in air; Equation 13 in the D-028 document is reproduced here as Equation (4). 96

2 2 2 2( ) ( )cX W cS b bau m u u m u u (4) 97

Here, u(mcX) is the standard uncertainty of the conventional result, uW is the standard uncertainty of the 98

weighing process, u(mcS) is the standard uncertainty of the conventional result of the reference weight, 99

ub is the standard uncertainty of the buoyancy correction, and uba is the standard uncertainty of the 100

balance. 101

Many labs report mass value (true mass) and not the conventional result. To obtain the conventional 102

result, these laboratories apply Equation (3) to the calibrated mass result. Equation (3) does not provide 103

guidance for reporting uncertainty. Equation (4) ignores the correlation between the buoyancy 104

correction in determining mass and the buoyancy correction in calculating the conventional result. This 105

problem was observed in 2013 [10]; Gläser [11], and later Bich and Malengo [12], stated that the 106

buoyancy corrections were correlated. The remainder of this paper illustrates some unintuitive 107

implications of the conventional result’s uncertainty. 108

The uncertainty of the measurement for the conventional value of the result is estimated using the 109

Guide to the expression of uncertainty in measurement, commonly referred to as the GUM [13]. NIST 110

Technical Note TN1297 [14] provides more detailed guidance than the GUM. The estimate of 111

measurement uncertainty must incorporate all influence factors into the evaluation of the uncertainty; 112

some of the evaluations are statistical, and others are non-statistical. The estimate then combines both 113

statistical and non-statistical evaluations. In general, having a measurement equation is a necessary 114

starting point for estimating measurement uncertainty. In the case of mass calibration, the 115

measurement equation for substitution weighing is Equation (1). If reporting the conventional result, 116

the measurement equation would be Equation (1) combined with Equation (3). At first glance, this 117

would appear to be:

1.21

( )1.2

18000

xcX S O Bm m

(5) 118

Section 5.1.2 of the GUM suggests combining the standard uncertainty for the measurement equation, 119

Equation (5), by calculating: 120

2

2 2

1

( ) ( )N

c i

i i

fu y u x

x

(6) 121

Where uc is the combined standard uncertainty, y is the measurand, f is the measurement equation, and 122

each xi is the individual influence factor in the measurement equation. The GUM points out that this 123

approach is valid only if the individual influence factors are uncorrelated. The GUM does provide 124

guidance for correlated uncertainties, based on adding covariance terms to Equation (6); however, this 125

can become mathematically tedious. References [11] and [12] have shown derivations of the analytical 126

solution. The GUM supplement [15] suggests using Monte Carlo methods to propagate uncertainties. 127

When using a Monte Carlo simulation, it is not necessary to calculate individual weighting factors, 128

because a properly performed simulation automatically assigns the correct weightings. 129

2. Calculating Uncertainties by Monte Carlo Simulation 130

The influence factors identified for the uncertainties of measuring true mass and the conventional result 131

of weighing in air are: 132

Uncertainty of the reference mass 133

Uncertainty of the reference weight’s density 134

Uncertainty of the unknown weight’s density 135

Uncertainty in measurement of air density 136

Uncertainty due to instrument readability 137

Uncertainty due to weighing process 138

These are summarized in Table 1. 139

[Insert Table 1 Here] 140

A Monte Carlo simulation is a virtual experiment: see for example Reference [16]. Monte Carlo methods 141

were initially developed to solve difficult physics problems where closed-form solutions were difficult to 142

obtain, and have become popular as computers have become more powerful. They are especially useful 143

in showing the effects of changing different variables on all of the results of a complex calculation, 144

without the cost of running an actual experiment multiple times. One has to be careful in running a 145

Monte Carlo simulation because the simulation relies on the quality of random numbers generated by 146

the computer (more properly, pseudorandom numbers) [17],[18]. The more important property of a 147

pseudorandom number generator for a Monte Carlo simulation is the distribution of the numbers [19]. 148

Many computer science courses demonstrate the Monte Carlo method by estimating the value of π. This 149

is illustrated in Figure 3 [20]. A good Monte Carlo simulation for estimating measurement uncertainty 150

not only requires the correct measurement equation, but also a good pseudorandom number generator. 151

For the purposes of this paper, Excel 2010 is used for Monte Carlo simulations. The simulation tool is 152

available in the online appendix. The Excel tool that the authors of this paper developed is illustrated in 153

Figure 4. 154

[Insert Figures 3 and 4 Here] 155

The statistical procedures and distributions of earlier versions of Excel are known to have inaccuracies 156

[21],[22],[23]. An evaluation of Excel 2010’s random number generator (RAND()) was more satisfactory 157

than those of previous versions [24]. While the evaluation was not comprehensive, due to the need to 158

test 235 pseudorandom numbers, they found that the 2010 pseudorandom number generator passed 159

the distribution tests. This evaluation concluded that the 2010 random number generator performs 160

similarly to a Mersenne Twister pseudorandom generator, which is considered adequate for Monte 161

Carlo simulations. 162

International Recommendation 111 (OIML R111) provides descriptions of weight classes suitable for 163

different purposes, ranging from reference weights used at national metrological institutes to field 164

weights used to calibrate balances in various commercial settings. There are other weight classes used; 165

American Society for Testing and Materials (ASTM) E617-13 is one such common alternative to the OIML 166

weights [25]. In both R111 and in ASTM E617, the standards specify acceptable limits for the properties 167

of the weight (properties primarily being allowable deviations from nominal mass, density, shape, and 168

surface finish). This paper will focus on the uncertainties for high-quality weights: This would be OIML E2 169

class or better. 170

Monte Carlo simulations were used to estimate the measurement uncertainty for calibrating 1 kg mass 171

standards with E2 characteristics, per OIML R111. It was assumed that the mechanical and magnetic 172

properties were acceptable, and a range of acceptable densities (7810 kg/m3 to 8210 kg/m3) were used, 173

along with a range of acceptable uncertainties of density per OIML R111 (0.08% to 1.8 % of actual 174

density, of an 8000 kg/m3 weight) and a range of air densities from near sea level (1.2 kg/m3) to the 175

highest capital city in the world (La Paz, Bolivia at an altitude of approximately 3600 m with air density 176

of approximately 0.77 kg/m3). The range 0.08% to 1.8% of the actual density is chosen based on the 177

density estimation methods allowable for an E2 class of weight per OIML R111-1 Tables B.8 and Section 178

B7.9.3. 179

The uncertainty of the measurement result in a Monte Carlo simulation is the standard deviation of all 180

experimentally determined calculations of that result. The supplement to the GUM suggests using on 181

the order of 106 simulations to get 2 significant figures in the uncertainty at a coverage factor k=2, or 182

doing a convergence study. For this calculation, 500,000 (0.5×106) simulations for each individual case 183

were run. While we did not perform a formal convergence study, at 500,000 simulations, the results 184

from one simulation to the next changed by between 0.04% and 0.1%, with the change usually a 185

difference of 1 or 2 in the third or fourth significant digit, and not in the first or second significant digit. 186

Table 2 shows a sample of ten simulation results at 500,000 trials. 187

[Insert Table 2 Here] 188

Based on the equipment and standards in our lab, and based on possible E2 class unknown weights, the 189

values of the uncertainties simulated are detailed in Table 1. 190

We simulated the uncertainties when performing double substitution per NIST SOP4. SOP4 provides 191

detailed guidance for the actual weighing observations and calculations of buoyancy observations. In 192

double substitution, four mass measurement observations are made. In the option chosen for our 193

laboratory, the first and last measurements are of the reference weight, and the second and third are of 194

the unknown weight. The average of the difference between the unknown and reference weights is 195

taken to correct for comparator drift. This comparison difference and the known mass of the reference, 196

together with corrections for the effects of buoyancy on both the reference weight and unknown 197

weight, are used to calculate the mass of the unknown. 198

The Monte Carlo simulations used the NIST SOP4 double substitution Equation 3.2.2.1, with correction 199

for buoyancy for true mass and omitting sensitivity weights. This is shown in Equation (7): 200

2 1 3 412

1

aS

S

X

a

X

O O O Om

m

(7) 201

Where mX is the measurement result, mS is the mass of the reference, a is the density of air where the 202

calibration is performed, S is the density of the reference, X is the density of the unknown, and O1 203

through O4 are the individual observations of the mass comparator. The simulation then applies a 204

correction to obtain the conventional value of the result of weighing in air, per Equation (3). Because of 205

the definition of the conventional result of weighing in air, the actual density of the unknown weight in 206

implementing equation (3) was used, but the uncertainty of the density of the unknown is used in the 207

calculation of true mass. 208

For each simulated measurement result (trial), the Monte Carlo method calculates the result using a 209

slightly different mass and density for the reference weight, a slightly different air density, and a slightly 210

different density of the unknown weight. As long as the different masses, densities, and instrument and 211

process effects are incorporated in each Monte Carlo trial, a large number of simulations will reflect the 212

actual statistical behavior of the measurement—without actually performing the physical 213

measurements—in the case of this paper, 500,000 trials per set of conditions (such as air density, or 214

density of the unknown, as listed in Table 1). In addition to the factors associated with the environment 215

and the weights, a value of 1 digit of resolution is added to each observation to simulate instrument 216

readability, and the historical process standard deviation of the laboratory is added to the overall 217

measurement result. The type of distribution for the random values is given in Table 1. In Excel, when 218

adding a variation uRECT due to a rectangular distribution, an equation of the form is used: 219

(2 RAND() 1)RECTu width (8) 220

Where width is the half-width of the uncertainty, e.g. for a ±1 kg/m3 rectangular uncertainty, width=1 221

kg/m3 is used. When adding a variation due to a standard uncertainty, uGAUSS, an equation of the form is 222

used: 223

NORM.INV(RAND(), , )GAUSSu x (9) 224

where x is the mean and is the standard deviation for the uncertainty. RAND() is a built-in Excel 225

function that returns a uniformly distributed pseudorandom number between 0 and 1, and NORM.INV is 226

a built-in Excel function that returns a pseudorandom number that matches a normal (Gaussian) 227

probability density function with the chosen mean and standard deviation. 228

The built-in Excel functions AVERAGE and STDEV.S were used to calculate the mean and the standard 229

deviation of the simulation results. Since 500,000 simulations are performed, the difference between 230

STDEV.S and STDEV.P (estimate of standard deviation of the sample and estimate of the standard 231

deviation of the population) is insignificant: 232

1STDEV.S: 0.00141421498...

499,999

1STDEV.P: 0.00141421356... =

500,000

(10) 233

STDEV.S was used on the results of 500,000 simulations, to calculate a standard uncertainty. 234

3. Results of the Monte Carlo Simulation 235

The calibration of a 1 kg unknown weight is simulated, where the true mass result is reported. Equation 236

(3) is then applied to obtain the conventional result. Note that the actual density of the unknown is used 237

in calculating both the true mass and the conventional result. Because 500,000 simulations were run, 238

the standard deviation of the results of all of the simulations represent the standard uncertainty of the 239

calibration. 240

The simulation results are presented in Table 3. The calibration uncertainties are standard uncertainties. 241

For any given set of master density, uncertainty of master density, uncertainty of unknown density, and 242

air density, the standard uncertainty when calculating the conventional result is always smaller than the 243

standard uncertainty of the true mass. This appears paradoxical, because one would expect that 244

applying one more equation (Equation (3)) to a result would add one more factor to the measurement 245

uncertainty. However, Equation (3) is correlated with Equation (7); Equation (6) is only applicable if the 246

uncertainty components are uncorrelated. 247

[Insert Table 3 Here – May need to be separate pages due to size of table] 248

To validate the results of the Monte Carlo simulation with published analytical uncertainty analysis, the 249

conditions of Example 2 from [12] were set as conditions for the simulation. Malengo and Bich found 250

uncertainties of 0.92 mg for the true mass and 0.058 mg for the conventional result, while the 251

simulation gave uncertainties of 0.91 mg and 0.058 mg. Some of the uncertainties the simulation 252

requires, such as the uncertainty of air density, or the uncertainty due to instrument readability, were 253

not included explicitly in the example, which might account for the small difference in uncertainties of 254

true mass. 255

4. Discussion 256

There is another way of considering the uncertainty when calculating uncertainty of the conventional 257

result. This method uses the volume of the reference, and the volume of the unknown, rather than their 258

densities. This may be a more intuitive approach. 259

To estimate the uncertainty in conventional result, consider the measurement model of Equation (1), 260

repeated here: 261

X S O Bm m (11) 262

Where mX is the measurement result, mS is the mass of the standard, O is the observed difference, and 263

B is the correction for buoyancy differences. Whether the calculation is for conventional result or true 264

mass, this equation applies. C is the specific correction to the conventional result, but is otherwise the 265

same as B. 266

If the local air density were 1.2 kg/m3, and the reference S had a density of 8000 kg/m3, C ≡ 0, because 267

of the definition of conventional mass. 268

The uncertainty components for Equation (11) still include the uncertainty mass and volume of S, the 269

uncertainty in volume of X (because the environment is not at 1.2 kg/m3), the uncertainty in density of 270

air, and the process standard deviation. However, the uncertainties associated with the volume of the 271

reference, the volume of the unknown, and the uncertainty of the density of air should be smaller for 272

the conventional result. 273

When calculating the conventional result for weighing in air, the effect of the volume of X (VX) is referred 274

to the nominal volume at 8000 kg/m3; similarly for VS. In addition, the effect of local air density is 275

relative to 1.2 kg/m3 rather than correcting to vacuum. 276

Consider a comparison of X with S in air at 1.2 kg/m3. For the moment, assume X has a density of 8000 277

kg/m3. The conventional result would be: 278

1.2 1.2( ) ( )

cX S O C

O X X S S

m m

m V m V

(12) 279

where 1.2 is 1.2 kg/m3 and the observed difference includes buoyancy terms. For true mass, it would be 280

necessary to perform a complete set of buoyancy corrections. For conventional result, one only corrects 281

if the density of S is not 8000 kg/m3. Assume that VS > V8000 (i.e. the density S < 8000 kg/m3). The 282

buoyant force on S is larger than it is in standard conventional conditions. S does not balance as much 283

weight as it should in conventional conditions, so a correction to mcX is needed to increase the weight of 284

mcX from the observed difference: 285

8000 1.2( )C SV V (13) 286

Now assume that the air density a is not 1.2 kg/m3; a < 1.2. The buoyant force on S is smaller than it is 287

in standard conditions, so S balances a larger weight than it should. A negative correction to C is 288

needed: 289

8000 1.2( )( )C S aV V (14) 290

Finally, assume VS=V8000, but VX > V8000 (X < 8000 kg/m3), and again, the air density is not 1.2 kg/m3; a < 291

1.2. In this case, the observation of X has a smaller buoyant force, add a positive correction to C is 292

needed: 293

8000 1.2( )( )C X aV V (15) 294

Combining Equations (13), (14), and (15): 295

8008000 1.2 8000 1.2

8000 1.2 1

0 1.2

.2

( )( ) ( ) (( )

) ( )( )

)

(

C X aS S a

S SC X a

V V V V

V V V V

V V

(16) 296

The first term in Equation (16), (V8000-VS) is the correction for mS to mcS to obtain the conventional result 297

for the standard S. 298

If the conventional value for the reference S is used in weighing X, the correction is simply: 299

1.2( )( )C X S aV V (17) 300

The uncertainties associated with the buoyancy correction are uncertainties in VS, VX, and a, the 301

uncertainty of the reference S, and the process standard deviation. 302

The sensitivity of C with respect to VS is now easy to calculate. By inspection, 303

1.2C

a

SV

(18) 304

Similarly, the sensitivity with respect to VX is 305

1.2C

a

XV

(19) 306

And the sensitivity with respect to a is 307

CX S

a

V V

(20) 308

The combined uncertainty due to the buoyancy correction for the conventional result is: 309

2 22

2 2 2 2 2

ProcessS X a S

C C CV V m

S X a

u u u u uV V

(21) 310

Note that the first three terms in Equation (21) are the uncertainties associated with buoyancy 311

corrections for the conventional result, and the magnitude of these terms are smaller than when 312

calculating true mass. 313

In summary, when calibrating true mass, a dominant term in the uncertainty is the uncertainty of the 314

volume of the unknown. When calibrating conventional result, the effect of the volume uncertainty of 315

the unknown is greatly reduced. In addition, the uncertainty of the volume of the reference is reduced 316

to the difference from the volume to the 8000 kg/m3 reference density. 317

5. Conclusions 318

When calibrating mass for low measurement uncertainties, it is generally necessary to correct for 319

volume differences between the reference and the unknown weights. When reporting the conventional 320

result, because the definition of conventional result explicitly states reference density and standard 321

conditions, the uncertainty of the buoyancy correction for the conventional result is smaller than the 322

uncertainty of the buoyancy correction for true mass. Because these uncertainties are correlated, they 323

cannot be added as independent terms. The covariance terms must be accounted for, and the 324

correlation is negative. 325

The use of conventional results can lead to an uncertainty evaluation that appears abnormally low; 326

however, proper application of the GUM requires this low uncertainty because of the correlation of the 327

volume of the unknown to both the calibration of true mass and the following conversion to 328

conventional result. This might lead to erroneous conclusions in interlaboratory comparisons, if the 329

requested measurand is a conventional value. This simulation tool has not been used to simulate an 330

interlaboratory comparison with the worst-case condition for assumed density at high altitude, or a 331

reference lab at sea level, but anyone can use this simulation tool to explore the possible Type B 332

uncertainties in calibrating mass and conventional results. 333

6. Acknowledgements 334

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia 335

Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of 336

Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. 337

Certain commercial equipment, instruments, or materials are identified in this paper in order to 338

adequately describe the experimental procedure. Such identification does not imply recommendation or 339

endorsement by the authors or by Sandia National Laboratories, nor does it imply that the materials or 340

equipment identified are the only or best available for the purpose. 341

The authors acknowledge the support of the Sandia STAR Fellowship Program from the Community 342

Involvement department at Sandia National Laboratories, and discussions with Rick Mertes, Eric Forrest, 343

and Sam Ramsdale. 344

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[25] ASTM, "Standard Specification for Laboratory Weights and Precision Mass Standards," ed: ASTM 396 International, 2013. 397

398

Appendix 1. Excel Simulation Tool 399

The Excel simulation tool is available in the online version of this article. The printed appendix shows the 400

variables used and the formulas used in the simulation. The computer used for these simulations was a 401

Dell OptiPlex 980, with Windows 7 Enterprise 64 bit, 4 GB RAM, and an Intel core i7 870 processor at 402

2.93 GHz. A simulation of 500,000 runs has a file size of approximately 60 MB. The version provided for 403

online access only has 22 runs, and is 16 KB. While the simulation only takes 2 seconds to run, opening 404

the file takes approximately 17 seconds, and saving the file takes approximately 9 seconds. When 405

debugging the simulation, it is advantageous to have a much smaller number of runs. Increasing the 406

number of runs is simple by using copy/paste and using Excel’s functions to go to a specific cell. 407

Table 4 shows the inputs into the simulation. For each simulation, a reference mass is used, mR, that is 408

varied with the uncertainty uR of one of the reference references, using a rectangular distribution to 409

reflect the expected variation over the calibration interval. The density R provided by the calibration 410

certificate is used with a standard uncertainty uR because densities are not expected to vary—the 411

density of the weight is not known perfectly, but the density will not change over time. T is the density 412

of the unknown, which is within the acceptable range for OIML E2 weights. The uncertainty uT of the 413

density of the unknown is based on the acceptable methods for obtaining density of an E2 weight. a is 414

the density of air in the laboratory where the calibration is performed; ua is the uncertainty of the 415

measurement of air density, based on the instruments used for measuring air pressure, temperature, 416

and relative humidity. uinstR is the uncertainty due to instrument readability, and uW is the uncertainty 417

due to the process (from the standard deviation control chart). O1 through O4 are simulated 418

observations. Table 5 shows five outputs of the simulation. 419

[Insert Tables 4 and 5 Here] 420

The actual reference mass mR used is calculated with: =mR+uR*(RAND()*2-1). The actual density of the 421

reference R is calculated with: =(NORM.INV(RAND(),ϱR,uϱR)). The actual density of the unknown T is 422

calculated with=(NORM.INV(RAND(),ϱT,uϱT)). The actual density of air a is calculated with: 423

=NORM.INV(RAND(),ϱa,uϱa). Calibrated mass is calculated with: =NORM.INV(RAND(),0,uW)+(F3*(1-424

I3/G3)+((((O2_+uinstR*(RAND()*2-1))-(O1_+uinstR*(RAND()*2-1)))+((O3_+uinstR*(RAND()*2-1))-425

(O4_+uinstR*(RAND()*2-1))))/2))/(1-I3/H3), where F3, I3, G3, and H3 are the values of mR , a, R, and T, 426

respectively, for this particular simulation. The row number, 3 in this case, changes depending on which 427

simulation is run. The conventional result is calculated with: =J3*(1-ϱ0/H3)/(1-ϱ0/8000), where J3 is the 428

calibrated mass calculation. Note that O1 in the Excel formula has a label O1_, to differentiate from the 429

cell address “O1,” as do the other observations. 430

Once all the calculations in the spreadsheet are validated, copy the calculations from the first row into a 431

number of rows equal to the desired number of simulations to be run. Automatic Recalculation should 432

be turned off. Manual recalculation can be used and the statistics of the result can be evaluated. With 433

500,000 simulations, the standard deviation changes in the fourth significant digit, which is consistent 434

with performing a convergence study per the GUM Supplement recommendation. MathType 435

Professional (the upgrade to Microsoft Word’s equation editor) can be used to check the nesting of 436

parentheses in Excel formulas. 437

All Figures and Tables 438

Figures 439

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/m3 and atmospheric

air density is exactly 1.2 kg/m3.

Reference Weight

Unknown Weight

440 441

Figure 2: Free Body Diagram showing forces acting upon a weight in a balance or mass comparator. The

net force measured by the instrument (Fnet) is the gravitational force on the weight (Fg) minus the

buoyancy force (Fb).

Fg = mA g

Fb = Vaa g

Fnet = Fg – Fb

Weight A

442 443

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.

444 445

446

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.

447 448

449

Tables 450

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 0.080 mg Rectangular

Density of reference (0.0675, 0.7) kg/m3 Standard (Gaussian)

Density of unknown (5, 10, 140, 170, 200) kg/m3 Rectangular

Measurement of air density 0.00023 kg/m3 Standard

Instrument readability 0.001 mg Rectangular

Weighing process 0.0015 mg Standard

451 452

Table 2: Informal convergence verification for 500,000 trials.

Standard Deviation of

mT (mg)

Standard Deviation of

mcT (mg)

Sim 1 1.97025803 0.04623833

Sim 2 1.97279429 0.04624658

Sim 3 1.96897933 0.04620828

Sim 4 1.96888547 0.04619908

Sim 5 1.96908214 0.04624285

Sim 6 1.96802939 0.04625341

Sim 7 1.97006932 0.04624521

Sim 8 1.96852055 0.04621532

Sim 9 1.96960627 0.04623035

Sim 10 1.97037043 0.04621815 453

454

455

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 456

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. 457

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 (%)

7879.466 0.068 0.00086% 7810 0.03% 2.5 1.2004 0.068 0.046 32%

7879.466 0.068 0.00086% 7810 0.03% 2.5 0.98635 0.061 0.047 23%

7879.466 0.068 0.00086% 7810 0.03% 2.5 0.7675 0.056 0.049 12%

7879.466 0.068 0.00086% 7810 0.13% 10.0 1.2004 0.20 0.046 77%

7879.466 0.068 0.00086% 7810 0.13% 10.0 0.98635 0.17 0.058 66%

7879.466 0.068 0.00086% 7810 0.13% 10.0 0.7675 0.13 0.085 37%

7879.466 0.068 0.00086% 7810 1.28% 100.0 1.2004 1.97 0.046 98%

7879.466 0.068 0.00086% 7810 1.28% 100.0 0.98635 1.6 0.35 78%

7879.466 0.068 0.00086% 7810 1.28% 100.0 0.7675 1.3 0.71 44%

7879.466 0.068 0.00086% 8030 0.03% 2.5 1.2004 0.066 0.046 29%

7879.466 0.068 0.00086% 8030 0.03% 2.5 0.98635 0.060 0.047 22%

7879.466 0.068 0.00086% 8030 0.03% 2.5 0.7675 0.055 0.049 11%

7879.466 0.068 0.00086% 8030 0.12% 10.0 1.2004 0.19 0.046 76%

7879.466 0.068 0.00086% 8030 0.12% 10.0 0.98635 0.19 0.046 76%

7879.466 0.068 0.00086% 8030 0.12% 10.0 0.7675 0.13 0.081 36%

7879.466 0.068 0.00086% 8030 0.87% 70.0 1.2004 1.303 0.046 96%

7879.466 0.068 0.00086% 8030 0.87% 70.0 0.98635 1.1 0.24 78%

7879.466 0.068 0.00086% 8030 0.87% 70.0 0.7675 0.84 0.47 43%

7879.466 0.068 0.00086% 8210 0.03% 2.5 1.2004 0.064 0.046 28%

7879.466 0.068 0.00086% 8210 0.03% 2.5 0.98635 0.059 0.047 20%

7879.466 0.068 0.00086% 8210 0.03% 2.5 0.7675 0.054 0.049 10%

7879.466 0.068 0.00086% 8210 0.12% 10.0 1.2004 0.18 0.046 75%

7879.466 0.068 0.00086% 8210 0.12% 10.0 0.98635 0.15 0.056 64%

7879.466 0.068 0.00086% 8210 0.12% 10.0 0.7675 0.12 0.079 36%

7879.466 0.068 0.00086% 8210 1.04% 85.0 1.2004 1.5 0.046 97%

7879.466 0.068 0.00086% 8210 1.04% 85.0 0.98635 1.2 0.27 78%

7879.466 0.068 0.00086% 8210 1.04% 85.0 0.7675 0.97 0.55 43%

8038.5 0.70 0.00871% 7810 0.03% 2.5 1.2004 0.069 0.048 30%

8038.5 0.70 0.00871% 7810 0.03% 2.5 0.98635 0.062 0.048 23%

8038.5 0.70 0.00871% 7810 0.03% 2.5 0.7675 0.057 0.050 11%

8038.5 0.70 0.00871% 7810 0.13% 10.0 1.2004 0.20 0.048 76%

8038.5 0.70 0.00871% 7810 0.13% 10.0 0.98635 0.17 0.059 65%

8038.5 0.70 0.00871% 7810 0.13% 10.0 0.7675 0.13 0.085 37%

8038.5 0.70 0.00871% 7810 1.28% 100.0 1.2004 1.97 0.048 98%

8038.5 0.70 0.00871% 7810 1.28% 100.0 0.98635 1.6 0.35 78%

8038.5 0.70 0.00871% 7810 1.28% 100.0 0.7675 1.3 0.71 44%

8038.5 0.70 0.00871% 8030 0.03% 2.5 1.2004 0.067 0.048 28%

8038.5 0.70 0.00871% 8030 0.03% 2.5 0.98635 0.061 0.048 21%

8038.5 0.70 0.00871% 8030 0.03% 2.5 0.7675 0.056 0.050 10%

8038.5 0.70 0.00871% 8030 0.12% 10.0 1.2004 0.19 0.048 75%

8038.5 0.70 0.00871% 8030 0.12% 10.0 0.98635 0.16 0.058 64%

8038.5 0.70 0.00871% 8030 0.12% 10.0 0.7675 0.13 0.082 36%

8038.5 0.70 0.00871% 8030 0.87% 70.0 1.2004 1.3 0.048 96%

8038.5 0.70 0.00871% 8030 0.87% 70.0 0.98635 1.1 0.24 78%

8038.5 0.70 0.00871% 8030 0.87% 70.0 0.7675 0.84 0.47 43%

8038.5 0.70 0.00871% 8210 0.03% 2.5 1.2004 0.065 0.048 27%

8038.5 0.70 0.00871% 8210 0.03% 2.5 0.98635 0.060 0.048 20%

8038.5 0.70 0.00871% 8210 0.03% 2.5 0.7675 0.055 0.050 10%

8038.5 0.70 0.00871% 8210 0.12% 10.0 1.2004 0.18 0.048 74%

8038.5 0.70 0.00871% 8210 0.12% 10.0 0.98635 0.15 0.057 63%

8038.5 0.70 0.00871% 8210 0.12% 10.0 0.7675 0.12 0.079 36%

8038.5 0.70 0.00871% 8210 1.04% 85.0 1.2004 1.5 0.048 97%

8038.5 0.70 0.00871% 8210 1.04% 85.0 0.98635 1.2 0.27 78%

8038.5 0.70 0.00871% 8210 1.04% 85.0 0.7675 0.97 0.55 44%

458

23 | P a g e

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.

mR true mass of ref. weight 1.00 kg

uR unc. of true mass of ref. weight 0.000000080 kg rectangular

mCR conv. mass of ref. weight 1.0000007 kg

umcR unc. of conv. mass of ref. weight 8.0006E-08 kg calculated

ϱR density of reference weight 8038.50 kg/m^3

uϱR unc. of density of ref. weight 0.70 kg/m^3 gaussian (k=1)

mT true mass of unknown weight 0.9999980 kg

mCT 1.000002 kg

ϱT density of unknown weight 8210.00 kg/m^3

uϱT unc. of density of unknown weight 170.00 kg/m^3 rectangular

ϱa density of air 0.77 kg/m^3

uϱa unc. of density of air 2.30E-04 kg/m^3 gaussian std unc per PSL analysis

ϱ0 air density in conv. conditions 1.20 kg/m^3

uinstR unc. due to instrument readability 1.00E-09 kg rectangular

uW unc. due to weighing process 1.50E-09 kg gaussian (standard)

O1 (R1) first observation of ref. weight 1.000000000 kg

O2 (T1) first observation of unknown weight 0.999999999 kg

O3 (T2) second observation of unknown weight 1.000000000 kg

O4 (R2) second observation of ref. weight 1.000000002 kg

459 460

461

24 | P a g e

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.

mR ϱR ϱT ϱa mT mCT

1.0000000 8.035983E+03 8.076613E+03 7.676704E-01 0.999999525 1.00000095

1.0000000 8.037909E+03 8.323287E+03 7.677918E-01 0.999996745 1.00000257

1.0000000 8.038325E+03 8.197844E+03 7.675997E-01 0.999998121 1.00000174

1.0000000 8.037398E+03 8.166938E+03 7.671671E-01 0.999998470 1.00000154

1.0000000 8.036978E+03 8.063591E+03 7.678058E-01 0.999999662 1.00000085 462

463