SOP No. 4

**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

1.0 Introduction

1.1. Purpose

The double substitution procedure is one in which a standard and an
unknown weight are intercompared twice to determine the average
difference between the two weights. Errors in any built‑in weights or in
the balance indications are eliminated by using the balance only as a
comparator and by calibrating the balance indications over the range of
use for the measurement with a sensitivity weight. Accordingly, the
procedure is especially useful for high accuracy calibrations. The
procedure does not incorporate measurement control steps to ensure the
validity of the standards and the measurement process; additional
precautions must be taken.

> 1.2. Prerequisites

1.2.1. Verify that valid calibration certificates are available for the
standards used in the test.

2.  Verify that the standards to be used have sufficiently small
    standard uncertainties for the intended level of calibration.
    Primary standards should not be used at this level.

3.  Verify that the balance is in good operating condition with
    sufficiently small process standard deviation as verified by a valid
    control chart or preliminary experiments to ascertain the
    performance quality when a new balance is put into service.

4.  Verify that the operator is experienced in precision weighing
    techniques and has had specific training in SOP 2, SOP 4, SOP 29,
    GMP 4, and GMP 10.

    1.2.5. Verify that the laboratory facilities comply with the
    following minimum conditions to meet the expected uncertainty
    possible with this procedure.

Table 1. Environmental conditions

+-------+----------------------------------------------+---------------+
| Ec    | ## Temperature                               | Relative      |
| helon |                                              | Humidity      |
|       |                                              |               |
|       |                                              | (percent)     |
+-------+----------------------------------------------+---------------+
| ## II | ## 20 °C to 23 °C,                           | 40 to 60 ± 10 |
|       |  a set point ± 2 °C, maximum change 1.0 °C/h | / 4 h         |
+-------+----------------------------------------------+---------------+
| #     | ## 18 °C to 27 °C, maximum change 2.0 °C/h   | 40 to 60 ± 20 |
| # III |                                              | / 4 h         |
+-------+----------------------------------------------+---------------+

2\. Methodology

2.1. Scope, Precision, Accuracy

This method is applicable to all weighings utilizing a mass comparator,
a single‑pan mechanical balance, a full electronic balance, or a balance
that combines digital indications with the use of built‑in weights
(combination balance). The precision will depend upon the sensitivity of
the balance and the care exercised to make the required weighings. The
accuracy achievable with this procedure depends on the accuracy of the
calibration of the working standards and the precision of the
intercomparison.

> 2.2. Summary
>
> The balance is adjusted if necessary, to obtain balance indications
> for all measurements that will be within the range of the optical
> scale or digital indications of the balance without changing the dial
> settings for the built‑in weights, if present. The standard and the
> test weight are each weighed. A small, calibrated weight, called a
> sensitivity weight, is added to the test weight and these are weighed.
> The standard and the same sensitivity weight are then weighed. The
> latter two weighings provide both second weighings of the standard and
> the test weight as well as a determination of the sensitivity of the
> balance under the load conditions at the time of the intercomparison.
> All weighings are made at regularly spaced time intervals to average
> out any effects due to instrument drift.
>
> The double substitution procedure is the same for all of the balances
> mentioned above, but the adjustment of the balance to prepare for the
> intercomparison and the selection of the sensitivity weight differs
> slightly depending upon the balance used. When steps specific to a
> particular balance are required, they are given in subsections of the
> procedure identified by a, b, and c along with the balance type.
>
> 2.3. Apparatus/Equipment Required
>
> 2.3.1. Precision balance with sufficient capacity and sensitivity for
> the calibrations planned.
>
> 2.3.2. Calibrated working standard and sensitivity weights with recent
> calibration values that are traceable to NIST.
>
> 2.3.3. Calibrated small standard weights with recent calibration
> certificates and values traceable to NIST to be used as tare weights.
>
> 2.3.4. Uncalibrated weights to be used to adjust the balance to the
> desired reading range.
>
> 2.3.5. Forceps to handle the weights, or gloves to be worn if the
> weights are moved by hand.
>
> 2.3.6. Stop watch or other timing device to observe the time of each
> measurement.

7.  Calibrated barometer accurate to ± 66.5 Pa (0.5 mm Hg) with recent
    calibration values traceable to NIST to determine air pressure.

8.  Calibrated thermometer accurate to ± 0.10 °C with recent calibration
    values traceable to NIST to determine air temperature.

    2.3.9. Calibrated hygrometer accurate to ± 10 percent with recent
    calibration values traceable to NIST to determine relative
    humidity.[^1]

```{=html}
<!-- -->
```
3.  Symbols

    **Table 2. Symbols used in this procedure**

  -----------------------------------------------------------------------
  Symbol   Description
  -------- --------------------------------------------------------------
  *S*      standard reference weight

  *X*      weight to be calibrated

  *t*      small calibrated tare weight, A subscript *s* or *x* is used
           to indicate the larger weight with which it is associated

  *sw*     small calibrated weight used to evaluate the sensitivity of
           the balance

  *M*      the mass (true mass) of a specific weight. Subscripts *s*,
           *x*, *t*, *sw* are used to identify the weight (equals Nominal
           plus Correction)

  *N*      the nominal value of a specific weight. Subscripts *s*, *x*,
           are used to identify the weight.

  *C*      the correction for a specific weight. Subscripts *s*, *x*, are
           used to identify the weight.

  *CM*     the conventional mass of a specific weight. Subscripts *s*,
           *x*, *t*, *sw* are used to identify the weight.

  *ρ~a~*   density of air at time of calibration

  *ρ~n~*   density of normal air (1.2 kg/m^3^)

  *ρ*      density of masses; subscripts *s*, *x*, *t~s~*, *t~x~*, *sw*
           are used to identify the weight.
  -----------------------------------------------------------------------

4.  Procedure

    1.  Preliminary Procedure

        1.  Place the test weight and standards in the balance chamber
            or near the balance overnight to permit the weights and the
            balance to attain thermal equilibrium.

        2.  Conduct preliminary measurements to obtain an approximate
            value for the difference between the standard and the
            unknown, to determine where the readings occur on the
            balance, to determine if tare weights are required, to
            determine the sensitivity weight that must be used, and to
            determine the time interval required for the balance
            indication to stabilize.

> Tare weights are rarely needed for high precision mass standards. If
> tare weights are required, carry tare weights, *t~s~* and *t~x~*, with
> the standard and the unknown, *S* and *X*, respec­tively. The tare
> weights must be calibrated standards with valid uncertainties that are
> evaluated in the process of determining calibration uncertainties. The
> standard and its tare weight, *S* + *t~s~*, should be \"nearly the
> same mass\" as the unknown with its tare weight, *X* + *t~x~*.
> \"Nearly the same mass\" depends upon the balance used (See GMP 14,
> Table 1). Select *t~s~* and *t~x~* such that the difference in mass
> between *S* + *t~s~* and *X* + *t~x~* is:

a.  Single‑pan mechanical balance ‑ less than $\frac{1}{\text{10}}$ the
    range of the optical scale.

b.  Full electronic balance ‑ less than 0.05 percent of the balance
    capacity.

c.  Combination balance ‑ less than $\frac{1}{\text{10}}$ the range of
    the digital indications.

d.  Mass comparator -- less than $\frac{1}{\text{10}}$ digital range

> A sensitivity weight must be used on equal-arm balances, and is
> normally used on single-pan mechanical and electronic balances, to
> ensure that the measured differences determined through the use of the
> optical scale or electronic range have valid accuracy and traceability
> (See GMP 14, Table 2). (e.g., The optical scale is *calibrated* each
> time the procedure is performed through the use of a sensitivity
> weight). The uncertainty of the sensitivity weight does not generally
> need to be included in calculations of uncertainty since the
> uncertainty value is distributed across its range of use.
>
> If a sensitivity weight will be used, select one that is:

a.  Single‑pan balance ‑ between ¼ and ½ the range of the optical scale,
    and at least 4 times the mass difference between *X* and *S*.

b.  Full electronic balance ‑ at least 4 times the mass difference
    between *X* and *S*, but not exceeding 1 percent of the balance
    capacity.

c.  Combination balance ‑ between ¼ and ½ the range of the digital
    indications, and at least 4 times the mass difference between *X*
    and *S*.

d.  Mass comparator -- at least 4 times the mass difference between *X*
    and *S*, but not exceeding ½ of the digital range.

> A sensitivity weight is not required if the electronic mass comparator
> that is used has been tested (with supporting data available) to
> determine that the balance has sufficient accuracy, resolution,
> repeatability, and stability so that no advantage is gained through
> the use of a sensitivity weight. For example, any possible errors must
> be less than what contributes to the uncertainty. When a mass
> comparator is used without a sensitivity weight, the sensitivity must
> be periodically verified and documented.
>
> 2.5.1.3. Determine which optional sequence will be used, A or B.
> Optional sequence A uses the standard on the balance for the first and
> fourth observations and the unknown on the balance for the second and
> third observations; this is often called the "*SXXS*" sequence.
> Optional sequence B starts with the unknown on the balance first and
> last with the standard on the balance for the second and third
> observations; this is often called the "*XSSX*" sequence. The primary
> advantage of sequence B is less handling of the mass standards. The
> advantage of sequence A is in the case where the unknown is a
> summation of weights that require careful arrangement on the balance
> pan only once.
>
> 2.5.1.4. Adjust the single pan balance or the combination balance so
> the first two readings of the double substitution fall in the first
> quarter of the optical scale or digital indications. The zero
> adjustment and tare adjustment may be used. Small weights may be
> placed on the balance pan to reach the desired reading range. These
> weights remain on the pan throughout the double substitution. Once the
> balance has been adjusted to the desired position, neither the balance
> dials, the zero and tare adjustments, nor the small weights placed on
> the balance pan, are to be changed during the measurement.
>
> 2.5.1.5. If the balance is equipped with a pan arrestment mechanism,
> arrest the pan between each observation.

2.5.2. Measurement Procedure, Optional Sequence A (*SXXS*)

Table 3. Optional Sequence A

+----------------------+-----------------------+-----------------------+
| Measurement No.      | Weights on Pan        | Observation           |
+----------------------+-----------------------+-----------------------+
| 1                    | *S + t~s~*            | *O~1~*                |
+----------------------+-----------------------+-----------------------+
| 2                    | *X + t~x~*            | *O~2~*                |
+----------------------+-----------------------+-----------------------+
| 3                    | *X + t~x~ + sw*       | ## *O~3~*             |
+----------------------+-----------------------+-----------------------+
| 4                    | *S + t~s~ + sw*       | *O~4~*                |
+----------------------+-----------------------+-----------------------+

> All observations should be recorded on suitable data sheets, such as
> those in the appendix. Record the laboratory ambient temperature,
> barometric pressure, and relative humidity.

#### 2.5.2.1. Observation 1. Place the standard weight(s), *S*, along with *t~s~* on the balance pan. If equipped with a pan arrest­ment mechanism, release the balance pan. When the pan is released, start the stop‑watch and record observation *O~1~* once the balance indication has stabilized.

#### 2.5.2.2. Observation 2. Remove weight(s) *S* and *t~s~* and replace with test weight *X* and its tare weight, *t~x~*. Release the pan, time the interval, and record observation *O~2~*.

#### 2.5.2.3. Observation 3. Add the sensitivity weight, *sw*, to the weights of observation 2. Release the pan, time the interval, and record observation *O~3~*.

#### 2.5.2.4. Observation 4. Remove weights *X* and *t~x~* and replace with *S* and *t~s~*. The sensitivity weight, *sw*, remains on the balance pan. Release the pan, time the interval, and record observation *O~4~*.

#### 

#### 2.5.2.5. Compare the two differences (*O~2~ -- O~1~*) and (*O~3~ -- O~4~*); they should not differ from one another by more than 2 standard deviations of the balance for this process and load. If this difference is exceeded, reject the data and redo the measurements. Investigate possible causes of excess variability if measurements do not agree within these limits.

#### 2.5.2.6. If repeated double substitutions are performed, the values between successive trials should not differ from one another by more than ± 2 standard deviations of the balance for this process and load. If this difference is exceeded, reject the data and take a new series of measurements that do so agree.

2.5.3. Measurement Procedure, Optional Sequence B (*XSSX*)

Table 4. Optional Sequence B

  ---------------------- ------------------------ ------------------------
  Measurement No.        Weights on Pan           Observation

  1                      *X + t~x~*               *O~1~*

  2                      *S + t~s~*               *O~2~*

  3                      *S + t~s~ + sw*          *O~3~*

  4                      *X + t~x~ + sw*          *O~4~*
  ---------------------- ------------------------ ------------------------

> Measurements for Option B are made as described in Option A except
> that *X*, *S*, *t~x~*, and *t~s~* are interchanged.

3\. Calculations

3.1. No air buoyancy correction. Calculate the conventional mass
correction, *C~x~*, for the test weight as follows, according to the
optional sequence used. In each case, the conventional mass corrections
for the standard weight(s), *C~s~*, the conventional mass of the tare
weights,$\text{CM}_{t_{s}}$ and$\text{CM}_{t_{x}}$, and the conventional
mass of the sensitivity weight, *CM~sw~*, are included. The symbols
*N~s~* and *N~x~* refer to the nominal values of *S* and *X*,
respectively. If no tare weights, and equal nominal values are used,
those terms may all be deleted from the equations.

.

1.  Optional Sequence A (*SXXS*)

> $C_{x} = \text{ C}_{s} + \text{ CM}_{t_{s}}\text{-  CM}_{t_{x}} + \left\lbrack \frac{\left( \text{ O}_{2}\text{ - O}_{1} \right) + \left( \text{ O}_{3}\text{ - O}_{4} \right)}{2} \right\rbrack\left\lbrack \frac{\text{CM}_{\text{sw}}}{O_{3}\text{ -  O}_{2}} \right\rbrack + \text{ N}_{s}\text{ - N}_{x}$

$C_{x} = \text{ C}_{s} + \text{ CM}_{t_{s}}\text{-  CM}_{t_{x}} + \left\lbrack \frac{\left( \text{ O}_{1}\text{ - O}_{2} \right) + \left( \text{ O}_{4}\text{ - O}_{3} \right)}{2} \right\rbrack\left\lbrack \frac{\text{CM}_{\text{sw}}}{\text{ O}_{3}\text{ -  O}_{2}} \right\rbrack + \text{ N}_{s}\text{ - N}_{x}$`<!-- -->`{=html}3.1.2.
Optional Sequence B (*XSSX*)

2.  Air Buoyancy Correction

    1.  Calculate the air density, *ρ~a~*, as described in the Appendix
        to SOP No. 2.

    2.  $M_{x}\text{  } = \text{   }\frac{M_{s}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{s}} \right)\text{  } + \text{  M}_{t_{s}}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{t_{s}}} \right)\text{  -  M}_{t_{x}}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{t_{x}}} \right)\text{  } + \text{  }\left\lbrack \text{  }\frac{\left( \text{ O}_{2}\text{  -  O}_{1} \right) + \text{  }\left( \text{ O}_{3}\text{  -  O}_{4} \right)}{2}\text{  } \right\rbrack\text{   }\left\lbrack \frac{M_{\text{sw}}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{\text{sw}}} \right)}{\text{ O}_{3}\text{  -  O}_{2}} \right\rbrack}{\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{x}} \right)}$Calculate
        the mass *M~x~* of the test weight, and its mass correction
        *C~x~* using the mass of the standard weight(s), the tare
        weights and the sensitivity weights according to the optional
        sequence used.

        1.  Optional Sequence A (*SXXS*)

#### 

#### 3.2.2.2. Optional Sequence B (*XSSX*)

$M_{x}\text{  } = \text{   }\frac{M_{s}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{s}} \right)\text{  } + \text{  M}_{t_{s}}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{t_{s}}} \right)\text{  -   M}_{t_{x}}\text{  }\left( \text{  }1\text{  -  }\frac{\rho_{a}}{\rho_{t_{x}}} \right)\text{   } + \text{   }\left\lbrack \text{  }\frac{\left( \text{ O}_{1}\text{ - O}_{2} \right) + \left( \text{ O}_{4}\text{ -  O}_{3} \right)}{2}\text{  } \right\rbrack\text{   }\left\lbrack \text{  }\frac{M_{\text{sw}}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{\text{sw}}} \right)}{\text{ O}_{3}\text{ - O}_{2}}\text{  } \right\rbrack}{\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{x}} \right)}$

#### 

3.  Calculate the mass correction *C~x~*, as follows:

> where *N~x~* is the nominal value for *X*.

3.2.4. Calculate the conventional mass[^2] of *X*, *CM~x~*. It is
recommended that the conventional mass be reported.

#### 3.2.4.1. Conventional mass

$\text{CM}_{x} = \text{   }\frac{M_{x}\text{  }\left( \text{ 1  -  }\frac{\rho_{n}}{\rho_{x}} \right)}{\left( \text{ 1  -  }\frac{\rho_{n}}{8\text{.}0} \right)}$

3.2.5. If requested, the apparent mass versus the reference density of
brass may be calculated. This value should only be used when calibrating
mechanical balances that have been adjusted to this reference density.

#### 

#### 3.2.5.1. Apparent mass versus brass

#### $\text{AM}_{\text{x  vs  brass}}\text{  } = \text{   }\frac{M_{x}\text{  }\left( \text{ 1  -  }\frac{\rho_{n}}{\rho_{x}} \right)}{\left( \text{ 1  -  }\frac{\rho_{n}}{8\text{.}\text{3909}} \right)}$

3.  Measurement Assurance

    1.  Duplicate the process with a suitable check standard (See GLP 1,
        SOP 9, SOP 30, and Sec. 7.4)

    2.  Plot the check standard value and verify that it is within
        established limits; a t-test may be incorporated to check
        observed value against accepted value.

    3.  The mean of the check standard is used to evaluate bias and
        drift over time.

    4.  Check standard observations are used to calculate the standard
        deviation of the measurement process, *s~p~*.

        5\. Assignment of Uncertainty

> The limits of expanded uncertainty, *U*, include estimates of the
> standard uncertainty of the mass standards used, *u~s~*, estimates of
> the standard deviation of the measurement process, *s~p~*, and
> estimates of the effect of other components associated with this
> procedure, *u~o~*~.~ These estimates should be combined using the
> root-sum-squared method (RSS), and the expanded uncertainty, *U*,
> reported with a coverage factor of two (k=2), to give us an
> approximate 95 percent level of confidence. See SOP 29 for the
> complete standard operating procedure for calculating the uncertainty.

1.  The expanded uncertainty for the standard, *U*, is obtained from the
    calibration report. The combined standard uncertainty, *u~c~*, is
    used and not the expanded uncertainty, *U*, therefore the reported
    uncertainty for the standard will usually need to be divided by the
    coverage factor k.

> 5.2. The value for *s~p~* is obtained from the control chart data for
> check standards using double substitution measurements. (See SOP No.
> 9.)

3.  Other standard uncertainties usually included at this calibration
    level include uncertainties associated with calculation of air
    density and standard uncertainties associated with the density of
    the standards used.

4.  The expanded uncertainty, *U*, must be ≤ 1/3 of the tolerance
    applicable as per ASTM E617-97 and OIML R111 to classify mass
    standards.

    6\. Report

Report results as described in SOP No. 1, Preparation of
Calibration/Test Reports.

Appendix

Double Substitution Data Sheet

(Optional Sequence A)

*SXXS*

**Laboratory data and conditions**:

  ----------------- ----------------- ----------------- -----------------
  Operator                                              

  Date                                Temperature       

  Balance                             Pressure          

  Nominal Load                        Relative Humidity 

  Standard                                              
  deviation of the                                      
  process, from                                         
  control chart,                                        
  *s~p~*                                                
  ----------------- ----------------- ----------------- -----------------

**Mass standard(s) data:**

+--------+----------+-------------+--------------+---------+----------+
| ID     | Nominal  | Mass        | Expanded     | Unc:    | Density  |
|        |          | C           | Unc:         |         |          |
|        |          | orrection\* |              | k       | g/cm^3^  |
|        |          |             | From cal.    | factor  |          |
|        |          |             | report       |         |          |
+--------+----------+-------------+--------------+---------+----------+
| ## *S* |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *X*    |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *sw*   |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *t~s~* |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *t~x~* |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
|        |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
|        |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+

\*Mass Correction = *True Mass* if using buoyancy correction. Mass
Correction = *Conventional Mass* if NOT using buoyancy correction.
Density is used only with buoyancy corrections.

Observations:

  ----------------- --------------- ------------------- -------------------
  Observation No.   Weights         Balance             
                                    Observations, Units 

  Time:                                                 

  1 (*O~1~*)        *S + t~s~*                          

  2 (*O~2~*)        *X + t~x~*                          

  3 (*O~3~*)        *X + t~x~ + sw*                     

  4 (*O~4~*)        *S + t~s~+ sw*                      

  Time:                                                 
  ----------------- --------------- ------------------- -------------------

Measurement Assurance (Duplication of the Process):

  ----------------- --------------- ------------------- -------------------
  Observation No.   Weights         Balance             
                                    Observations, Units 

  Time:                                                 

  1 (*O~1~*)        *S + t~s~*                          

  2 (*O~2~*)        *S~c~ + t~Sc~*                      

  3 (*O~3~*)        *S~c~ + t~Sc~ +                     
                    sw*                                 

  4 (*O~4~*)        *S + t~s~+ sw*                      

  Time:                                                 
  ----------------- --------------- ------------------- -------------------

Note: dotted line represents decimal point

## Appendix

Double Substitution Data Sheet

(Optional Sequence B)

*XSSX*

**Laboratory data and conditions**:

  ----------------- ----------------- ----------------- -----------------
  Operator                                              

  Date                                Temperature       

  Balance                             Pressure          

  Load                                Relative Humidity 

  Process standard                                      
  deviation from                                        
  control chart,                                        
  *s~p~*                                                
  ----------------- ----------------- ----------------- -----------------

**Mass standard(s) data:**

+--------+----------+-------------+--------------+---------+----------+
| ID     | Nominal  | Mass        | Expanded     | Unc:    | Density  |
|        |          | C           | Unc:         |         |          |
|        |          | orrection\* |              | k       | g/cm^3^  |
|        |          |             | From Cal.    | factor  |          |
|        |          |             | report       |         |          |
+--------+----------+-------------+--------------+---------+----------+
| ## *X* |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *S*    |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *sw*   |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *t~x~* |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *t~s~* |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
|        |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
|        |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+

\*Mass Correction = *True Mass* if using buoyancy correction. Mass
Correction = *Conventional Mass* if NOT using buoyancy correction.
Density is used only with buoyancy corrections.

Observations:

  ----------------- --------------- ------------------- -------------------
  Observation No.   Weights         Balance             
                                    Observations, Units 

  Time:                                                 

  1 (*O~1~*)        *X + t~x~*                          

  2 (*O~2~*)        *S + t~s~*                          

  3 (*O~3~*)        *S + t~s~+ sw*                      

  4 (*O~4~*)        *X + t~x~ + sw*                     

  Time:                                                 
  ----------------- --------------- ------------------- -------------------

Measurement Assurance (Duplication of the Process):

  ----------------- --------------- ------------------- -------------------
  Observation No.   Weights         Balance             
                                    Observations, Units 

  Time:                                                 

  1 (*O~1~*)        *S~c~ + t~Sc~*                      

  2 (*O~2~*)        *S + t~s~*                          

  3 (*O~3~*)        *S + t~s~+ sw*                      

  4 (*O~4~*)        *S~c~ + t~Sc~ +                     
                    sw*                                 

  Time:                                                 
  ----------------- --------------- ------------------- -------------------

Note: dotted line represents decimal point.

Example: With Buoyancy Corrections

Double Substitution Data Sheet

(Optional Sequence A)

*SXXS*

**Laboratory data and conditions**:

  ----------------- ----------------- ----------------- -----------------
  Operator                            HO                

  Date              8/24/86           Temperature       22.3 °C

  Balance           M5SA              Pressure          753.5 mm Hg

  Nominal Load      10 g              Relative Humidity 45 %

  Standard                                              0.002 9 mg
  deviation of the                                      
  process, from                                         
  control chart,                                        
  *s~p~*                                                
  ----------------- ----------------- ----------------- -----------------

**Mass standard(s) data:**

+--------+----------+-------------+--------------+---------+----------+
| ID     | Nominal  | Mass        | Expanded     | Unc:    | Density  |
|        |          | C           | Unc:         |         |          |
|        |          | orrection\* |              | k       | g/cm^3^  |
|        |          |             | From Cal.    | factor  |          |
|        |          |             | Rpt. (mg)    |         |          |
+--------+----------+-------------+--------------+---------+----------+
| ## *S* | 10 g     | -0.679 mg   | 0.014 mg     | 3       | 8.00     |
+--------+----------+-------------+--------------+---------+----------+
| *X*    | 10 g     | TBD         | TBD          | 2       | 7.84     |
+--------+----------+-------------+--------------+---------+----------+
| *sw*   | 5 mg     | -0.0227 mg  | 0.000 28     | 2       | 8.5      |
+--------+----------+-------------+--------------+---------+----------+
| *t~s~* | \-\--    | \-\--       | \-\--        | \-\--   | \-\--    |
+--------+----------+-------------+--------------+---------+----------+
| *t~x~* | \-\--    | \-\--       | \-\--        | \-\--   | \-\--    |
+--------+----------+-------------+--------------+---------+----------+
| *S~c~* | 10 g     | 0.321 mg    | 0.025 mg     | 2       | 8.0      |
+--------+----------+-------------+--------------+---------+----------+
|        |          |             |              |         |          |
+--------+----------+-------------+--------------+---------+----------+

\*Mass Correction = *True Mass* if using buoyancy correction. Mass
Correction = *Conventional Mass* if NOT using buoyancy correction.
Density is used only with buoyancy corrections.

Observations:

  ----------------- --------------- ------------------- -------------------
  Observation No.   Weights         Balance             
                                    Observations, Units 
                                    mg                  

  Time: 8:35 AM                                         

  1 (*O~1~*)        *S + t~s~*      1                   268

  2 (*O~2~*)        *X + t~x~*      1                   821

  3 (*O~3~*)        *X + t~x~ + sw* 6                   798

  4 (*O~4~*)        *S + t~s~+ sw*  6                   245

  Time: 8:47 AM                                         
  ----------------- --------------- ------------------- -------------------

Measurement Assurance (Duplication of the Process):

  ----------------- --------------- ------------------- -------------------
  Observation No.   Weights         Balance             
                                    Observations, Units 

  Time: 9:00 AM                                         

  1 (*O~1~*)        *S + t~s~*      1                   270

  2 (*O~2~*)        *S~c~ + t~Sc~*  2                   271

  3 (*O~3~*)        *S~c~ + t~Sc~ + 7                   248
                    sw*                                 

  4 (*O~4~*)        *S + t~s~+ sw*  6                   248

  Time: 9:10 AM                                         
  ----------------- --------------- ------------------- -------------------

Note: dotted line represents decimal point

Calculate the air density (SOP 2):

*ρ~a~ = 1.179 5* mg/cm³ *= 0.001 179 5* g/cm³

Use equation 3.2.2.1 for optional sequence A (*SXX*S) with buoyancy
corrections[^3]:

$M_{x}\text{  } = \text{   }\frac{M_{s}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{s}} \right)\text{  } + \text{  M}_{t_{s}}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{t_{s}}} \right)\text{  -  M}_{t_{x}}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{t_{x}}} \right)\text{  } + \text{  }\left\lbrack \text{  }\frac{\left( \text{ O}_{2}\text{  -  O}_{1} \right) + \text{  }\left( \text{ O}_{3}\text{  -  O}_{4} \right)}{2}\text{  } \right\rbrack\text{   }\left\lbrack \frac{M_{\text{sw}}\text{  }\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{\text{sw}}} \right)}{\text{ O}_{3}\text{  -  O}_{2}} \right\rbrack}{\left( 1\text{  -  }\frac{\rho_{a}}{\rho_{x}} \right)}$

$$\text{M }_{X}\text{   } = \text{    }\frac{9\text{.}\text{999  321    }\left( \text{  1   -   }\frac{0\text{.}\text{0011795}}{8\text{.}0}\text{    } \right)\text{  } + \text{  0   -   0  } + \text{  }\left\lbrack \text{  }\frac{\left( \text{ 1}\text{.}\text{821  -  1}\text{.}\text{268  } \right)\text{  } + \text{   }\left( \text{ 6}\text{.}\text{798  -  6}\text{.}\text{245  } \right)}{2}\text{  } \right\rbrack\text{   }\left\lbrack \text{  }\frac{0\text{.}\text{004  977 3     }\left( \text{  1  -  }\frac{0\text{.}\text{0011795}}{8\text{.}5}\text{   } \right)}{\text{ 6}\text{.}\text{798  -  1}\text{.}\text{821}} \right\rbrack}{\left( \text{  1  -  }\frac{0\text{.}\text{0011795}}{7\text{.}\text{84}}\text{   } \right)}$$

$$M_{x} = \frac{(\text{ 9}\text{.}\text{997 846 1 } + \text{ 0}\text{.}\text{000 552 957 })}{0\text{.}\text{99984949}} = \text{ 9}\text{.}\text{999 904 1 g}$$

Calculate the mass (*true mass*) correction:

*C~x~ = M~x~ -- N~x~*

*C~x~ = 9.999 904 1* g *-- 10* g *= - 0.000 095 9* g *= - 0.095 9* mg

Calculate the conventional mass value:

$$\text{CM}_{x} = \frac{M_{x}\text{  }\left( \text{  1  -  }\frac{\rho_{n}}{\rho_{x}}\text{  } \right)}{\left( \text{1  -   }\frac{\rho_{n}}{8\text{.}0} \right)}$$

$$\text{CM}_{x}\text{  } = \text{  }\frac{9\text{.}\text{999 904 1  }\left( \text{  1  -  }\frac{0\text{.}\text{001 2}}{7\text{.}\text{84}}\text{  } \right)}{0\text{.}\text{999}\text{850}}$$

$$\text{CM}_{x}\text{  } = \text{  }\frac{9\text{.}\text{999 904 1   }\left( \text{ 0}\text{.}\text{999 846 9 } \right)}{0\text{.}\text{999}\text{850}}\text{  } = \text{  9}\text{.}\text{999 873 51 g}$$

*C~x~ = CM ~x~ -- N~x~*

*C~x~ = 9.999 873 51* g *-- 10* g

*C~x~ = - 0.000 126 49* g *= - 0.126 49* mg

Calculate the uncertainty for the calibration:

$$u_{c}\text{  } = \text{   }\sqrt{\text{  u}_{s^{2}}\text{  } + \text{  s}_{p^{2}}\text{  } + \text{  u}_{o^{2}}}$$

The uncertainty for the standard, *U*, must be divided by the k factor
to determine the *u~s~*.

$$u_{c} = \sqrt{(\text{ 0}\text{.}\text{004 667 })^{2} + (\text{ 0}\text{.}\text{002 9 })^{2} + (\text{ 0}\text{.}\text{000 000 032 })^{2}}$$

### 

### *U = 0.005 494 623 6 \* 2 = 0.010 989 247 3 mg*

**Uncertainty Statement**

The uncertainty reported is the root sum square of the standard
uncertainty of the standard, the standard deviation of the process, and
an uncorrected systematic error for lack of buoyancy corrections,
multiplied by a coverage factor of 2 (k=2) for an approximate 95 percent
confidence interval. Factors not considered in the evaluation: magnetism
(weights are considered to meet magnetism specifications unless
measurement aberrations are noted), balance eccentricity and linearity
(these factors are considered as a part of the measurement process when
obtaining the standard deviation of the process).

**Compliance Evaluation**

We have to evaluate the correction with its expanded uncertainty to
determine if the weight is in tolerance or not. The magnitude of the
expanded uncertainty has to be less than 1/3 of the tolerance to be able
to perform that evaluation, according to ASTM E617-97 and OIML R111.

Load = 10 g

+-----------------+-----------------+-----------------+-----------------+
| ASTM E617       |                 | ## OIML R111    |                 |
+-----------------+-----------------+-----------------+-----------------+
| Class           | Tolerance (mg)  | Class           | Tolerance (mg)  |
+-----------------+-----------------+-----------------+-----------------+
| 0               | 0.025           | E~1~            | 0.020           |
+-----------------+-----------------+-----------------+-----------------+
| 1               | 0.050           | E~2~            | 0.060           |
+-----------------+-----------------+-----------------+-----------------+
| 2               | 0.054           | F~1~            | 0.20            |
+-----------------+-----------------+-----------------+-----------------+

If we look at three times the uncertainty: 0.011 mg x 3 = 0.033 mg, we
realize that the uncertainty complies with the 1/3 rule for ASTM classes
1, 2, and OIML classes E~2~, F~1~.

> Next, we look at the correction with the uncertainty: - *0.126* mg *±
> 0.011* mg.

We can see that the absolute value of the correction is within: *0.115 ≤
C~x~ ≤ 0.137*, therefore, it only complies with OIML class F~1~.

**Reporting**

The conventional mass correction and uncertainty would be reported as
follows:

## Example: Without Buoyancy Corrections

Double Substitution Data Sheet

(Optional Sequence B)

*XSSX*

**Laboratory data and conditions**:

  ----------------- ----------------- ----------------- -----------------
  Operator                            HO                

  Date              8/24/86           Temperature       22.3 °C

  Balance           CB 100            Pressure          753.5 mm Hg

  Load              30 g & 1 troy oz  Relative Humidity 45 %

  Process standard                                      0.018 mg
  deviation from                                        
  control chart,                                        
  *s~p~*                                                
  ----------------- ----------------- ----------------- -----------------

**Mass standard(s) data:**

+--------+----------+-------------+--------------+---------+----------+
| ID     | Nominal  | Mass        | Expanded     | Unc:    | Density  |
|        |          | C           | Unc:         |         |          |
|        |          | orrection\* |              | k       | g/cm^3^  |
|        |          |             | From cal.    | factor  |          |
|        |          |             | report       |         |          |
+--------+----------+-------------+--------------+---------+----------+
| ## *S* | 30 g     | 0.407 mg    | 0.022 mg     | 3       | 8        |
+--------+----------+-------------+--------------+---------+----------+
| *X*    | 1 t oz   | TBD         | TBD          | 2       | 7.84     |
+--------+----------+-------------+--------------+---------+----------+
| *sw*   | 50 mg    | -0.084 00   | 0.000 65 mg  | 2       | 8.5      |
|        |          | mg          |              |         |          |
+--------+----------+-------------+--------------+---------+----------+
| *t~s~* | 1.1 g    | 0.359 6 mg  | 0.006 3 mg   | 3       | 8.04     |
+--------+----------+-------------+--------------+---------+----------+
| *t~x~* | None     | \-\--       | \-\--        | \-\--   | \-\--    |
+--------+----------+-------------+--------------+---------+----------+
| *S~c~* | 30 g     | 0.907 mg    | 0.030 mg     | 2       | 8        |
+--------+----------+-------------+--------------+---------+----------+

\*Mass Correction = *True Mass* if using buoyancy correction. Mass
Correction = *Conventional Mass* if NOT using buoyancy correction.
Density is used only with buoyancy corrections.

Observations:

  ----------------- --------------- ------------------- -------------------
  Observation No.   Weights         Balance             
                                    Observations, Units 
                                    [mg]{.underline}    

  Time: 9:00 AM                                         

  1 (*O~1~*)        *X + t~x~*      20                  93

  2 (*O~2~*)        *S + t~s~*      17                  21

  3 (*O~3~*)        *S + t~s~+ sw*  67                  08

  4 (*O~4~*)        *X + t~x~ + sw* 70                  81

  Time: 9:05 AM                                         
  ----------------- --------------- ------------------- -------------------

Measurement Assurance (Duplication of the Process):

  ----------------- --------------- ------------------- -------------------
  Observation No.   Weights         Balance             
                                    Observations, Units 

  Time: 9:15 AM                                         

  1 (*O~1~*)        *S + t~s~*      20                  95

  2 (*O~2~*)        *S~c~ + t~Sc~*  21                  45

  3 (*O~3~*)        *S~c~ + t~Sc~ + 71                  32
                    sw*                                 

  4 (*O~4~*)        *S + t~s~+ sw*  70                  83

  Time: 9:20 AM                                         
  ----------------- --------------- ------------------- -------------------

Note: dotted line represents decimal point

Use equation 3.1.2 for optional sequence B (*XSSX*) with NO buoyancy
corrections[^4]:

$$C_{x}\text{  } = \text{  C}_{s}\text{  } + \text{  CM}_{t_{s}}\text{  -  CM}_{t_{x}}\text{  } + \text{  }\left\lbrack \frac{\left( \text{ O}_{1}\text{  -  O}_{2} \right)\text{  } + \text{  }\left( \text{ O}_{4}\text{  -  O}_{3} \right)}{2} \right\rbrack\text{  }\left\lbrack \frac{\text{CM}_{\text{sw}}}{O_{3}\text{ -  O}_{2}} \right\rbrack\text{  } + \text{  N}_{s}\text{  -  N}_{x}$$

Note: be careful to combine like units only!

$$C_{x}\text{  } = 0\text{.}\text{407}\text{ mg  } + \text{  1}\text{.}\text{100 359 6 g -  0  } + \text{  }\left\lbrack \frac{\left( \text{ 20}\text{.}\text{93 - 17}\text{.}\text{21 } \right)\text{  } + \text{  }\left( \text{ 70}\text{.}\text{81 - 67}\text{.}\text{08 } \right)}{2} \right\rbrack\text{  }\left\lbrack \frac{\text{49}\text{.}\text{916 mg}}{\text{ 67}\text{.}\text{08  -   17}\text{.}\text{21 }} \right\rbrack\text{  } + \text{  30 g  -  1 t oz}$$

$$C_{x}\text{  } = 0\text{.}\text{407}\text{ mg  } + \text{  1}\text{.}\text{100 359 6 g  -  0  } + \text{  3}\text{.}\text{728 435 9 mg  } + \text{  30 g  -  31}\text{.}\text{103 47 68 g}$$

*C~x~ = 1 104.495 036* mg *-- 1.103 476 8* g

*C~x~ = 1.018 236* mg

Calculate the uncertainty for the calibration:

$$u_{c}\text{  } = \text{  }\sqrt{\text{  u}_{s^{2}}\text{  } + \text{  s}_{p^{2}}\text{  } + \text{  u}_{o^{2}}}$$

The uncertainty for the standard, *U*, must be divided by the k factor
for the standard and the tare weight to determine each *u~s~*. The
additional uncertainty for not performing the air buoyancy correction
can be determined using the magnitude of the air buoyancy correction
from SOP 2.

$$u_{c} = \sqrt{(\text{ 0}\text{.}\text{007 33 })^{2} + (\text{ 0}\text{.}\text{002 1 })^{2} + (\text{ 0}\text{.}\text{018 })^{2} + (\text{ 0}\text{.}\text{001 6 })^{2}}$$

*u~c~ =* 0.019 617 15 mg

### *U =* 0.019 617 15 \* 2 = 0.039 234 mg

The conventional mass correction and uncertainty would be reported as
follows:

*C~x~ =* 1.018 mg ± 0.039 mg

OR

*C~x~ =* 0.000 032 7 t oz  0.000 001 3 t oz

[^1]: The barometer, thermometer, and hygrometer are used to determine
    the air density at the time of the measurement. The air density is
    used to make an air buoyancy correction. The accuracies specified
    are recommended for high precision calibration. Less accurate
    equipment can be used with only a small degradation in the overall
    accuracy of the measurement.

[^2]: Conventional Mass: "The conventional value of the result of
    weighing a body in air is equal to the mass of a standard, of
    conventionally chosen density, at a conventionally chosen
    temperature, which balances this body at this reference temperature
    in air of conventionally chosen density." The conventions are:
    reference density 8.0 g/cm^3^; reference temperature 20 °C; *normal*
    air density 0.0012 g/cm3. Conventional mass was formerly called
    "Apparent Mass versus 8.0 g/cm^3^" in the United States. *See OIML
    IR 33 (1973, 1979), under revision.*

[^3]: Keep in mind that these equations may be truncated for the purpose
    of this example and minor differences may be seen in the ending
    decimal places due to the use of calculators or spreadsheets.

[^4]: Keep in mind that these equations may be truncated for the purpose
    of this example and minor differences may be seen in the ending
    decimal places due to the use of calculators or spreadsheets.