SOP No. 7

**Recommended Standard Operations Procedure**

**for**

**Weighing by Single 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

In the single substitution procedure a standard and an unknown weight
are intercompared once to determine the difference in 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. This procedure is suitable for calibration when moderate
accuracy is required and as a single substitution, does not eliminate
errors due to drift. 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 that is used is in good operating condition
    with sufficiently small process standard deviation as verified by a
    valid control chart or preliminary experiments to ascertain its
    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 7, SOP 29,
    GMP 4, and GMP 10.

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

Table 1. Environmental conditions

+-------+----------------------------------------------+---------------+
| Ec    | ## Temperature                               | Relative      |
| helon |                                              | Humidity      |
|       |                                              |               |
|       |                                              | (%)           |
+-------+----------------------------------------------+---------------+
| ## II | ## 20 C to 23                                | 40 to 60 10 / |
|       | C, 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 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 depends upon the sensitivity of
> the balance and the care exercised in making 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 single 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 differ
> 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
> values that are 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 that are traceable to NIST to determine air
    pressure.

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

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

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3.  Symbols

    Table 2. Symbols used in this procedure

+-------+--------------------------------------------------------------+
| ## S  | Description                                                  |
| ymbol |                                                              |
+-------+--------------------------------------------------------------+
| *S*   | standard weight                                              |
+-------+--------------------------------------------------------------+
| *X*   | weight 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.              |
+-------+--------------------------------------------------------------+
| *~a~* | density of air at time of calibration                        |
+-------+--------------------------------------------------------------+
| *~n~* | density of normal air (1.2 kg/m^3^)                          |
+-------+--------------------------------------------------------------+
|       | 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 % of the balance capacity.

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

> 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 differences determined through the use of the optical
> scale or electronic range have valid accuracy and traceability. (e.g.,
> The optical scale is *calibrated* each time the procedure is used
> through the use of a sensitivity weight). The uncertainty of the
> sensitivity weight does not need to be included in calculations of
> uncertainty since the uncertainty value is distributed across the
> range of use.
>
> If a sensitivity weight will be used, select one that is (See GMP 14,
> Table 2):

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 % 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*.

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1.  Determine whether optional sequence A or B will be used. Optional
    sequence A uses the standard on the balance for the first
    observation and the unknown on the balance for the second and third
    observations; this is often called the "*SXX*" sequence. Optional
    sequence B starts with the unknown on the balance first and with the
    standard on the balance for the second and third observations; this
    is often called the "*XSS*" sequence.

2.  Adjust the single pan balance or the combination balance so the
    first two readings of the single 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 single 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.

3.  If the balance is equipped with a pan arrestment mechanism, arrest
    the pan between each observation.

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1.  Measurement Procedure, Optional Sequence A (*SXX*)

> 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~*
  ---------------------- ------------------------ ------------------------

> 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.

#### 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.

#### 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~*.

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

#### If repeated single substitutions are performed, the values between successive trials should not differ from one another by more than ± 2 sd of the balance. If this difference is exceeded, reject the data and take a new series of measurements that will so agree.

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

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~*
  ---------------------- ------------------------ ------------------------

> 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.

.

> 3.1.1. Optional Sequence A (*SXX*)

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

2.  Optional Sequence B (*XSS*)

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

2.  Air Buoyancy Correction

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

    2.  Calculate the mass of the test weight, *M~x~*, 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 (*SXX*)

#### 

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

#### 

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

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

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

#### 

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

#### 

> 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.

####  

#### $\text{AM}_{\text{x  vs  brass}}\text{  } = \text{   }\frac{M_{x}\text{  }\left( \text{ 1  -  }\frac{\rho_{n}}{\rho_{x}} \right)}{\left( \text{ 1  -  }\frac{\rho_{n}}{8\text{.}\text{3909}} \right)}$`<!-- -->`{=html}3.2.5.1. Apparent mass versus brass

#### 

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 % 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 single 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.  Evaluation of compliance. The expanded uncertainty, *U*, must be 1/3
    of the tolerance applicable as per ASTM E 617-97 and OIML R 111 if
    compliance statements are used

    6\. Report

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

## **Appendix**

**Single Substitution Data Sheet**

**(Optional Sequence A)**

***SXX***

> **Laboratory data and conditions**:

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

  Date                                  Temperature        

  Balance                               Pressure           

  Load                                  Relative Humidity  

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

> **Mass standard(s) data:**

+----------+-----------+-----------+------------+---------+----------+
| ID       | Nominal   | Mass      | Unc:       | Unc:    | Density  |
|          |           | C         |            |         |          |
|          |           | orrection | From cal.  | k       | g/cm^3^  |
|          |           |           | report     | factor  |          |
+----------+-----------+-----------+------------+---------+----------+
| ## *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:

  ------------------- ------------- -------------------- ------------------
  Measurement \#      Weights       Balance              
                                    Observations, Units  
                                    \_\_\_\_\_\_\_\_\_   

  Time:                                                  

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

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

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

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

> Measurement Assurance (Duplication of the Process):

  ------------------- ------------- -------------------- ------------------
  Measurement \#      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*                        

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

Note: dotted line represents decimal point.

## **Appendix**

**Single Substitution Data Sheet**

**(Optional Sequence B)**

***XSS***

> **Laboratory data and conditions**:

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

  Date                                  Temperature        

  Balance                               Pressure           

  Load                                  Relative Humidity  

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

> **Mass standard(s) data:**

+----------+-----------+-----------+------------+---------+----------+
| ID       | Nominal   | Mass      | Unc:       | Unc:    | Density  |
|          |           | C         |            |         |          |
|          |           | orrection | From cal.  | k       | g/cm^3^  |
|          |           |           | report     | factor  |          |
+----------+-----------+-----------+------------+---------+----------+
| ## *X*   |           |           |            |         |          |
+----------+-----------+-----------+------------+---------+----------+
| *S*      |           |           |            |         |          |
+----------+-----------+-----------+------------+---------+----------+
| *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:

  ------------------- ------------- -------------------- ------------------
  Measurement \#      Weights       Balance              
                                    Observations, Units  
                                    \_\_\_\_\_\_\_\_\_   

  Time:                                                  

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

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

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

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

> Measurement Assurance (Duplication of the Process):

  ------------------- ------------- -------------------- ------------------
  Measurement \#      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*                        

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

Note: dotted line represents decimal point.

**Example: Without Buoyancy Corrections**

**Single Substitution Data Sheet**

**(Optional Sequence A)**

***SXX***

> **Laboratory data and conditions**:

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

  Date               8/27/86              Temperature        22.6 °C 

  Balance            H 20                 Pressure           751.7   
                                                             mm Hg   

  Load               50 g                 Relative Humidity  50 %    

  Standard deviation                                         0.018   
  of the process,                                            mg      
  from control                                                       
  chart, *s~p~*                                                      
  ------------------ ----------------- -- ------------------ ------- ----------------

> **Mass standard(s) data:**

+----------+-----------+-----------+------------+---------+----------+
| ID       | Nominal   | Mass      | Unc:       | Unc:    | Density  |
|          |           | C         |            |         |          |
|          |           | orrection | From cal.  | k       | g/cm^3^  |
|          |           |           | report     | factor  |          |
+----------+-----------+-----------+------------+---------+----------+
| ## *S*   | 50 g      | 0.255 mg  | 0.033 mg   | 3       | 8.00     |
+----------+-----------+-----------+------------+---------+----------+
| *X*      | 50 g      |           |            |         | 7.95     |
+----------+-----------+-----------+------------+---------+----------+
| *sw*     | 50 mg     | -0.084 mg | 0.000 85   | 2       | 8.5      |
|          |           |           | mg         |         |          |
+----------+-----------+-----------+------------+---------+----------+
| *t~s~*   |           |           |            |         |          |
+----------+-----------+-----------+------------+---------+----------+
| *t~x~*   |           |           |            |         |          |
+----------+-----------+-----------+------------+---------+----------+
| *S~c~*   | 50 g      | 0.315 mg  | 0.045 mg   | 2       | 8.00     |
+----------+-----------+-----------+------------+---------+----------+
|          |           |           |            |         |          |
+----------+-----------+-----------+------------+---------+----------+

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:

  ------------------- ------------- ------------------------------------ ------------------
  Measurement \#      Weights       Balance Observations, Units          
                                    \_\_\_\_\_[mg]{.underline}\_\_\_\_   

  Time:               3:40 PM                                            

  1 (*O~1~*)          *S + t~s~*    12                                   62

  2 (*O~2~*)          *X + t~x~*    12                                   51

  3 (*O~3~*)          *X + t~x~ +   62                                   37
                      sw*                                                

  Time:               3:45 PM                                            
  ------------------- ------------- ------------------------------------ ------------------

> Measurement Assurance (Duplication of the Process):

  ------------------- ------------- ------------------------------ ------------------
  Measurement \#      Weights       Balance Observations, Units    
                                    \_\_\_[mg]{.underline}\_\_\_   

  Time:               3:50 PM                                      

  1 (*O~1~*)          *S + t~s~*    12                             67

  2 (*O~2~*)          *S~c~ +       12                             73
                      t~Sc~*                                       

  3 (*O~3~*)          *S~c~ +       62                             60
                      t~Sc~ + sw*                                  

  Time:               3:55 PM                                      
  ------------------- ------------- ------------------------------ ------------------

Note: dotted line represents decimal point.

**Calculation of Conventional Mass**

Use equation 3.1.1 for optional sequence A (*SXX*) without buoyancy
corrections[^3]:

Since no tare weights were used and equal nominal values were used, the
equation may be simplifies as follows:

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

$$C_{x} = \text{  0}\text{.}\text{255 mg  } + \text{  }(\text{ 12}\text{.}\text{51  -  12}\text{.}\text{62 })\text{  }\left\lbrack \frac{\text{49}\text{.}\text{916 mg}}{\left( \text{ 62}\text{.}\text{37 -  12}\text{.}\text{51 } \right)} \right\rbrack$$

*C~x~ = 0.255* mg *+ (-0.110 124)* mg

*C~x~ = 0.144 876* mg

**Calculation of Uncertainty**

Calculate the uncertainty for the calibration:

$$\text{U } = \text{ u}_{c}\text{ * 2}$$

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

The uncertainty for the standard from the calibration report (or data
sheet) must be divided by the k factor to determine the *u~s~*. Refer to
SOP 29 for the use of multiple standards. An additional uncertainty for
not performing the air buoyancy correction can be determined using the
magnitude of the air buoyancy correction from SOP 2.

$$u_{c} = \sqrt{(\text{ 0}\text{.}\text{011 })^{2} + (\text{ 0}\text{.}\text{018 })^{2} + (\text{ 0}\text{.}\text{0010 })^{2}}$$

### 

### *U = 0.021 330 7 \* 2 = 0.042 661 5* 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 %
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**

You may need to evaluate the conventional mass correction and its
uncertainty to determine if a weight is or is not within specified
tolerances. The magnitude of the expanded uncertainty must be less than
1/3 of the tolerance to be able to perform that evaluation, according to
ASTM E 617‑97 and OIML R111 documentary standards.

Load = 50 g

+-----------------+-----------------+-----------------+-----------------+
| ASTM E 617      |                 | ## OIML R111    |                 |
+-----------------+-----------------+-----------------+-----------------+
| Class           | Tolerance (mg)  | Class           | Tolerance (mg)  |
+-----------------+-----------------+-----------------+-----------------+
| 1               | 0.12            | E~2~            | 0.10            |
+-----------------+-----------------+-----------------+-----------------+
| 2               | 0.25            | F~1~            | 0.30            |
+-----------------+-----------------+-----------------+-----------------+

If we look at three times the calculated expanded uncertainty: 0.043 mg
x 3 = 0.129 mg, we observe that the uncertainty complies with the 1/3
rule for ASTM Class 2, and OIML Class F~1~ but not ASTM Class 1 or OIML
Class E~2~.

Next, the weight value is considered to be within tolerance when the
absolute value of its error plus its uncertainty, do not exceed the
tolerance established for the particular class of weight.

Value and uncertainty: *0.145* mg *0.043* mg.

0.145 mg + 0.043 mg = 0.188 mg (upper limit of the value, or error bar
if graphed)

0.145 mg -- 0.043 mg = 0.102 mg (lower limit of the value, or error bar
if graphed)

We can see that the correction of 0.145 is within these limits: *0.102
C~x~ 0.188*, therefore, the value also complies with ASTM Class 2 and
OIML Class F~1~, and a compliance statement may be included on the
calibration report.

**Reporting**

The conventional mass correction and uncertainty are reported as
follows:

C~x~ = *0.145* mg * 0.043* mg

[^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.