**SOP 5**

**Recommended Standard Operations Procedure**

**for**

**Using a 3‑1 Weighing Design**

Number range level0Number range level1Number range level2Number range
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1.1. Purpose

The 3‑1 weighing design is a combination of three double substitution
comparisons of three weights of equal nominal value; a standard, an
unknown weight, and a second standard called a check standard. (The
check standard may be made up of a summation of weights.) The weights
are compared using an equal‑arm, single‑pan mechanical, full electronic,
or a combination balance utilizing built‑in weights and a digital
indication. The specific SOP for the double substitution procedure for
each balance is to be followed. The 3‑1 weighing design provides two
methods of checking the validity of the measurement using an F-test to
check the measurement process and a t-test to evaluate the stability of
the standard and check standard. Hence, the procedure is especially
useful for high accuracy calibrations in which it is critical to assure
that the measurements are valid and well documented. This procedure is
recommended as a minimum for precision calibration of laboratory working
standards that are subsequently used for lower level calibrations and
for routine calibration of precision mass standards used for balance
calibration. For surveillance of reference and working mass standards
and calibration of precision mass standards used to calibrate other mass
standards, see SOP 28 for the use of higher level weighing designs.

1.2 Prerequisites

1.2.1 Calibrated mass standards, traceable to NIST, with valid
calibration certificates must be available with sufficiently small
standard uncertainties for the intended level of calibration. Reference
standards should only be used to calibrate the next lower level of
working standards in the laboratory and should not be used to routinely
calibrate customer standards.

1.2.2 The balance used must be in good operating condition with
sufficiently small process standard deviation as verified by F-test
values, pooled short term standard deviations, and by a valid control
chart for check standards or preliminary experiments to ascertain its
performance quality when new balances are put into service.

1.2.3 The operator must be experienced in precision weighing techniques.
The operator must have specific training in SOP 2, SOP 4, SOP 5, SOP 29,
and be familiar with the concepts in GMP 10.

1.2.4 The laboratory facilities must meet the following minimum
conditions to meet the expected uncertainty possible with this
procedure:

**Table 1. Environmental** conditions

+-------+-----------------------------------------------+-------------+
| Ec    | # Temperature                                 | # Relat     |
| helon |                                               | ive Humidit |
|       |                                               | y (percent) |
+-------+-----------------------------------------------+-------------+
| I     | 20 °C to 23 °C, allowable variation of ± 1 °C | 40 to 60 ±  |
|       |                                               | 5           |
|       | Maximum change of 0.5 °C/h                    |             |
+-------+-----------------------------------------------+-------------+
| II    | 20 °C to 23 °C, allowable variation of ± 2 °C | 40 to 60 ±  |
|       |                                               | 10          |
|       | Maximum change of 1.0 °C/h                    |             |
+-------+-----------------------------------------------+-------------+

2Number range level1Number range level2 Methodology 2Number range
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2.1 Scope, Precision, Accuracy

This method can be performed on any type of balance using the
appropriate double substitution SOP for the particular balance. Because
considerable effort is involved, this weighing design is most useful for
calibrations of the highest accuracy. The weighing design utilizes three
double substitutions to calibrate a single unknown weight. This
introduces redundancy into the measurement process and permits two
checks on the validity of the measurement; one on accuracy and stability
of the standard and the other on process repeatability. A least‑squares
best fit analysis is done on the measurements to assign a value to the
unknown weight. The standard deviation of the process depends upon the
resolution of the balance and the care exercised to make the required
weighings. The accuracy will depend upon the accuracy and uncertainty of
the calibration of the standard weights and the precision of the
comparison.

2.2 Summary

A standard weight, S, an unknown weight, X, and a check standard, S~c~
are intercompared in a specific order using the double substitution
procedure. The balance and the weights must be prepared according to the
appropriate double substitution SOP for the particular balance being
used. Once the balance and weights have been prepared, all readings must
be taken from the reading scale of the balance without adjusting the
balance or weights in any way. Within a double substitution all
weighings are made at regularly spaced time inter­vals to average out any
effects due to instrument drift. Because of the amount of effort
required to perform the 3‑1 weighing design, the procedure includes the
air buoyancy correction.

2.3 Apparatus/Equipment Required

2.3.1 Precision analytical balance or mass comparator with sufficient
capacity and resolution for the calibrations planned.

2.3.2 Working standard weights and sensitivity weights with valid
calibrations traceable to NIST.

2.3.3 Small working standards with valid calibrations 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 or adequate optical or electronic range for the
intended load and 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 or the operator is experienced with determining a stable
indication. If an electronic balance is used that has a means for
indicating a stable reading, the operator may continue to time readings
to ensure consistent timing that can minimize errors due to linear
drift.

2.3.7 Thermometer accurate to 0.10 °C to determine air temperature.

2.3.8 Barometer accurate to 0.5 mm of mercury (66.5 Pa) to determine air
pressure.

2.3.9 Hygrometer accurate to 10 percent to determine relative humidity.

2.4Number range level2 Procedure

2.4.1 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. The equilibration time is particularly important
with weights larger than 1 gram. Conduct preliminary measurements to
determine the tare weights that may be required, the size of the
sensitivity weight required, adjust the balance to the appropriate
reading range of the balance indications, and to exercise the balance.
Refer to the appropriate double substitution SOP for details.

2.4.2 Weighing Design Matrix

The following table shows the intercomparisons to be made in the 3-1
design, in a matrix format as shown in NBS Technical Note 952, Designs
for the Calibration of Standards of Mass, J. M. Cameron, M. C. Croarkin,
and R. C. Raybold, 1977.:

+----------------------+---------------+---------------+---------------+
| Weight ID            | S             | X             | Sc            |
|                      |               |               |               |
| Comparison           |               |               |               |
+======================+===============+===============+===============+
| a~1~                 | \+            | \-            |               |
+----------------------+---------------+---------------+---------------+
| a~2~                 | \+            |               | \-            |
+----------------------+---------------+---------------+---------------+
| a~3~                 |               | \+            | \-            |
+----------------------+---------------+---------------+---------------+
| Standard             | \+            |               |               |
+----------------------+---------------+---------------+---------------+
| Check Standard       |               |               | \+            |
+----------------------+---------------+---------------+---------------+

This design is represented as design ID "A.1.1"in Technical Note 952,
with the exception that the design order is reversed and Restraint B is
used. The restraint is another name for the "standard" used in the
comparison that may be found in NBS Technical Note 952. This matrix may
be useful for anyone using the NIST Mass Code for data reduction. When
creating a data file for this design, the design matrix will appear as
follows:

  ------------------------- --------------- -------------- ---------------
  restraint                 1               0              0

  Check                     0               0              1

  following series sum      0               0              0

  Report                    0               1              1

  1st double sub            1               -1             0

  2nd double sub            1               0              -1

  3rd double sub            0               1              -1
  ------------------------- --------------- -------------- ---------------

2.4.3 Measurement Procedure

Record the pertinent information for the standard, S, unknown, X, and
check standard, S~c~, as indicated on a suitable data sheet such as the
one in the Appendix of this SOP. Record the laboratory ambient
temperature, barometric pressure, and relative humidity. Perform the
measurements in the order shown in the following table.

+-----------------+-----------------+------------------+--------------+
| Double          | Measurement     | Weights          | Observation  |
|                 |                 |                  |              |
| Substitution    | Number          | on Pan           |              |
+-----------------+-----------------+------------------+--------------+
| a~1~: S vs X    | 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~         |
+-----------------+-----------------+------------------+--------------+
| a~2~: S vs S~c~ | 5               | S + t~s~         | O~5~         |
+-----------------+-----------------+------------------+--------------+
|                 | 6               | S~c~ + t~sc~     | O~6~         |
+-----------------+-----------------+------------------+--------------+
|                 | 7               | S~c~ + t~sc~ +   | O~7~         |
|                 |                 | sw               |              |
+-----------------+-----------------+------------------+--------------+
|                 | 8               | S + t~s~ + sw    | O~8~         |
+-----------------+-----------------+------------------+--------------+
| a~3~: X vs S~c~ | 9               | X + t~x~         | O~9~         |
+-----------------+-----------------+------------------+--------------+
|                 | 10              | S~c~ + t~sc~     | O~10~        |
+-----------------+-----------------+------------------+--------------+
|                 | 11              | S~c~ + t~sc~ +   | O~11~        |
|                 |                 | sw               |              |
+-----------------+-----------------+------------------+--------------+
|                 | 12              | X + t~x~ + sw    | O~12~        |
+-----------------+-----------------+------------------+--------------+

where:

  ----------------- -----------------------------------------------------
  Variable          Description

  t~s~              calibrated tare weights carried with S

  t~x~              calibrated tare weights carried with X

  t~sc~             calibrated tare weights carried with S~c~

  sw                calibrated sensitivity weight
  ----------------- -----------------------------------------------------

3Number range level1Number range level2 Calculations 3Number range
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3.1 Calculate the air density, ρ~A~, as described in the Appendix to SOP
No. 2.

3.2 Calculate the measured differences, a~1~, a~2~, and a~3~, for the
weights used in each double substitution using the following formula
(note: do not confuse this formula with the calculations used in SOP 4;
the signs will be opposite from Option A of SOP 4):

$\text{a }_{x}\text{= }\frac{(O_{1}\text{ - }O_{2}\text{ + }O_{4}\text{ - }O_{3})}{2}\text{  }\frac{M_{\text{sw}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{\text{sw}}} \right)}{O_{3}\text{ - }O_{2}}$

where:

  ----------------- -----------------------------------------------------
  Variable          Description

  M~sw~             mass of the sensitivity weight

  ρ~sw~             density of the sensitivity weight
  ----------------- -----------------------------------------------------

3.3 Calculate the short term within process standard deviation, s~w~,
for the 3‑1 weighing design. This standard deviation has one degree of
freedom.

s~w~ = 0.577(a~1~ ‑ a~2~ + a~3~)

3.4 Compute the F statistic which compares the short term within process
standard deviation, s~w~, to the pooled within process standard
deviation. (See chapter 8.4 and 8.5 for a discussion of the statistics
used in weighing designs.)

$\text{F-statistic = }\frac{{s_{w}}^{2}}{(\text{ Pooled }s_{w})^{2}}$

> The F‑statistic so computed must be less than the F‑value obtained
> from an F‑table at 99 % confidence level (Table 9.5) to be acceptable.
> The F-value is obtained from the F-table for numerator degrees of
> freedom equal one, and denominator degrees of freedom equal to the
> number of degrees of freedom in the pooled within process standard
> deviation. If the data fails the F‑test and the source of the error
> cannot be determined conclusively, the measurement must be repeated.

3.5 Compute the observed mass value of the check standard.

Compute the least‑squares measured difference d~sc~ for S~c~.

$d_{s_{c}}\text{ = }\frac{- a_{1}\text{ - 2}a_{2}\text{ - }a_{3}}{3}$

3.6 Compute the observed mass of S~c~, M~sc~.

> $M_{\text{sc}}\text{ = }\frac{M_{s}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{s}} \right)\text{ + }d_{\text{sc}}\text{ + }M_{t_{s}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{t_{s}}} \right)\text{ - }M_{t_{\text{sc}}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{t_{\text{sc}}}} \right)}{\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{\text{sc}}} \right)}$`<!-- -->`{=html}3.7
> Evaluation of the observed mass of S~c~, M~sc~ .

The mass determined for the check standard should be plotted on the
control chart and must lie within the control limits. If it does not,
and the source of error cannot be found, the measurement must be
repeated. The 'Accepted M~Sc~' is the mean of the historically observed
mass values for the check standard.

> \|Observed M~Sc~ -- Accepted M~Sc~\| \> 3sd Status: Out of Control
>
> 2sd \< \|Observed M~Sc~ -- Accepted M~Sc~\| \< 3sd Status: In
> Control\*Warning
>
> \|Observed M~Sc~ -- Accepted M~Sc~\| \< 2sd Status: In Control
>
> Perform an E~normal~ test to compare the mean value of the M~sc~ value
> from the 3-1 design to a calibration value that has demonstrated
> measurement traceability for the check standard.
>
> $$E_{n} = \frac{\left| {\overline{M}}_{\text{Sc}} - \text{CalibratedM}_{\text{Sc}} \right|}{\sqrt{U_{M_{\text{sc}}}^{2} + U_{\text{CalibratedM}_{\text{sc}}}^{2}}}$$
>
> The E~normal~ value must be less than one to pass.

3.8 Compute the least‑squares measured difference, d~x~, for X.

$d_{x}\text{ = }\frac{\text{-2}a_{1}\text{ - }a_{2}\text{ + }a_{3}}{3}$

> 3.9 Compute the mass of X, M~x~.

$M_{x}\text{ = }\frac{M_{s}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{s}} \right)\text{ + }d_{x}\text{ + }M_{t_{s}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{t_{s}}} \right)\text{ - }M_{t_{x}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{t_{x}}} \right)}{\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{x}} \right)}$

where:

  ----------------- -----------------------------------------------------
  Variable          Description

  ρ~A~              air density

  M~i~              mass for weight I

  ρ~i~              reference density for weight I
  ----------------- -----------------------------------------------------

3.10 Calculate the conventional mass of X versus the desired reference
density of 8.0 g/cm^3^ or apparent mass of brass (8.3909 g/cm^3^). It is
recommended that the conventional mass versus 8.0 g/cm^3^ be reported
unless otherwise requested. The density of X, ρ~x~, must be entered in
g/cm^3^. (See SOP No. 2)

3.10.1 Conventional mass

$\text{CM}_{x}\text{ vs}\text{.}\text{ 8}\text{.}\text{0 = }\frac{M_{x}\left( \text{ 1 - }\frac{0\text{.}\text{0012}}{\rho_{x}} \right)}{0\text{.}\text{999850}}$

> 3.10.2 Apparent mass versus brass (8.3909 g/cm3 at 20 °C)

$\text{AM}_{x}\text{ vs}\text{.}\text{ 8}\text{.}\text{4 = }\frac{M_{x}\left( \text{ 1 - }\frac{0\text{.}\text{0012}}{\rho_{x}} \right)}{\left( \text{ 1 - }\frac{0\text{.}\text{0012}}{8\text{.}\text{3909}} \right)}$

4 Number range level1Number range level2Assignment of Uncertainty4Number
range level1Number range level2

The limits of expanded uncertainty, U, include estimates of the standard
uncertainty of the mass standards used, u~s~, plus the uncertainty of
measurement, u~m~, at the 95 percent level of confidence. See SOP 29,
"Standard Operating Procedures for the Assignment of Uncertainty", for
the complete standard operating procedure for calculating the
uncertainty. When the 3-1 weighing design is used in conjunction with
the Mass Code for data reduction, see SOP 28, "Recommended Standard
Operating Procedure for Using Advanced Weighing Designs", for detailed
instructions on calculating the uncertainty components which are
required by the Mass Code program.

4.1 The standard uncertainty for the standard, u~s~, 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 need to be divided by the coverage factor k. Since
only one standard is used as the restraint for the 3-1 weighing design,
the uncertainty of the check standard is not included in assigning an
uncertainty to the unknown mass.

4.2 Standard deviation of the measurement process from control chart
performance (See SOP No. 9.)

The value for s~p~ is obtained from the control chart data for check
standards using 3-1 weighing designs. Statistical control must be
verified by the measurement of the check standard in the 3-1 design.

4.3 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.

5Number range level1 Report

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

Appendix

3-1 Weighing Design When Tare Weights Are Used

(Densities used to compute air buoyancy correction)

(Air buoyancy correction on the tare weights)

**Laboratory data and conditions:**

  ------------------------- -------------- ------------------- ------------
  Date                                     Temperature         

  Balance                                  Pressure            

  Load                                     Relative Humidity   

  Pooled within process                    Calculated Air      
  s.d., s~w~=                              Density             

  Check standard s.d., s~p~                                    
  =                                                            
  ------------------------- -------------- ------------------- ------------

**Mass standard(s) data:**

+-----+------------+-------+------+-----+------------+------+-------+
| ID  | Mass = N + | De    | Un   | ID  | Mass = N + | Den  | Unc~( |
|     | C          | nsity | c~(k |     | C          | sity | k=1)~ |
|     |            |       | =1)~ |     |            |      |       |
|     | \(g\)      | (g/c  |      |     | \(g\)      | (    | (mg)  |
|     |            | m^3^) | (mg) |     |            | g/cm |       |
|     |            |       |      |     |            | ^3^) |       |
+=====+============+=======+======+=====+============+======+=======+
| N   |            |       |      | t   |            |      |       |
| ~x~ |            |       |      | ~x~ |            |      |       |
+-----+------------+-------+------+-----+------------+------+-------+
| M   |            |       |      | t   |            |      |       |
| ~s~ |            |       |      | ~s~ |            |      |       |
+-----+------------+-------+------+-----+------------+------+-------+
| M~  |            |       |      | t~  |            |      |       |
| sc~ |            |       |      | sc~ |            |      |       |
+-----+------------+-------+------+-----+------------+------+-------+
| sw  |            |       |      |     |            |      |       |
+-----+------------+-------+------+-----+------------+------+-------+

N = Nominal, C = Correction, M = *True Mass*

**Laboratory observations:**

  -------------- ------ ------ ---------- ------ ------ ---------- ------ ------
  Balance                                                                 
  Observations                                                            

  S - X = a~1~                 S - S~c~ =               X - S~c~ =        
                               a~2~                     a~3~              

  S + t~s~                     S + t~s~                 X + t~x~          

  X + t~x~                     S~c~ +                   S~c~ +            
                               t~sc~                    t~sc~             

  X + t~x~ + sw                S~c~ +                   S~c~ +            
                               t~sc~ + sw               t~sc~ + sw        

  S + t~s~ + sw                S + t~s~ +               X + t~x~ +        
                               sw                       sw                

  a~1~ =                       a~2~ =                   a~3~ =            
  -------------- ------ ------ ---------- ------ ------ ---------- ------ ------

Note: dotted line represents decimal point.

Calculate "a" values:
$a_{x}\text{ = }\frac{(O_{1}\text{ - }O_{2}\text{ + }O_{4}\text{ - }O_{3}}{2}\text{  }\frac{M_{\text{sw}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{\text{sw}}} \right)}{O_{3}\text{ - }O_{2}}$

Calculate short term within process standard deviation and conduct
F-test:

s~w~ = .577(a~1~ - a~2~ + a~3~)

$$\text{F-statistic = }\frac{s_{w}^{2}}{(\text{ Pooled }s_{w})^{2}}\text{ < value}\underset{}{}\text{F-table 9}\text{.}5$$

F-test passes? Yes No

Evaluate check standard (by plotting on a control chart or with a
t-test):

$$d_{s_{c}}\text{ = }\frac{- a_{1}\text{ - 2}a_{2}\text{ - }a_{3}}{3}$$

$$M_{s_{c}}\text{ = }\frac{M_{s}\left( \text{1-}\frac{\rho_{A}}{\rho_{s}} \right)\text{ + }d_{s_{c}}\text{ + }M_{t_{s}}\left( \text{1-}\frac{\rho_{A}}{\rho_{t_{s}}} \right)\text{ - }M_{t_{s_{c}}}\left( \text{1-}\frac{\rho_{A}}{\rho_{t_{s_{c}}}} \right)}{\text{1-}\frac{\rho_{A}}{\rho_{s_{c}}}}$$

Check standard passes? Yes No

If both F-test and check standard pass the tests, calculate the mass of
the unknown test item:

$$d_{x}\text{ = }\frac{\text{-2}a_{1}\text{ - }a_{2}\text{ + }a_{3}}{3}$$

$$M_{x}\text{ = }\frac{M_{s}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{s}} \right)\text{ + }d_{x}\text{ + }M_{t_{s}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{t_{s}}} \right)\text{ - }M_{t_{x}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{t_{x}}} \right)}{\text{1 - }\frac{\rho_{A}}{\rho_{x}}}$$

$$\text{CM}_{x}\text{ vs }\rho_{\text{ref}} = \frac{M_{x}\left( \text{1-}\frac{0\text{.}\text{0012}}{\rho_{x}} \right)}{\text{1-}\frac{0\text{.}\text{0012}}{8\text{.}0}}$$

where ρ~ref~ refers to the reference density.

Example

3-1 Weighing Design When Tare Weights Are Used

(Densities used to compute air buoyancy correction)

(Air buoyancy correction on the tare weights)

**Laboratory data and conditions:**

  ------------------------- -------------- ------------------- ------------
  Date                      8/18/96        Temperature         21.7

  Balance                   AT 1005        Pressure            753.5

  Load                      1 kg           Relative Humidity   45

  Pooled within process     0.023 mg       Calculated Air      1.182
  s.d., s~w~=                              Density             mg/cm^3^

  Check standard s.d., s~p~ 0.10 mg                            
  =                                                            
  ------------------------- -------------- ------------------- ------------

**Mass standard(s) data:**

+-------+----------+-----+--------+-----+------------+-----+--------+
| ID    | Mass =   | D   | Unc~   | ID  | Mass = N + | D   | Unc~   |
|       | N + C    | ens | (k=1)~ |     | C          | ens | (k=1)~ |
|       |          | ity |        |     |            | ity |        |
|       |          |     | (mg)   |     | \(g\)      |     | (mg)   |
|       |          | (g/ |        |     |            | (g/ |        |
|       |          | cm^ |        |     |            | cm^ |        |
|       |          | 3^) |        |     |            | 3^) |        |
+=======+==========+=====+========+=====+============+=====+========+
| N~x~  | 1000 g   | 7   | TBD    | t   | NA         |     |        |
|       |          | .84 |        | ~x~ |            |     |        |
+-------+----------+-----+--------+-----+------------+-----+--------+
| M~s~  | 9        | 8   | 0.0327 | t   | NA         |     |        |
|       | 99.99850 |     | mg     | ~s~ |            |     |        |
|       | g        |     |        |     |            |     |        |
+-------+----------+-----+--------+-----+------------+-----+--------+
| M~sc~ | 1        | 8   | 0.0327 | t~  | NA         |     |        |
|       | 000.0023 |     | mg     | sc~ |            |     |        |
|       | g        |     |        |     |            |     |        |
+-------+----------+-----+--------+-----+------------+-----+--------+
| sw    | 50.086   | 8   | 0.0010 |     |            |     |        |
|       | mg       | .41 | mg     |     |            |     |        |
+-------+----------+-----+--------+-----+------------+-----+--------+

N = Nominal, C = Correction, M = *True Mass*

**Laboratory observations:**

  -------------- ------ ------ ---------- ------ ------ ---------- ------ ------
  Balance                                                                 
  Observations                                                            

  S - X = a~1~                 S - S~c~ =               X - S~c~ =        
                               a~2~                     a~3~              

  S + t~s~       10     00     S + t~s~   10     30     X + t~x~   15     50

  X + t~x~       15     30     S~c~ +     14     00     S~c~ +     14     10
                               t~sc~                    t~sc~             

  X + t~x~ + sw  65     30     S~c~ +     64     10     S~c~ +     64     00
                               t~sc~ + sw               t~sc~ + sw        

  S + t~s~ + sw  60     10     S + t~s~ + 60     40     X + t~x~ + 65     60
                               sw                       sw                

  a~1~ = -                     a~2~ = -                 a~3~ =            
  5.25829                      3.69845                  1.50538           
  -------------- ------ ------ ---------- ------ ------ ---------- ------ ------

Note: dotted line represents decimal point.

Calculate "a" values:

$$\text{a = }\frac{(O_{1}\text{ - }O_{2}\text{ + }O_{4}\text{ - }O_{3}}{2}\text{  }\frac{M_{\text{sw}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{\text{sw}}} \right)}{O_{3}\text{ - }O_{2}}$$

Calculate short term within process standard deviation and conduct
F-test:

s~w~ = .577(a~1~ - a~2~ + a~3~) = -0.03142

$$\text{F-statistic = }\frac{{s_{w}}^{2}}{(\text{ Pooled }s_{w})^{2}}\text{ < value}\underset{}{}\text{F-table 9}\text{.}5$$

$$\text{F-statistic = }\frac{{\text{-0}\text{.}\text{03142}}^{2}}{{0\text{.}\text{023}}^{2}}\text{ = 1}\text{.}\text{87  <  7}\text{.}\text{31value}\underset{}{}\text{F-table 9}\text{.}\text{5 d}\text{.}f\text{.}\text{  = 40}$$

F-test passes? **[Yes]{.underline}** No

Evaluate check standard (by plotting on a control chart or with a
t-test):

$$d_{s_{c}}\text{ = }\frac{- a_{1}\text{ - 2}a_{2}\text{ - }a_{3}}{3}\text{ = }\frac{\text{-5}\text{.}\text{25829 - 2}(\text{-3}\text{.}\text{69845})\text{ - 1}\text{.}\text{50538}}{3}\text{  = 3}\text{.}\text{71660 mg}$$

$$M_{s_{c}}\text{ = }\frac{M_{s}\left( \text{1-}\frac{\rho_{A}}{\rho_{s}} \right)\text{ + }d_{s_{c}}\text{ + }M_{t_{s}}\left( \text{1-}\frac{\rho_{A}}{\rho_{t_{s}}} \right)\text{ - }M_{t_{s_{c}}}\left( \text{1-}\frac{\rho_{A}}{\rho_{t_{s_{c}}}} \right)}{\text{1-}\frac{\rho_{A}}{\rho_{s_{c}}}}$$

$$M_{s_{c}}\text{ =  }\frac{\text{999}\text{.}\text{9985 g }\left( \text{1 - }\frac{0\text{.}\text{001182}}{8} \right)\text{ + 0}\text{.}\text{00371660}}{\text{1-}\frac{0\text{.}\text{001182}}{8}}\text{ = 1000}\text{.}\text{002217 g}$$

Check standard passes? **[Yes]{.underline}** No

If both F-test and check standard pass the tests, calculate the mass of
the unknown test item:

$$d_{x}\text{ = }\frac{\text{-2}a_{1}\text{ - }a_{2}\text{ + }a_{3}}{3}\text{ = }\frac{\text{-2}(\text{-5}\text{.}\text{25829})\text{ - }(\text{-3}\text{.}\text{69845})\text{ + 1}\text{.}\text{50538}}{3}\text{ = 5}\text{.}\text{24014  mg}$$

$$M_{x}\text{ = }\frac{M_{s}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{s}} \right)\text{ + }d_{x}\text{ + }M_{t_{s}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{t_{s}}} \right)\text{ - }M_{t_{x}}\left( \text{ 1 - }\frac{\rho_{A}}{\rho_{t_{x}}} \right)}{\text{1 - }\frac{\rho_{A}}{\rho_{x}}}$$

$$M_{x}\text{ = }\frac{\text{999}\text{.}\text{9985 g }\left( \text{ 1 - }\frac{0\text{.}\text{001182}}{8} \right)\text{ + 0}\text{.}\text{00524014 g}}{\text{1 - }\frac{0\text{.}\text{001182}}{7\text{.}\text{84}}}\text{ = 1000}\text{.}\text{006757 g}$$

$$\text{CM}_{x}\text{ = }\frac{M_{x}\left( \text{1-}\frac{0\text{.}\text{0012}}{\rho_{x}} \right)}{\text{1-}\frac{0\text{.}\text{0012}}{\rho_{\text{ref}}}}\text{ = }\frac{\text{1000}\text{.}\text{006757 g }\left( \text{1-}\frac{0\text{.}\text{0012}}{7\text{.}\text{84}} \right)}{\text{1-}\frac{0\text{.}\text{0012}}{8}}\text{ = 1000}\text{.}\text{003695 g}$$

where ρ~ref~ refers to the reference density 8.0 g/cm^3^, or
conventional mass.

Uncertainty:

$$\text{U = 2 * }U_{c}\text{ = 2 * }\sqrt{{u_{s}}^{2}\text{ + }{s_{p}}^{2}\text{ + }{u_{o}}^{2}}\text{ = 2 * }\sqrt{\left( \frac{0\text{.}\text{098 mg}}{3} \right)^{2}\text{ + }{0\text{.}\text{10 mg}}^{2}\text{ + }{0\text{.}\text{005 mg}}^{2}}$$

U = 0.210638 mg = 0.21 mg

C~x~ = 3.70 mg ± 0.21 mg (Conventional Mass vs 8.0 g/cm^3^).