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SOP No. 10 

Recommended Standard Operating Procedure 

for 

Calibration of Rigid Rules 

1. Introduction 

1.1. Purpose 

This SOP describes the procedure to be followed for the calibration of rigid rules 

by comparison to the 18 inch metal rule issued to each State as the State reference 

standard which is calibrated in the interval from 1 inch to 13 inches. Equations 

provided here assume that the rigid rules being compared have the same 

coefficient of expansions. If rigid rules having different coefficients of expansion 

are compared at a temperature other than 20 °C the procedure and equations of 

SOP 11 must be used. 

The maximum length of rule that can be directly compared to the standard rule is 

12 inches. However, longer lengths can be calibrated in segments of 12 inches 

with reference to the standard rule. 

1.2. Prerequisites 

1.2.1. Valid calibration certificates with appropriate values and uncertainties 

must be available for all of the standards used in the calibration. All 

standards must have demonstrated metrological traceability to the 

international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST. 

1.2.2. The ocular microscope used in measuring differences in lengths must be in 

good operating condition and must be equipped with a graduated reticle 

having established traceability. 

1.2.3. The operator must be trained and experienced in precision measuring 

techniques with specific training in GMP 2, GMP 8, GMP 9, SOP 10, and 

SOP 29. 

1.2.4. Laboratory facilities must comply with following minimum conditions to 

meet the expected uncertainty possible with this procedure. Equilibration 

of supporting surface, standard rule and rule to be calibrated requires 

environmental stability of the laboratory within the stated limits for a 

minimum of 24 hours before a calibration. 
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Table 1. Environmental conditions. 

Temperature Requirements During a Calibration Relative Humidity 

(%) 

Lower and upper limits: 18 °C to 22 °C 

Maximum changes: < ± 1 °C / 24 h and ± 0.5 °C / 1 h 40 to 60 ± 10 / 4 h 

2. Methodology 

 2.1. Scope, Precision, Accuracy 

The accuracy of calibration of standard rules is possible to within 0.000 1 inch, 

provided suitable calibration of standards are obtained. The precision of 

intercomparison and the accuracy of the standard limit the uncertainty of 

calibration to 0.002 inch under optimum conditions. 

2.2. Summary 

A rigid rule (test rule) is calibrated by comparing intervals on it with intervals of 

the standard rule. A reticle eye piece (ocular micrometer) is used for this purpose. 

Test rules longer than the standard rule may be calibrated in segments, using the 

last calibrated graduation as the zero graduation mark for the succeeding 

segments. Deviations from nominal that are calculated for each successive 

interval are cumulative. 

2.3. Apparatus/Equipment Required 

2.3.1. Length bench or similar flat surface on which to lay the test rule and the 

standard rule. 

2.3.2. Calibrated standard rule. 

2.3.3. Microscope with calibrated graduated reticle having graduations spaced at 

intervals no greater than 0.002 inch. 
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2.4. Symbols 

Table 2. Symbols used in this procedure. 

Symbol Description 

A, B, C, D Calculated centers of graduations 

S Standard 

X Unknown 

X0L Left edge of zero on unknown 

X0R Right edge of zero on unknown 

S0L Left edge of zero on standard 

S0R Right edge of zero on standard 

Xm Center of graduation of unknown 

Sm Center of graduation of standard 

di Difference between X and S. The subscript i, designates the trial number, 1 or 2. 

LS Calibrated length of standard 

LX Calibrated length of test rule 

2.5. Procedure 

2.5.1. Both the test rule and the standard rule must be in temperature equilibrium 

with the environment of the length laboratory. 

2.5.2. Place the test rule and the standard rule on the length bench or similar flat 

surface, parallel to one another with the reading edges adjacent. It is not 

necessary to have the “zero” graduations in coincidence. 

 Ordinarily, this will require that one rule reads left-to-right (increasing 

units) while the other reads right-to-left (decreasing units). In this case, for 

convenience of calculation, the standard rule is placed in the right-to-left 

position. A worksheet to reverse the calibration on the standard rule is 

provided at the end of this SOP. Place shims under the rules as necessary 

so that the upper surfaces of both are in the same plane. 

2.5.3. Place the microscope on the bench in the vicinity of the zero position and 

align it so that its reticle scale is parallel to the scales under test. (See 

GMP No. 2 for instructions on reading graduations with the microscope.) 

2.5.3.1. Observe readings of left and right sides of the corresponding 

graduation of the standard rule and record to the nearest 

0.001 inch. The average of these readings will give a value for A. 

2.5.3.2. Observe readings of left and right sides of the corresponding 

graduation of the test rule and record to the nearest 0.001 inch. 

The average of these readings will give a value for B. If the 
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“zero” graduation is the end of the rule, only the reading for the 

end of the rule is taken. 

2.5.4. Move the microscope successively to each graduation to be calibrated and 

record readings as described in 2.5.3.1 and 2.5.3.2 identifying the readings 

for the standard rule as C and for the unknown as D. CAUTION! Be 

certain that the rules are not disturbed during movement of the 

microscope. 

2.5.5. Return the microscope to the zero graduation and repeat readings (2.5.3.1. 

and 2.5.3.2.) to verify that the rules have not been disturbed. Accept all 

previous data if the initial zero readings do not disagree with the final zero 

readings by more than 0.002 inch; otherwise, discard all previous data and 

repeat entire sequence of readings until a satisfactory set of zero readings 

are obtained. 

2.5.6. Repeat 2.5.3. thru 2.5.5. for every additional segment of the test rule 

requiring calibration. This will require repositioning the rules, aligning the 

last measured interval graduation on the test rule with the initial 

graduation of the standard rule. Deviations from nominal for successive 

segments are cumulative. 

2.5.7. Move the rules and reposition, making a second set of measurements as 

directed in 2.5.3, 2.5.4, and 2.5.5 for Trial 2. 

. 

3. Calculations 

3.1. Calculate the difference between the Initial (A, B) and Final (A1, B1) zero 

measurements to ensure that the initial and final readings agree within 0.002 inch. 

(2.5.5.) 

 |A – B – A1 + B1| (1) 

3.2. Calculate the center of graduation for unknown, Xm, and standard, Sm, for each set of 

measurements and each scale interval. The center of the “starting” graduation on the 

standard rule is A; for the test rule it is B. The subsequent centers of graduations for the 

standard and the test rule are recorded as C and D, respectively, for each interval 

measured. For each trial, the values for A and B will remain constant for all measured 

intervals on that trial, and will be used to compute the measured differences between 

the test rule and the standard. 

A C S S S L R

m , 2

( )    (2) 
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B D X X X L R

m , 2

( )    (3) 

3.3. Calculate the differences, d1, and d2 between X and S for each scale interval and 

each trial. (The subscript i is associated with the trial number.) 

di  A  B Ci  Di (4) 

3.4. Obtain the calibrated length of the standard, LS, for the measured interval from the 

calibration certificate for the standard rule. 

3.5. Calculate the length of the test rule, LX, for each interval measured using the mean 

of the measured differences. 

 

X LS

d d L   2

1 2 (5) 

4. Measurement Assurance 

4.1. Duplicate the process with a suitable check standard or have a suitable range of 

check standards for the laboratory. See NISTIR 7383 SOP 17, SOP 20 and 

NISTIR 6969 SOP 30. Plot the check standard length and verify it is within 

established limits OR a t-test may be incorporated to check the observed value 

against an accepted value. The mean of the check standard observations is used to 

evaluate bias and drift over time. Check standard observations are used to 

calculate the standard deviation of the measurement process which contributes to 

the Type A uncertainty components. 

4.2 If a standard deviation chart is used for measurement assurance, the standard 

deviation of each combination of Trial 1 and Trial 2 is calculated and the pooled 

(or average) standard deviation is used as the estimate of variability in the 

measurement process. Note: the pooled or average standard deviation over time 

will reflect varying conditions of test items that are submitted to the laboratory. A 

standard deviation chart will be needed for each interval calibrated so that the 

variability resulting from transfers will be measured (the number of charts may be 

adjusted through analysis using F-tests). 

5. Assignment of Uncertainty 

5.1. The limits of expanded uncertainty, U, include estimates of the standard 

uncertainty of the length standards used, us, estimates of the standard deviation of 

the measurement process, sp, and estimates of the effect of other components 

associated with this procedure, uo. These estimates should be combined using the 

root-sum-squared method (RSS), and the expanded uncertainty, U, reported with 

a coverage factor to be determined based on degrees of freedom, which if large 
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enough will be 2, (k = 2), to give an approximate 95 percent level of confidence. 

See NISTIR 6969, SOP 29 (Standard Operating Procedure for the Assignment of 

Uncertainty) for the complete standard operating procedure for calculating the 

uncertainty. 

5.1.1 The expanded uncertainty for the standard, U, is obtained from the 

calibration report. The combined standard uncertainty, uc, 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. When 

transfers are used, us for values after the transfer are dependent and 

cumulative. See NISTIR 6969, SOP 29 for handling of dependent 

uncertainties. 

5.1.2. The standard deviation of the measurement process, sp, is taken from a 

control chart for a check standard or standard deviation charts. See 

NISTIR 7383, SOP 17, SOP 20, and NISTIR 6969, SOP 30. 

5.1.3. Uncertainty associated with bias, ud. Any noted bias that has been 

determined through analysis of control charts and round robin data must 

be less than limits provided in SOP 29 and included if corrective action is 

not taken. See SOP 29 for additional details 

5.1.4. Other standard uncertainties usually included at this calibration level 

include uncertainties associated with the ability to read the graduated 

reticle, only part of which is included in the process variability due to 

parallax and visual capabilities, and uncertainties associated with the 

graduations of the reticle. 

 Table 3. Example uncertainty budget table. 

Component Description Reference 

us Standard uncertainty for standards Calibration report, divide by k 

sp Standard uncertainty for the process Measurement assurance process; 

range charts 

ugr Standard uncertainty for graduated 

reticle 

Must be assessed experimentally or 

from a calibration certificate 

ud Standard uncertainty for disparity due 

to drift/bias 

Rectangular distribution and 

reasons, 0.577 d, 0.29 d; 

SOP 29 (NISTIR 6969) 

ures Standard uncertainty due to resetting of 

the rules 

Must be assessed experimentally; if 

an interval based standard deviation 

chart is used ures will be included in 

the control chart standard deviation 

value. 

uo Standard uncertainty for other factors 

6. Report 

Report results as described in SOP No. 1 Preparation of Calibration Certificates. 

 
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Appendix A 

Rigid Rule Calibration Data Sheet 

Date Environmental parameters 

Metrologist Before After Unc/ability to measure 

Test No. Temperature °C °C 

sp in Pressure mmHg mmHg 

df Humidity % % 

Based on NISTIR 6969, SOP 29, Appendix A at 95.45 % probability distribution: k factor 

 ID Range 

S 

X 

 Trial 1 Trial 2 

Rule Graduation Left Right Center of 

graduation 

Left Right Center of 

graduation 

S A = A = 

X B = B = 

S1 C1 = C1 = 

X1 D1 = D1 = 

d1 = A – B – C1 + D1 = d2 = A – B – C1 + D1 = 

 Average d = (d1 + d2) / 2 = LS = 

 Length of X, LX = average d + LS = 

S2 C2 = C2 = 

X2 D2 = D2 = 

d1 = A – B – C2 + D2 = d2 = A – B – C2 + D2 = 

 Average d = (d1 + d2) / 2 = LS = 

 Length of X, LX = average d + LS = 

S3 C3 = C3 = 

X3 D3 = D3 = 

d1 = A – B – C3 + D3 = d2 = A – B – C3 + D3 = 

 Average d = (d1 + d2) / 2 = LS = 

 Length of X, LX = average d + LS = 

S A1 = A1 = 

X B1 = B1 = 

Trial 1: Absolute difference between initial 

and final zero measurements = |A – B – A1 + B1|= 

Is result ≤ 0.002 inch? Yes/No If YES, accept all data. 

Trial 2: Absolute difference between initial 

and final zero measurements =|A – B – A1 + B1|= 

Is result ≤ 0.002 inch? Yes/No If YES, accept all data. 

 
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Form A-2. Worksheet to Reverse the Calibration of the Rigid Rule Standard 

Nominal 

Length of 

Interval 

(inches) 

Calculations to be performed Length of Interval 

(inches) 

1 L1 = C13 – C12 = - 

2 L2 = C13 – C11 = - 

3 L3 = C13 – C10 = - 

4 L4 = C13 – C9 = - 

5 L5 = C13 – C8 = - 

6 L6 = C13 – C7 = - 

7 L7 = C13 – C6 = - 

8 L8 = C13 – C5 = - 

9 L9 = C13 – C4 = - 

10 L10 = C13 – C3 = - 

11 L11 = C13 – C2 = - 

12 L12 = C13 – C1 = - 

To reverse the calibration on the State standard rigid rule, subtract the lengths for each calibrated 

interval from the length from the 1 inch to the 13 inch graduation. 

Example: 

Length from 13 to 12 = Length from 1 to 13 – Length from 1 to 12 

Length from 13 to 12 = 11.999 8 – 10.999 7 = 1.000 1 inch 

 
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Appendix B 

Calibration of a Metric Rule Using a Standard Indicating in Inches 

A. Work with the customer to determine the intervals they wish to have calibrated. 

Convert those interval to inches (E.g., 10 mm / 25.4 mm/in = 0.3937 in). It is helpful 

to make the conversions for all the interested intervals at one time. Round these 

conversions to the nearest 0.01 in. 

B. For the first calibration interval “a”, refer to the group of intervals from 1 in to 12.5 in 

and 1 in to 13 in on the calibration report for the lab’s standard rule. Select the 

report’s interval with the same numbers after the decimal point as the selected 

customer interval “a”. (E.g., for the customer’s 2.94 in interval, refer to the 1 in to 

12.94 in interval on the report.) An interval “a” which ends in a number less than 0.5 

should have a report interval selected which is 0.5 more than the listed value. (E.g., 

for the customer’s 0.39 in interval, refer to the 1 in to 12.89 in interval on the report.) 

The 12.xx value which has been selected will be the point on the standard which is 

lined up with the 0 calibration point on the customer’s rule. Record the appropriate 

standard length value for the referred to interval. (E.g., 11.888 4 for the 1 in to 12.89 

in interval.) 

C. Subtract the desired interval “a” from the 12.xx figure which was selected in step B. 

This difference “b” is the point on the standard rule which will be lined up with the 

desired calibration point on the customer’s rule. Refer to the calculated difference “b” 

on the calibration report for the laboratory 18 in standard rule. Record the appropriate 

standard length value for the interval “b”. (E.g., for the interval 0.39 in, subtract 

0.39 in from 12.89 in to obtain a difference of 12.50 in. The 12.5 in mark on the 

standard rule will be lined up with an interval which is 0.39 in (10 mm) from the 0 

point on the customer’s rule. Refer to the 1 in to 12.50 in interval on the data sheet 

and record a value of 11.498 1.) 

D. Subtract the value recorded in step C from the one recorded in step B to obtain the 

standard’s length for the interval “a” which will be observed. (E.g., subtract 11.498 1 

from 11.888 4 and you get a standard length of 0.390 3 in.) Use this value as your LS

in your calculations. 

E. Repeat steps B through D for all of the intervals to be calibrated. 

F. Set up the rules, take readings, and make calculations for each interval in accordance 

with this SOP. After calculating LX in inches, convert it to metric, and report the 

metric values to the customer.