How to Calculate Calibration Uncertainty

Short answer: Calibration uncertainty is calculated by combining the CMC uncertainty, UUT resolution, and UUT repeatability using the square root of the sum of squares (RSS) method per the JCGM 100:2008 section 5.1.2 and ILAC P14 section 5.3. The combined uncertainty is expanded to a 95.45 % confidence interval using an appropriate coverage factor (k), rounded to two significant digits, and reported in calibration certificates with its associated results.

 
 

Calculate Calibration Uncertainty Step-by-Step

Calculate calibration uncertainty per the JCGM 100:2008 and ILAC P14 following the instructions below.

  1. Find the CMC Uncertainty from the Scope of Accreditation.
  2. Find the UUT resolution (i.e. smallest incremental change).
  3. Determine the UUT repeatability (i.e. standard deviation of repeated measurements).
  4. Convert each uncertainty contributor to a standard uncertainty.
    1. CMC Uncertainty: Divide uncertainty by k=2.
    2. UUT Resolution: Divide uncertainty by 2√3.
    3. UUT Repeatability: Divide uncertainty by √n (i.e. observations in final result).
  5. Calculate the combined standard uncertainty.
    1. Square each uncertainty (i.e. convert to variance).
    2. Add together the squared uncertainties (i.e. combined variance).
    3. Calculate the square root of the combined variance.
  6. Determine the coverage factor (k) for a 95.45 % confidence interval.
    1. Use k=2 from JCGM 100:2008 Table G.1, or
    2. Use effective degrees of freedom and JCGM 100:2008 Table G.2.
  7. Calculate the expanded uncertainty – Multiple the coverage factor and the combined standard uncertainty.

 
 

Calibration Uncertainty Calculation Example

The table below shows an example uncertainty budget for calculating calibration uncertainty per ILAC P14.

Example budget calculating calibration uncertainty
Description Symbol Sensitivity
Coefficient		 Uncertainty
Value		 Unit Distribution Divisor Standard
Uncertainty		 Unit
CMC Uncertainty Ucmc 1 0.250 mg Normal 2 0.1250 mg
UUT Resolution Ud 1 0.010 mg Rectangular 2√3 0.0029 mg
UUT Repeatability ur 1 0.079 mg Normal 1 0.079 mg
Combined Uncertainty uc 0.1479 mg
Coverage Factor k 2.000
Expanded Uncertainty Ucal 0.30 mg

Download: ISOBudgets Calibration Uncertainty Calculator – Version 1.0

ISOBudgets ILAC P14 Calibration Uncertainty Calculator Version 1,0 - Free

 
 

FAQ

What is calibration uncertainty?

Calibration uncertainty is the measurement uncertainty expressed in calibration reports at a 95 % confidence interval, where k=2, with two significant digits. It is calculated by combining the laboratory’s CMC uncertainty with the Unit Under Test (UUT) resolution and repeatability per ILAC P14.

 
 

What are ILAC P14’s Requirements for Reporting Calibration Uncertainty in Certificates?

Per ILAC P14 Section 5, calibration uncertainty reported in certificates must meet the following requirements:

  • Must report in compliance with the GUM (JCGM 100:2008).
  • Must be reported as the result and its uncertainty (y ± U).
  • Must state the coverage factor and coverage probability (i.e. 95 % C.I., k=2).
  • Must be rounded to, at most, two-significant digits.
  • Must include the CMC Uncertainty and UUT resolution and repeatability.
  • Must not be less than the CMC in the lab’s Scope of Accreditation.
  • Must be reported as an absolute uncertainty or relative uncertainty.

 
 

What Significant Contributors are included in Calibration Uncertainty?

The calibration uncertainty is a combination of the lab’s Calibration and Measurement Capability (CMC) uncertainty and contributors relevant to the Unit Under Test (UUT).

The three uncertainty contributors must be evaluated are:

  • CMC Uncertainty: The Calibration and Measurement Capability of the laboratory.
  • UUT Resolution: The smallest incremental change the Unit Under Test can display.
  • UUT Repeatability: The variability of the Unit Under Test after repeated measurements.

 
 

How to round uncertainty to two significant digits?

Round uncertainty to two significant digits per JCGM 100:2008 and ILAC P14 following the instructions below.

  1. Review the numerical value of measurement uncertainty.
  2. Find the first significant digit (i.e. first non-zero number).
  3. Find the third significant digit (i.e. two numbers immediately to the right).
  4. Round the second significant digit using an appropriate method (JCGM 100:2008 section 7).
    1. Conventional rounding (i.e. Round up n ≥ 5, Round down n < 5)
    2. Round up to the next larger number.
    3. Round down to the next smaller number.
    4. Round to nearest even digit.

 
 

What is difference between CMC Uncertainty and Calibration Uncertainty?

CMC Uncertainty is reported in the scope of accreditation and represents the laboratory’s measurement capability offered to customers under typical conditions while calibration uncertainty is reported in the calibration report and represents the expanded measurement uncertainty associated with the reported result.

The CMC Uncertainty in the scope of accreditation can represent the calibration uncertainty for a best existing device per ILAC P14 section 5.3. However, the actual UUT resolution and reproducibility must replace the best existing device at the time of calibration and be reported in the calibration certificate. Otherwise, contributions from the best existing device can be omitted from the CMC uncertainty so long as it is disclosed in the scope of accreditation (typically with a footnote).

 
 

What is CMC Uncertainty?

According to the proceedings from the 96th meeting of the CIPM (2007), CMC uncertainty is a calibration and measurement capability available to customers under normal conditions:

  1. As published in the BIPM key comparison database (KCDB) of the CIPM MRA; or
  2. As described in the laboratory’s scope of accreditation granted by an ILAC-MRA signatory.

 
The term was approved by the ILAC General Assembly in October 2007 according to the proceedings from the 96th meeting of the CIPM and is considered identical to the previously used term “Best Measurement Capability” or BMC.

For more information, refer to ILAC P14 policy, appendix A.

 
 

What is Test Uncertainty Ratio for Calibration?

Test uncertainty ratio is ratio of the tolerance or specification interval in relation to the expanded uncertainty interval associated with a measurement process or value.

It is defined in the ANSI Z540.3 Handbook. Additionally, an identical term “measurement capability index” is defined in the JCGM 100 106:2012.

It is commonly used to evaluate the measurement risk or suitability of a measurement process. However, it does not take into account the specific risk (e.g. PFA or PFR) associated with a measurement result.

Common test uncertainty ratios recommended by many standard methods and guidance documents are between 3:1 and 5:1. The most commonly accepted or referenced test uncertainty ratio is a 4:1 ratio.

It should be noted that calibration labs are not allowed to report accredited results with a TUR ratio less than 1:1.

 
 

Glossary

CMC Uncertainty
calibration and measurement capability available to customers under normal conditions: As published in the BIPM key comparison database (KCDB) of the CIPM MRA or as described in the laboratory’s scope of accreditation granted by an ILAC-MRA signatory. (Source: CIPM 96th Meeting)
Measurement Uncertainty
non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used. (JCGM 200:2012, 2.26)
Standard Measurement Uncertainty
measurement uncertainty expressed as a standard deviation. (Source: JCGM 200:2012, 2.30)
Expanded Measurement Uncertainty
the product of a combined standard measurement uncertainty and a factor larger than the number one. (Source: JCGM 200:2012, 2.35)
Combined Standard Measurement Uncertainty
standard measurement uncertainty that is obtained using the individual standard measurement uncertainties associated with the input quantities in a measurement model. (Source: JCGM 200:2012, 2.31)
Coverage Factor
number larger than one by which a combined standard measurement uncertainty is multiplied to obtain an expanded measurement uncertainty. (Source: JCGM 200:2012, 2.38)
Level of Confidence
the likelihood that a set of measurement values are contained within a specified coverage interval. (Source: JCGM 200:2012, 2.37)
Significant Figure
each of the digits of a number that are used to express it to the required degree of accuracy, starting from the first nonzero digit. (Source: Oxford Languages and Google)
Measurement Repeatability
measurement precision under a set of repeatability conditions of measurement (Source: JCGM 200:2012, 2.21)
Repeatability Condition of Measurement
condition of measurement, out of a set of conditions that includes the same measurement procedure, same operators, same measuring system, same operating conditions and same location, and replicate measurements on the same or similar objects over a short period of time (Source: JCGM 200:2012, 2.20)
Resolution
smallest change in a quantity being measured that causes a perceptible change in the corresponding indication (JCGM 200:2012, 4.14)
Relative Measurement Uncertainty
measurement uncertainty expressed in a term relative to the measurand.
Absolute Measurement Uncertainty
measurement uncertainty expressed in the same unit of measurement as the measurand.