Short Answer
Rule Statement: The GUM Reporting Framework
In a calibration certificate, significant figures are governed by the uncertainty, not by the raw output of the instrument. The expanded uncertainty U defines the last meaningful decimal place of the measured value. The GUM recommends that U be reported to at most two significant digits, and the measured value be rounded to the same decimal place. This is the central reporting rule for calibration certificates.
The rule applies to both standard uncertainty uc(y) and expanded uncertainty U. For calibration certificates, expanded uncertainty is typical. You must also state the coverage factor k (usually k = 2 for approximately 95 % coverage) and the units. ISO/IEC 17025:2017 Clause 7.8.2.1 requires calibration certificates to include measurement uncertainty. ILAC P14:2020 Section 5.4 reinforces the two-significant-figure convention.
‘The numerical values of the estimate y and its standard uncertainty uc(y) or expanded uncertainty U should not be given with an excessive number of digits. It is usually appropriate to give uc(y) or U to at most two significant digits.’ — JCGM 100:2008 (GUM), Clause 7.2.6
Core rule in practice
- Determine the expanded uncertainty U from the uncertainty budget.
- Round U to at most two significant figures. Do not round it to one significant figure unless the uncertainty is so small that two figures would be misleading or the standard you follow permits it.
- Round the measured value to the same decimal place as the rounded U.
- Report the result as: measured value ± U (units), with k and coverage probability.
- If the uncertainty is used for further calculations, retain more digits in intermediate steps; round only for the final certificate per GUM 7.2.7.
Standards Citation: GUM, ILAC, NIST, ISO
Several standards converge on the same reporting discipline. The table below summarizes the key clauses and their requirements.
| Standard | Clause | Key requirement |
|---|---|---|
| JCGM 100:2008 (GUM) | 7.2.6 | Report uc(y) or U to at most two significant digits; avoid excessive digits. |
| JCGM 100:2008 (GUM) | 7.2.4 | State the coverage factor k when reporting expanded uncertainty. |
| JCGM 100:2008 (GUM) | 7.2.7 | Retain more digits when uncertainty is used in subsequent calculations. |
| ILAC P14:2020 | Section 5.4 | Report expanded uncertainty to at most two significant figures; round result to same decimal place. |
| NIST TN 1297 | Section 7.2 | Expanded uncertainty to no more than two significant digits; result to same decimal place. |
| ISO/IEC 17025:2017 | 7.8.2.1 | Calibration certificates shall include measurement uncertainty. |
| ISO 80000-1:2009 | 5.3.2 | Rounding conventions for significant digits; ties-to-even is common in ISO practice. |
| ASTM E29-22 | Sections 5–7 | Standard practice for using significant digits and rounding in test data. |
Worked Examples
Example 1: Dimensional calibration
Raw result: y = 10.2345 mm, U = 0.1234 mm (k = 2).
- Round U to two significant figures: 0.1234 mm → 0.12 mm.
- Identify the last decimal place of U: hundredths (0.01 mm).
- Round y to the same decimal place: 10.2345 mm → 10.23 mm.
- Report: 10.23 mm ± 0.12 mm, k = 2 (approximately 95 % coverage).
Example 2: Temperature calibration
Raw result: y = 23.456 °C, U = 0.47 °C (k = 2).
- U already has two significant figures: 0.47 °C.
- Last decimal place of U: hundredths (0.01 °C).
- Round y to hundredths: 23.456 °C → 23.46 °C.
- Report: 23.46 °C ± 0.47 °C, k = 2.
Example 3: Electrical calibration with small uncertainty
Raw result: y = 1.234567 V, U = 0.001234 V (k = 2).
- Round U to two significant figures: 0.001234 V → 0.0012 V.
- Last decimal place of U: ten-thousandths (0.0001 V).
- Round y to ten-thousandths: 1.234567 V → 1.2346 V.
- Report: 1.2346 V ± 0.0012 V, k = 2.
Example 4: One-significant-figure case
Raw result: y = 12.34 °C, U = 0.3 °C (k = 2).
- U has one significant figure: 0.3 °C. This is acceptable when the uncertainty is already at the limit of meaningful resolution.
- Last decimal place of U: tenths (0.1 °C).
- Round y to tenths: 12.34 °C → 12.3 °C.
- Report: 12.3 °C ± 0.3 °C, k = 2.
Counter-Examples: Common Misapplications
These examples show what not to do and why.
- Excessive digits in both value and uncertainty: 10.2345 mm ± 0.1234 mm. The uncertainty is over-reported, and the value implies precision that the calibration does not support. Correct: 10.23 mm ± 0.12 mm.
- Misaligned decimal places: 10.23 mm ± 0.1 mm. The uncertainty is in tenths, so the value should not have hundredths. Correct: 10.2 mm ± 0.1 mm.
- Over-rounding the uncertainty: 0.14 mm → 0.1 mm. This loses information and may understate the uncertainty. Keep two significant figures unless a standard or contract requires otherwise.
- Rounding the result before the uncertainty: If you first round 10.2345 mm to 10.2 mm and then align to 0.12 mm, you may introduce double-rounding error. Round U first, then round y once.
- Omitting k: Reporting ‘10.23 mm ± 0.12 mm’ without k is incomplete. GUM 7.2.4 requires the coverage factor for expanded uncertainty.
- Using significant figures as a substitute for uncertainty: A value of 10.23 mm does not, by itself, state an uncertainty. Calibration certificates must report U explicitly.
Convention Comparison Table
| Convention | Uncertainty digits | Result rounding | Rounding method | Notes |
|---|---|---|---|---|
| GUM | At most 2 sig figs | Same decimal place as U | Not specified | State k; retain more digits for intermediate calculations. |
| ILAC P14 | At most 2 sig figs | Same decimal place as U | Not specified | Calibration-specific policy; widely used by accredited labs. |
| NIST TN 1297 | No more than 2 sig figs | Same decimal place as U | Not specified | NIST guidance for expressing uncertainty. |
| ISO 80000-1 | Depends on context | Depends on context | Ties-to-even common | General quantity and unit rules; supports consistent precision. |
| ASTM E29 | Depends on test method | Same place as uncertainty | Half-up and ties-to-even variants | Standard practice for significant digits in test data. |
Quick Reference Table: Reporting Steps
| Step | Action | Example |
|---|---|---|
| 1 | Start with measured value y and expanded uncertainty U. | y = 10.2345 mm, U = 0.1234 mm |
| 2 | Round U to at most two significant figures. | U = 0.12 mm |
| 3 | Find the last decimal place of rounded U. | Hundredths (0.01 mm) |
| 4 | Round y to that same decimal place. | y = 10.23 mm |
| 5 | State coverage factor k and units. | k = 2, mm |
| 6 | Final certificate statement. | 10.23 mm ± 0.12 mm, k = 2 |
Common Mistakes
- Reporting too many digits: Instrument displays often show 6–7 digits, but the calibration uncertainty may support only 2–3. The certificate should not simply copy the display.
- Using decimal places instead of significant figures: ‘Two significant figures’ is not the same as ‘two decimal places’. 0.0012 V has two significant figures; 0.00120 V has three.
- Forgetting to align the result: If U = 0.12 mm, the result should end in hundredths, e.g., 10.23 mm, not 10.234 mm.
- Rounding uncertainty down too aggressively: Rounding 0.14 to 0.1 can understate uncertainty. Two significant figures are preferred.
- Not stating the coverage factor: Expanded uncertainty without k is ambiguous. Always report k (and νeff when relevant).
- Double rounding: Rounding intermediate values and then rounding again can shift the last digit. Round once at the final reporting stage.
- Ignoring customer or accreditation requirements: Some contracts, ASTM methods, or accreditation bodies may require more digits or a specific rounding rule. Check the applicable specification.
Related Rules and Internal Resources
For deeper reference, see our articles on rounding rules, significant figures, measurement uncertainty, GUM Guide, rounding vs significant figures, and standard deviation reporting. These pages cover the foundational conventions that support calibration certificate reporting.
Sources & Further Reading
- JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM), Clause 7.2.6.
- ILAC P14:09/2020, ILAC Policy for Uncertainty in Calibration, Section 5.4.
- NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Section 7.2.
- ISO/IEC 17025:2017, General requirements for the competence of testing and calibration laboratories, Clause 7.8.2.1.
- ISO 80000-1:2009, Quantities and units — Part 1: General, Clause 5.3.2.
- ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
FAQ
How many significant figures should uncertainty have in a calibration certificate?
Usually two significant figures. GUM 7.2.6 says at most two significant digits for u_c(y) or U. One significant figure is sometimes acceptable for very small uncertainties, but two is the standard default.
Should I round the measured value to the same decimal place as the uncertainty?
Yes. After rounding U, round the measured value to the same last decimal place. For example, if U = 0.12 mm, report the value to hundredths: 10.23 mm, not 10.234 mm.
Do I need to report the coverage factor k?
Yes, when reporting expanded uncertainty. GUM 7.2.4 requires the coverage factor. Most calibration certificates use k = 2, corresponding to approximately 95 % coverage for a normal distribution.
Can I report more than two significant figures for uncertainty?
Only when necessary for subsequent calculations, or when a specific standard, contract, or accreditation requirement calls for it. GUM 7.2.7 permits more digits in intermediate calculations. For the final certificate, two significant figures is the norm.
What rounding method should I use?
GUM does not prescribe a single rounding method. ISO 80000-1:2009 Clause 5.3.2 and ASTM E29-22 provide guidance. Common practice is round-half-up or ties-to-even; be consistent and document your laboratory's rule.
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