Short Answer
Rule Statement
In a GUM uncertainty budget, an order-of-magnitude estimate is acceptable for screening, scoping, and Type B bounding, but it is not a formal substitute for an evaluated standard uncertainty in the final uncertainty statement. The practical rule is: use order-of-magnitude reasoning when a contributor is expected to be at least ten times smaller than the dominant combined standard uncertainty, and document the assumption. If the contributor is within a factor of about three of the dominant term, evaluate it properly.
This site is a precision-and-rounding reference, not merely a calculator. The calculator can check significant figures and rounding, but the reference material explains the metrological reasoning—why a small term may be omitted, how Type B bounds become standard uncertainties, and how GUM reporting rules interact with order-of-magnitude screening.
Standards Citation
- JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM), Clause 3.3.6 defines standard uncertainty; Clause 4.1 introduces Type A and Type B evaluation; Clause 4.3 covers Type B judgment; Clause 5.1.2 gives the law of propagation of uncertainty; Clause 7.2.2 recommends at most two significant digits for reported uncertainty; Clause 7.2.6 requires consistent rounding of the estimate and its uncertainty.
- NIST TN 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, 1994, Section 2.3 and Appendix A discuss Type B evaluations based on scientific judgment and available information.
- ISO/IEC Guide 98-3:2008 is the ISO publication of the GUM; its Clause 4.3 and 5.1.2 carry the same technical requirements.
- ISO 80000-1:2022, Quantities and units — Part 1: General, Clause 7.3 addresses rounding, and Clause 7.4 addresses significant figures.
- ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, provides rounding conventions for conformance testing.
GUM 5.1.2: The combined standard uncertainty uc(y) is the positive square root of the combined variance obtained from all uncertainty components, however they were evaluated.
Worked Examples
Example 1: Screening a temperature effect
A 10 V reference is calibrated with a dominant standard uncertainty u1 = 120 µV. The temperature coefficient is expected to be ±0.5 µV/°C, and the laboratory temperature may vary by ±2 °C. The half-width is a = (0.5 µV/°C)(2 °C) = 1 µV. Assuming a rectangular distribution, u2 = a/√3 ≈ 0.58 µV. The ratio u2/u1 ≈ 0.0048, far below 0.1. The combined uncertainty without the temperature term is 120 µV; with it, √(120² + 0.58²) ≈ 120.001 µV. The difference is negligible at two significant digits. Order-of-magnitude screening justifies omission.
Example 2: Type B order-of-magnitude bound
A digital multimeter has a resolution of 1 mV. The half-width is a = 0.5 mV, and a rectangular Type B evaluation gives u = 0.5/√3 ≈ 0.29 mV. If the dominant combined standard uncertainty is 2.0 mV, the ratio is 0.145. That is larger than 0.1, so the resolution term should not be dismissed by order-of-magnitude reasoning; use the numerical value in the budget.
Example 3: When order-of-magnitude becomes final
If the final expanded uncertainty is U ≈ 5 mV and will be reported to one significant digit, contributions below about 0.5 mV may be treated by order-of-magnitude screening. The rationale must still be recorded, and the dominant terms must be evaluated normally.
Counter-Examples
- Using order of magnitude to hide a dominant uncertainty. If a contribution is the same order as the largest term, it must be evaluated fully. Calling it “about the same” is not a GUM uncertainty evaluation.
- Treating a bound as a standard uncertainty. A statement such as “within 0.1 mV” is a half-width, not u. Convert it using an assumed distribution, e.g., u = a/√3 for rectangular.
- Assuming order-of-magnitude components cancel. Independent uncertainties add in quadrature; they do not cancel like systematic biases.
- Double rounding. Rounding a component to one significant figure, combining, and then rounding again can introduce avoidable error. See Double Rounding Error.
- Forgetting sensitivity coefficients. A small input uncertainty can have a large effect if its sensitivity coefficient is large. Use Error Propagation before dismissing it.
Convention Comparison Table
| Context | GUM treatment | Order-of-magnitude role | Reporting consequence |
|---|---|---|---|
| Preliminary budget | Not a formal evaluation stage | Identify dominant terms and likely negligible terms | Internal only |
| Type B evaluation | JCGM 100:2008 Clause 4.3 | Estimate bounds when data are scarce | Document distribution and divisor |
| Formal uncertainty budget | Clause 5.1.2 quadrature | Only for components screened as negligible | Do not omit material components |
| Reporting | Clauses 7.2.2 and 7.2.6 | Not a substitute for uc or U | Round y and u to the same decimal place |
Common Mistakes
- Confusing order of magnitude with significant figures. Order of magnitude is a base-10 scale; significant figures express reported precision. See Orders of Magnitude and Significant Figures.
- Dropping a component with a 0.2 ratio because it “looks small.” A 0.2 ratio can change uc by about 2%, which matters for two-digit reporting.
- Using order-of-magnitude reasoning for Type A evaluation. Type A requires repeated observations and statistical analysis; see Measurement Uncertainty.
- Ignoring the divisor for a stated confidence interval. A 95% interval is not a standard uncertainty until divided by the appropriate coverage factor.
- Rounding uncertainty to one significant digit too early. GUM 7.2.2 generally permits two significant digits; use Rounding Rules.
- Applying an order-of-magnitude threshold without documenting it. A defensible budget records why a term was omitted.
Quick Reference Table
| Ratio ui/udominant | Approximate effect on uc | Practical order-of-magnitude decision |
|---|---|---|
| ≥ 1 | Dominant | Evaluate fully |
| 0.3–1 | About 1–10% | Evaluate numerically |
| 0.1–0.3 | About 0.1–1% | Use a numerical estimate if possible |
| 0.03–0.1 | About 0.01–0.1% | Order-of-magnitude screening often acceptable |
| < 0.03 | < 0.01% | Usually negligible; document omission |
For independent components, when ui is much smaller than uc, the relative increase in combined standard uncertainty is approximately (1/2)(ui/uc)².
Discipline Note
In metrology and calibration laboratories, order-of-magnitude reasoning is common during method scoping but final uncertainty statements follow ISO/IEC 17025 and the GUM. In physics, order-of-magnitude estimates are used for feasibility and sanity checks. In chemistry, method validation and regulatory work often require a complete numerical budget. Regardless of discipline, the site’s GUM Guide and Measurement Uncertainty references provide clause-level guidance.
Sources & Further Reading
- JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement, Clauses 3.3.6, 4.1, 4.3, 5.1.2, 7.2.2, and 7.2.6.
- NIST TN 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, 1994, Section 2.3 and Appendix A.
- ISO/IEC Guide 98-3:2008, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement.
- ISO 80000-1:2022, Quantities and units — Part 1: General, Clause 7.3 and Clause 7.4.
- ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
Related reading: GUM Guide, Measurement Uncertainty, Error Propagation, Orders of Magnitude, and Significant Figures.
FAQ
Can I use order of magnitude for a final uncertainty statement?
Only if the final expanded uncertainty is reported to one or two significant digits and the order-of-magnitude approximations are documented as Type B bounds. GUM 7.2.2 does not permit omitting material components.
How small must a component be to ignore?
A common screening threshold is ten times smaller than the dominant standard uncertainty, which changes u_c by about 0.5%. For stricter work, use thirty times smaller, about 0.05%.
Does GUM mention order of magnitude?
Not as a formal evaluation method. GUM provides Type B evaluation based on scientific judgment in Clause 4.3; order-of-magnitude reasoning is an informal aid within that judgment.
What if the component is between 0.1 and 0.3 of the dominant term?
Evaluate it numerically. At a 0.3 ratio, the contribution to combined variance is about 9%, which can affect the second significant digit of the reported uncertainty.
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