Short Answer
Rule Statement: What Most Teachers Expect (and What Standards Say)
A measurement result should be reported as a value plus or minus an uncertainty: x ± U. The uncertainty tells the reader where the last meaningful digit is. The most common grading rule is simple: round the uncertainty to one or two significant figures, then round the measured value to the same decimal place as the uncertainty. For example, 12.3456 ± 0.0234 becomes 12.346 ± 0.023 if two significant figures are kept for the uncertainty. The result is not 12.3456 ± 0.0234.
This is not arbitrary. The Guide to the Expression of Uncertainty in Measurement (GUM) states in JCGM 100:2008, Clause 7.2.6 that the numerical values of the estimate and its standard or expanded uncertainty should not be given with excessive digits, and it is preferable to give uncertainty to at most two significant digits. ISO 80000-1:2022, Clause 6.4 and 6.5, gives rounding and significant-figure conventions. ASTM E29-22, Sections 6–7, covers rounding test data for conformance. NIST TN 1297, Section 7, gives similar reporting advice for NIST measurement results.
Teachers mark uncertainty wrong for three main reasons: the uncertainty has too many significant figures, the value and uncertainty are misaligned, or the student used significant-figure rules for raw measurements without propagating uncertainty. See our related rules on significant figures and error propagation.
Worked Examples
Example 1: Simple rounding and alignment
Raw result: 25.3456 mm, uncertainty 0.01234 mm. Round uncertainty to two significant figures: 0.012. The uncertainty is in the thousandths place, so round the value to three decimal places: 25.346 mm. Report: 25.346 ± 0.012 mm.
Example 2: Density with propagation
Mass m = 2.3456 g ± 0.012 g; volume V = 1.234 mL ± 0.005 mL. Density ρ = m/V = 1.900 g/mL. Relative uncertainties: 0.012/2.3456 = 0.00512 and 0.005/1.234 = 0.00405. Combine in quadrature: sqrt(0.00512² + 0.00405²) = 0.00653. Standard uncertainty u = 1.900 × 0.00653 = 0.0124. Expanded uncertainty U = 2u = 0.025 (k = 2). Report: 1.900 ± 0.025 g/mL.
Example 3: When the teacher wants one significant figure
A student writes 9.81 ± 0.2 m/s². If the uncertainty is 0.2 (tenths place), the value must be 9.8 ± 0.2 m/s². The extra 1 in 9.81 is not supported. If the uncertainty was known as 0.20, then 9.81 ± 0.20 m/s² is acceptable. Always match the decimal place, not just the number of digits.
Counter-Examples
- Wrong: 4.567 ± 0.12. Why: uncertainty is in hundredths, value is in thousandths. Correct: 4.57 ± 0.12.
- Wrong: 12.345 ± 0.0234. Why: uncertainty has three significant figures (2, 3, 4) and extends to ten-thousandths. Correct with two sig figs: 12.345 ± 0.023.
- Wrong: 0.00567 ± 0.000123. Why: uncertainty has three sig figs. Correct: 0.00567 ± 0.00012 (two sig figs in uncertainty, value to five decimal places).
- Wrong: 1000 ± 1.000. Why: uncertainty is over-precise and the value has ambiguous trailing zeros. Better: (1.000 ± 0.001) × 10³.
- Wrong: adding independent uncertainties linearly. Correct: combine in quadrature when independent.
Convention Comparison Table
| Convention/Source | Uncertainty digits | Rounding method | Value alignment | Notes |
|---|---|---|---|---|
| JCGM 100:2008 (GUM), 7.2.6 | At most two significant digits; often two | Round normally; do not over-round intermediate values | Same decimal place as uncertainty | Expanded uncertainty U = k u_c; state k |
| NIST TN 1297, Section 7 | Usually one or two significant digits | Round to nearest; avoid excessive digits | Match value and uncertainty | Common in NIST reports |
| ISO 80000-1:2022, 6.4–6.5 | Significant figures defined for quantities | Half-even (round to even) when exactly halfway | Decimal place of last significant digit | General quantities and units standard |
| ASTM E29-22, Sections 6–7 | Based on specification limits | Follows standard rounding practice; check edition | Round test result to specification limit decimal place | Used for conformance testing |
| Typical classroom | Often one or two; some teachers require two | Often half-up | Same decimal place | Follow the instructor’s stated convention |
Standards Citation
- JCGM 100:2008 (GUM), Clause 7.2.6: The numerical values of the estimate and its standard or expanded uncertainty should not be given with excessive digits; it is preferable to give uncertainty to at most two significant digits.
- JCGM 100:2008, Clause 7.2.2: Expanded uncertainty U is obtained by multiplying the standard uncertainty u_c(y) by a coverage factor k. For approximately 95% coverage in many cases, k = 2.
- JCGM 200:2012 (VIM): Defines measurement uncertainty, standard uncertainty, expanded uncertainty, and coverage factor.
- ISO 80000-1:2022, Clause 6.4 and 6.5: Rounding rules and significant figures for quantities and units.
- ASTM E29-22, Sections 6–7: Using significant digits in test data to determine conformance with specifications.
- NIST TN 1297 (1994), Section 7: Reporting uncertainty; NIST SP 811, Section 7.9: rounding values.
Common Mistakes
- Too many significant figures in the uncertainty. 0.1234 should usually be 0.12 or 0.1, depending on convention.
- Mismatched decimal places. 4.567 ± 0.12 is misaligned; use 4.57 ± 0.12.
- Using significant-figure counting for everything. Raw measurements have uncertainty; derived values need propagation.
- Forgetting the coverage factor. U = 2u_c is not the same as u_c. State k = 2 if you use expanded uncertainty.
- Rounding intermediate results too early. Keep extra digits until the final report; see double rounding error.
- Double rounding. Rounding twice can change the last digit. Round once at the end.
- Treating exact numbers as measured. Counting numbers and defined conversions have no uncertainty. See exact numbers.
- Using half-up when the standard uses half-even. ISO 80000-1 uses half-even for exact halves. Many classrooms use half-up. Follow the course convention.
- Ambiguous trailing zeros. 1000 ± 1 is clearer than 1000 with no uncertainty. Use scientific notation if needed. See ambiguous trailing zeros.
- Adding independent uncertainties linearly. Use quadrature for independent random errors unless your teacher specifies worst-case addition.
Practice Problems
- Round 3.4567 ± 0.0234 to a standard two-significant-figure uncertainty.
Answer: 3.457 ± 0.023. - Round 0.012347 ± 0.00067.
Answer: uncertainty 0.00067 has two significant figures. The value rounds to five decimal places: 0.01235 ± 0.00067. - Mass 5.00 ± 0.01 g, volume 2.00 ± 0.01 mL. Find density and report with U = 2u_c.
Answer: ρ = 2.50 g/mL. Relative uncertainties: 0.002 and 0.005; combined = 0.005385. u = 0.01346; U = 0.0269. With two significant figures in U: 2.500 ± 0.027 g/mL. With one significant figure: 2.50 ± 0.03 g/mL. - A student writes 9.81 ± 0.2 m/s². What is the likely mistake?
Answer: The uncertainty is in the tenths place, so the value should be 9.8 ± 0.2 m/s², unless the uncertainty is actually 0.20. - Which is better: 1000 ± 10 or (1.00 ± 0.01) × 10³?
Answer: Both can be correct, but the scientific-notation form removes ambiguity about trailing zeros.
Quick Reference Table
| Task | Rule | Example |
|---|---|---|
| Uncertainty significant figures | 1–2, usually 2 | 0.0234 → 0.023 |
| Value decimal places | Match uncertainty | 12.3456 ± 0.023 → 12.346 ± 0.023 |
| Expanded uncertainty | U = k u_c, k = 2 often | u_c = 0.0124 → U = 0.025 |
| Addition/subtraction | Combine absolute uncertainties in quadrature | sqrt(u₁² + u₂²) |
| Multiplication/division | Combine relative uncertainties in quadrature | sqrt((u₁/x₁)² + (u₂/x₂)²) |
| Rounding method | ISO: half-even; classrooms often half-up | 0.0125 → 0.012 (half-even) |
| Exact numbers | No uncertainty, infinite sig figs | 12 counted items |
Sources & Further Reading
- JCGM 100:2008, Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement (GUM), Clause 7.2.6.
- JCGM 200:2012, International Vocabulary of Metrology (VIM).
- ISO 80000-1:2022, Quantities and units — Part 1: General, Clause 6.4 and 6.5.
- ASTM E29-22, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
- NIST TN 1297 (1994), Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Section 7.
- NIST SP 811, Guide for the Use of the International System of Units (SI), Section 7.9.
FAQ
Why did my teacher mark 4.567 ± 0.12 wrong?
Because the uncertainty is in the hundredths place, but the value is written to the thousandths place. The value should be rounded to 4.57 ± 0.12.
How many significant figures should uncertainty have?
The GUM recommends at most two significant digits, and many fields use two. Some instructors accept one significant figure, especially when the leading digit is 1 or 2. Always follow your course convention.
Do I always round uncertainty up?
No. Standard uncertainty is rounded according to the normal rounding rule. In some safety or compliance contexts, people round uncertainty up to avoid understating risk, but that is a policy choice, not the general GUM rule.
Why did half-even rounding change my last digit?
ISO 80000-1 uses round-half-to-even for exact halves. Many classrooms use round-half-up. Both conventions exist; the mismatch is a common grading error.
What if my teacher uses significant figures instead of uncertainty propagation?
Some introductory courses use simplified significant-figure rules. If your teacher uses that system, follow it for the course. For professional work, use uncertainty propagation and report x ± U.
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