Short Answer
This article is part of a precision-and-rounding reference, not merely a calculator. It explains how GUM uncertainty rounding interacts with round half to even when you report a measurement result. If you need to verify significant digits, our significant figures calculator is a useful check, but the rules and caveats below are the reference.
Rule Statement
For a measurement result, the GUM (JCGM 100:2008) recommends reporting an estimate y and an expanded uncertainty U with a controlled number of digits. The practical rule is:
- Compute the expanded uncertainty, usually U = k uc(y), with k = 2 for roughly 95% coverage.
- Round U to at most two significant digits. GUM 7.2.2 prefers two significant digits.
- Round y to the same decimal place as the rounded U.
- When a rounding decision is an exact tie — the first discarded digit is 5 and all following discarded digits are zero — round to the nearest even retained digit, known as round half to even. Otherwise, use ordinary nearest rounding: discard < 5, round down; discard > 5, round up; discard 5 followed by any nonzero digit, round up.
Round half to even is not applied to every digit 5. It is a tie-break for exact halfway values only. The purpose is to avoid a systematic upward or downward bias when many halfway values are rounded in a series.
Standards Citation
The key normative sources for GUM uncertainty rounding are:
- JCGM 100:2008, GUM 7.2.2 — numerical values of y and u(y) or U should not be given with excessive digits; two significant digits for U is preferable, and y should be rounded to the same decimal place. GUM refers to ISO 31-0:1992, Annex B for rounding, now superseded by ISO 80000-1.
- ISO 80000-1:2022, Annex B.3 — if the digit following the last digit to be retained is exactly 5 and no nonzero digits follow it, increase the retained digit only if it is odd; leave it unchanged if it is even.
- ASTM E29-13, Section 6 — standard practice for significant digits and rounding includes tie-to-even when the discarded portion is exactly 5 followed by zeros.
- NIST SP 811 (2008), Section 10.5.2 — SI rounding conventions, including round half to even for exact ties.
- NIST TN 1297 (1994), Section 7.2 — NIST guidance on expressing uncertainty; expanded uncertainty is normally reported to two significant digits.
Convention Comparison Table
| Convention | Tie rule for exact 5 | Typical use | GUM/ISO status |
|---|---|---|---|
| Round half to even | Nearest even retained digit | Metrology, statistics, decimal finance | Default via ISO 80000-1 B.3 |
| Round half up | Always upward | School arithmetic, some reporting | Not ISO/GUM default |
| Round half away from zero | Up in magnitude | Excel ROUND, MATLAB round | Not GUM default |
| Round half toward zero | Toward zero | Special programming contexts | Not GUM default |
| Round uncertainty up | Increase uncertainty | Conservative laboratory policies | Additional convention, not ISO nearest rounding |
Worked Examples
Assume U is the expanded uncertainty and y is the measured estimate. Round U first, then match y to the same decimal place.
Example 1: Exact tie with even retained digit
U = 0.0125 V to two significant figures. The retained digits are 1 and 2. The next digit is exactly 5 with no nonzero remainder. The retained digit 2 is even, so U = 0.012 V. The estimate y = 1.2345 V must be rounded to three decimal places: 1.2345 becomes 1.234 V because the retained digit 4 is even. Report 1.234 V ± 0.012 V.
Example 2: Exact tie with odd retained digit
U = 0.0135 V to two significant figures. The retained digits are 1 and 3. The next digit is exactly 5. The retained digit 3 is odd, so U = 0.014 V. The estimate y = 1.2345 V is again rounded to three decimal places: 1.234 V. Report 1.234 V ± 0.014 V.
Example 3: 5 followed by nonzero digits
U = 0.0125001 V to two significant figures. The next digit is 5, but it is followed by nonzero digits, so this is not an exact tie. Round up: U = 0.013 V. The estimate y = 1.2345 V becomes 1.235 V at three decimal places. Report 1.235 V ± 0.013 V.
Example 4: Digit less than 5
U = 0.012499 V to two significant figures. The next digit is 4, so U = 0.012 V. The estimate y = 1.2345 V becomes 1.234 V at three decimal places. Report 1.234 V ± 0.012 V.
Counter-Examples
- Wrong: 1.2451 rounded to two decimals becomes 1.24 because of half-even. Correct: 1.25, because the 5 is followed by 1 and is not an exact tie.
- Wrong: round U to 0.0125 V and y to 1.235 V, mixing half-even and half-up. Correct: apply one rounding convention consistently to final reported values.
- Wrong: round intermediate uncertainty components before combining them. Correct: keep full precision during the uncertainty budget and propagation; round only the final U and y.
- Wrong: report U = 0.01234 V with four significant figures. Correct: GUM prefers two significant digits for U.
Common Mistakes
- Assuming GUM always rounds uncertainty upward. It does not; ISO 80000-1 uses nearest-even for exact ties.
- Applying round half to even to any number containing 5, even when nonzero digits follow.
- Confusing significant digits with decimal places. U is rounded to one or two significant digits, then y is rounded to the same decimal place.
- Double rounding, such as rounding first to three significant digits and then to two.
- Using binary floating-point without a decimal context, where 0.0125 may not be exactly representable.
Software Behavior Note
Software often disagrees on ties, so do not assume a generic round function matches GUM/ISO behavior.
| Environment | Tie behavior | GUM-compliant? |
|---|---|---|
Python round() |
Half-even for binary floats | Use Decimal with ROUND_HALF_EVEN for exact decimal safety |
Python decimal |
ROUND_HALF_EVEN available | Yes, when explicitly selected |
R round() |
IEC 60559 round to even | Generally yes for decimal contexts |
Excel ROUND |
Half away from zero | No native half-even for standard rounding |
Google Sheets ROUND |
Typically half away from zero | Usually not half-even |
MATLAB round |
Ties away from zero | Not half-even |
JavaScript Math.round |
Half up toward +∞ | Not half-even |
| SQL databases | Varies by product | Check the numeric rounding mode |
For standards work, use decimal arithmetic with an explicit ROUND_HALF_EVEN mode, or a precision calculator that preserves exact decimal ties.
Quick Reference Table
| Digit after last retained | Action | Example to 2 significant figures |
|---|---|---|
| < 5 | Round down | 0.0124 → 0.012 |
| > 5 | Round up | 0.0126 → 0.013 |
| = 5, followed by nonzero | Round up | 0.01251 → 0.013 |
| = 5, followed only by zeros | Round to even | 0.0125 → 0.012; 0.0135 → 0.014 |
After rounding U, round y to the same number of decimal places as U.
Related Rules
See our guides on Banker’s rounding, half-up rounding, rounding vs significant figures, ISO 80000, ASTM E29, and standard deviation reporting.
FAQ
Does GUM explicitly require round half to even?
GUM 7.2.2 controls the number of digits and decimal-place matching, and for rounding it refers to ISO 31-0, now ISO 80000-1. ISO 80000-1 Annex B.3 uses round half to even for exact ties, so GUM-compliant result expression inherits that tie rule.
Should uncertainty always be rounded up?
No. Some laboratories adopt a conservative upward rounding policy, but that is an additional convention. The default ISO/GUM mathematical rule is nearest rounding with half-even at exact ties.
How many significant digits should U have?
GUM 7.2.2 says two significant digits is preferable. One significant digit may be used for concise reporting, but two is the usual metrology default.
Do I round intermediate uncertainties?
No. Keep full precision through the uncertainty budget and propagation. Round only the final reported U and y.
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