Short Answer
Rule Statement
Percent error is a dimensionless comparison between an experimental or measured value and an accepted, true, or reference value. The common formula is:
% error = |measured − accepted| ÷ |accepted| × 100%
Rounding a percent error is not merely a cosmetic step. It communicates how much precision the result deserves. The governing principle is simple: the final percent error should not imply more precision than the input measurements or their uncertainty support.
In practice, most scientific and engineering reporting uses one or two significant figures for percent error, especially when the result functions as an uncertainty-like quantity. The GUM recommends rounding standard uncertainties and expanded uncertainties to at most two significant figures (JCGM 100:2008, Clause 7.2.6). ASTM E29 and ISO 80000-1 likewise require that reported values be rounded to a number of digits consistent with the data and specification.
Key operational rules:
- Carry at least one or two guard digits through intermediate calculations.
- Round only the final percent error result, not intermediate differences or ratios.
- Use the least precise input to set the approximate precision limit; then reduce further if uncertainty analysis supports fewer digits.
- For uncertainty-like percent errors, report one or two significant figures, e.g., 2.4%, 0.61%, or 12%.
- For exact accepted values with high-precision measurements, follow the significant figures of the measured value and the measurement uncertainty.
- State the rounding convention when a result falls exactly on a tie; half-up and half-even are both used in different contexts.
Worked Examples
Example 1: Three-significant-figure inputs
A student measures a density as 2.68 g/cm³. The accepted value is 2.70 g/cm³.
- Difference: |2.68 − 2.70| = 0.02 g/cm³.
- Relative error: 0.02 ÷ 2.70 = 0.007407407…
- Percent error before rounding: 0.7407407…%.
- Both inputs have three significant figures, so the pure significant-figures result would be 0.741%.
- Because the difference itself is limited to one significant figure in the hundredths place, a conservative uncertainty-style report is 0.7% or 0.74%. Many instructors would accept 0.74% for a three-significant-figure calculation, while a metrology report might prefer 0.7%.
Recommended report: 0.74% (3 s.f.) or 0.7% (1 s.f.) depending on the uncertainty policy; avoid 0.7407407%.
Example 2: Uncertainty-limited result
A calibrated mass is measured as 10.02 g with an expanded uncertainty of ±0.05 g. The accepted value is 10.00 g.
- Difference: 0.02 g.
- Percent error: (0.02 ÷ 10.00) × 100% = 0.2%.
- The difference is much smaller than the uncertainty (±0.05 g). The result is statistically indistinguishable from zero within uncertainty.
- Reporting 0.200% or 0.2000% would be false precision.
Recommended report: 0.2% or, better, “0.2% (not significant compared with the ±0.05 g measurement uncertainty).”
Counter-Examples
The following are common rounding failures for percent error:
- False precision: Reporting 1.234567% when the inputs have three significant figures. The digits after 1.23 are unsupported.
- Intermediate rounding: Rounding the error to 1.2, then dividing by the accepted value. This can shift the final digit.
- Ignoring the accepted value’s precision: Treating 9.81 m/s² as an exact number with infinite digits when it is a three-significant-figure reference value.
- Double rounding: Rounding 2.449% to 2.45% and then to 2.5%. Round once, at the end.
- Zero-error overclaim: Reporting 0.00% when the measurement resolution cannot support that claim.
- Percent difference confusion: Using percent difference when the accepted value is known, or percent error when neither value is a reference.
- Tie-breaking inconsistency: Using half-up in one row and half-even in the next without documenting the convention.
Convention Comparison Table
| Context | Typical digits for percent error | Tie-breaking | Authority or practice |
|---|---|---|---|
| General science education | 1–2 significant figures | Half-up common | Textbook convention |
| Measurement uncertainty | 1–2 significant figures for uncertainty-like quantities | Not mandated; document it | GUM Clause 7.2.6 |
| NIST uncertainty reporting | At most 2 significant figures | Follow NIST policy | NIST TN 1297, Section 7 |
| ISO quantity reporting | Consistent with data precision | Half-even in many ISO contexts | ISO 80000-1 Clause 7.3.3 |
| ASTM conformance testing | Round per specification limits | ASTM E29 rounding procedure | ASTM E29-13 Sections 6–7 |
| Computational software | Depends on displayed precision | Excel half-away-from-zero; Python/R half-even; MATLAB often half-away | Software documentation |
Standards Citation
- GUM (JCGM 100:2008), Clause 7.2.6: Numerical values of uncertainty should be rounded to at most two significant digits. This is the strongest general guide when percent error is used as an uncertainty-like statement.
- NIST TN 1297, Section 7: Guidelines for evaluating and expressing uncertainty at NIST specify that uncertainties be reported to at most two significant figures and that the measurement result be rounded consistently.
- ISO 80000-1:2009, Clause 7.3.3: Provides rules for rounding numerical values and emphasizes that the number of digits retained should reflect the accuracy of the value.
- ASTM E29-13, Sections 6–7: Standard practice for using significant digits in test data and determining conformance with specifications, including rounding procedures for reported values.
These standards do not all prescribe one identical percent-error format, because percent error is often a derived educational metric rather than a formal uncertainty. They do agree on the central point: do not report digits that the measurement cannot support.
Common Mistakes
- Rounding before propagation. Keep guard digits until the final percent error is computed.
- Using too many significant figures. A result like 4.23719% is rarely justified.
- Using too few significant figures too early. Rounding 0.046% to 0.05% before further analysis can distort conclusions.
- Forgetting zeros. In 0.050%, the trailing zero after the 5 is significant; the leading zeros are not. The value has two significant figures.
- Mixing percent and decimal forms. 0.024 is 2.4%, not 0.024%.
- Reporting signed error inconsistently. Percent error may be absolute or signed; state which one you use.
- Assuming all calculators round the same. Tie-breaking varies.
Software Behavior Note
Spreadsheets and programming languages compute at full floating-point precision and only round for display unless you explicitly call a rounding function. That hidden precision is useful, but it can mask the need for significant-figure rounding.
- Excel and Google Sheets:
ROUNDtypically rounds half away from zero. Display formatting may show fewer digits without changing the stored value. - Python: The built-in
round()uses round-half-to-even, also called banker’s rounding. - R:
round()uses IEC 60559 rounding, generally half-to-even. - MATLAB:
round()generally rounds half away from zero. - TI-84 and Casio fx-991: Display rounding may use half-up or a calculator-specific mode; always check the manual and the active significant-figure setting.
Our significant figures calculator is designed to apply standard significant-figure and rounding rules consistently, but the reference material on this site is the primary resource: use it to choose the right convention before trusting any single button.
Quick Reference Table
| Situation | Recommended rounding | Example |
|---|---|---|
| Typical lab report with 3-s.f. inputs | 2–3 s.f.; often 1–2 s.f. for error | 0.741% or 0.74% |
| Uncertainty-like percent error | 1–2 s.f. | 2.4% |
| GUM uncertainty statement | At most 2 s.f. | 0.61% |
| Very small error | Use scientific notation or ppm | 3.2 × 10⁻⁴ % or 3.2 ppm |
| Exact accepted value | Limited by measured value and uncertainty | Depends on instrument resolution |
| Tie case | Document half-up or half-even | 2.45% → 2.4% or 2.5% |
Related Rules
- Rounding Rules — basic digit-dropping and tie-breaking conventions.
- Significant Figures — how input precision limits calculated results.
- Percent Error — formula, interpretation, and common variants.
- Error Propagation — combining uncertainties before final rounding.
- Rounding vs Significant Figures — why the two ideas are related but not identical.
- Why Sig Figs Are an Approximation — limitations of significant-figure arithmetic.
- ASTM E29 and ISO 80000 — standards for reporting rounded test data.
FAQ
How many significant figures should a percent error have?
Usually one or two significant figures. If the percent error is acting like an uncertainty, GUM Clause 7.2.6 limits it to at most two. In a classroom significant-figures exercise, follow the least precise input, which may allow two or three significant figures. Avoid more than three unless there is a specific uncertainty justification.
Should percent error be rounded up or down?
Round to the nearest representable value. For exact ties, the convention matters: half-up is common in education and spreadsheets, while half-even is common in some computational and ISO contexts. Document the tie-breaking rule you use.
Can percent error be negative?
Yes, if you preserve the sign to show direction: negative means the measured value is below the accepted value. Many reports use the absolute value and call it percent error. State whether your result is signed or absolute.
Can percent error exceed 100%?
Yes. If the measured value is more than twice the accepted value, or far below it, the absolute percent error can exceed 100%. This is mathematically valid, though it may indicate a calibration or experimental problem.
Does percent error have units?
No. Percent error is dimensionless because the units cancel in the ratio. It is expressed with the percent sign.
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