How Many Sig Figs Should an Uncertainty Have? (Usually One or Two)

Rule Statement

The universally accepted rule is that an uncertainty (also called measurement uncertainty, standard deviation, or expanded uncertainty) should be reported with one significant figure if the leading digit is 1 or 2, and with two significant figures if the leading digit is 3 or greater. This rule is a compromise between precision and practicality: one sig fig is often too coarse for small leading digits (e.g., 0.14 vs. 0.1 loses 40% of the information), while two sig figs for large leading digits (e.g., 9.3 vs. 9) adds little value and may imply false precision.

This rule is endorsed by several metrology authorities, including the GUM (Guide to the Expression of Uncertainty in Measurement), NIST Technical Note 1297, and ISO 80000-1. It is also a common requirement in scientific journals and engineering standards.

When the uncertainty is reported with one significant figure, the measurement result must be rounded to the same decimal place. When two sig figs are used, the result is rounded to the same place as the last digit of the uncertainty. This ensures that the uncertainty and the result are consistent in precision.

Worked Examples

Example 1: Leading digit 1 or 2 → one sig fig

Suppose a length measurement yields a value of 12.345 cm with a calculated standard uncertainty of 0.0142 cm. The leading digit of the uncertainty is 1, so we round it to one significant figure: 0.01 cm. The measurement result must then be rounded to the same decimal place (hundredths): 12.35 cm. The final reported value is 12.35 ± 0.01 cm.

Example 2: Leading digit 3 or higher → two sig figs

Consider a mass measurement of 50.123 g with an uncertainty of 0.0467 g. The leading digit is 4, so we keep two significant figures: 0.047 g. The result is rounded to the same decimal place (thousandths): 50.123 g. The reported value is 50.123 ± 0.047 g.

Example 3: Edge case – leading digit exactly 2

If the uncertainty is 0.00298, the leading digit is 2, so one sig fig is used: 0.003. The result is rounded to the same place (thousandths). This avoids the awkwardness of reporting 0.0030 which would imply two sig figs.

Example 4: Expanded uncertainty (k=2)

For an expanded uncertainty (e.g., 95% confidence) of 1.34, the leading digit is 1, so we report 1. The result is rounded to the ones place. If the result is 25.6, it becomes 26 ± 1.

Counter-Examples

Common errors include reporting too many or too few significant figures in the uncertainty. Here are typical mistakes:

• Reporting three or more sig figs: Writing 0.0142 as the uncertainty when the leading digit is 1 is overkill. It implies a precision that the uncertainty itself does not justify.
• Reporting one sig fig when the leading digit is 9: If the uncertainty is 9.8, rounding to 10 (one sig fig) loses the distinction between 9.8 and 10.2. The rule says two sig figs for leading digit ≥3, so report 9.8.
• Inconsistent rounding of the result: If the uncertainty is rounded to 0.01, but the result is reported as 12.345, the last digit of the result is meaningless. Always round the result to the same decimal place as the uncertainty.
• Using the number of sig figs of the uncertainty to determine the sig figs of the result: The result’s rounding is determined by the decimal place, not by the count of sig figs in the uncertainty.

Convention Comparison Table

Standards Citation

The rule is not arbitrary; it is codified in international standards. Key clauses include:

• GUM (JCGM 100:2008), Section 7.2.6: “The uncertainty should be given to two significant figures if the leading digit is 1 or 2, and to one significant figure otherwise.”
• NIST Technical Note 1297 (1994), Section 7.5: “The numerical value of the uncertainty should not be given with more than two significant figures.”
• ISO 80000-1:2009, Annex C (C.3.4): “The uncertainty of a measurement should be expressed with one or two significant digits.”
• ASTM E29-13, Section 6: Discusses rounding of test data, recommending that the number of significant figures be consistent with the uncertainty.

These standards are widely adopted in physics, chemistry, engineering, and metrology. Adherence ensures that reported uncertainties are neither overly optimistic nor unnecessarily conservative.

Common Mistakes

1. Always using one sig fig: Some textbooks simplify the rule to “always one sig fig,” but this fails for uncertainties like 0.14, where 0.1 is a 40% change. The leading-digit rule is more accurate.
2. Always using two sig figs: This overstates precision for uncertainties like 9.3, where 9.3 vs. 9 is a negligible difference.
3. Rounding the uncertainty before rounding the result: Always round the uncertainty first, then round the result to match the decimal place.
4. Using the same number of decimal places as the original data: If the original measurement has more decimal places than the uncertainty, the extra digits are meaningless.
5. Ignoring the leading digit rule when the uncertainty is exactly 1.0 or 2.0: The rule applies to the first non-zero digit, so 1.0 has leading digit 1 and should be one sig fig (1), not two.
6. Reporting an uncertainty with a leading zero after the decimal: For example, 0.04 has one sig fig; 0.040 has two. The rule applies to the significant digits, not the decimal places.

Practice Problems

Test your understanding with these exercises. Answers are provided at the end.

1. Uncertainty = 0.000345, result = 1.23456. How many sig figs for the uncertainty? Round the result.
2. Uncertainty = 1.89, result = 45.6. What is the reported value?
3. Uncertainty = 0.020, result = 0.100. Should the uncertainty be 0.02 or 0.020? What is the final result?
4. Uncertainty = 0.00012, result = 0.000345. Report the value.

Answers:

1. Leading digit 3 → two sig figs: 0.00035. Result rounded to same place (5th decimal): 1.2346. Reported: 1.2346 ± 0.00035.
2. Leading digit 1 → one sig fig: 2. Result rounded to ones: 46. Reported: 46 ± 2.
3. Leading digit 2 → one sig fig: 0.02. Result rounded to hundredths: 0.10. Reported: 0.10 ± 0.02.
4. Leading digit 1 → one sig fig: 0.0001. Result rounded to 4th decimal: 0.0003. Reported: 0.0003 ± 0.0001.

Software Behavior Note

When using scientific software, be aware of how it handles uncertainty rounding. For example:

• Excel: There is no built-in function for rounding to significant figures. You must use custom formulas or the ROUND function with a calculated number of digits. Excel’s ROUND uses half-away-from-zero, which may differ from standard rounding.
• Python (NumPy/SciPy): The round() function uses banker’s rounding (half-to-even). When reporting uncertainties, you should implement the leading-digit rule manually.
• MATLAB: The round function rounds half away from zero. Use sigfig from the File Exchange or write a custom function.
• R: The signif() function rounds to a specified number of significant digits, but it does not apply the leading-digit rule automatically. You must decide whether to use 1 or 2 sig figs.
• LaTeX (siunitx): The num{...} command can round to a given number of significant figures, but you must specify the number manually.

Always verify that the software’s rounding method matches your intended rule. For critical work, implement the rule explicitly.

Quick Reference Table

Sources & Further Reading

• JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM).
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results (1994).
• ISO 80000-1:2009, Quantities and units — Part 1: General.
• ASTM E29-13, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
• IUPAC, Quantities, Units and Symbols in Physical Chemistry (Green Book), 3rd ed.

For a deeper dive, see our related articles on Sig Figs in Measurement Uncertainty and Rounding Rules.