From Significant Figures to Combined Standard Uncertainty: A GUM-Based Approach

Significant figures are a compact convention for expressing measurement precision. They are not a full uncertainty model. The Guide to the Expression of Uncertainty in Measurement (GUM) provides the rigorous path: assign standard uncertainties to input quantities, propagate them through the measurement function, and report a combined standard uncertainty. This reference explains how sig figs relate to GUM, where they agree, and where they fail. For quick arithmetic, use our significant figures calculator; for the rules behind it, see Significant Figures Rules.

Rule Statement: From Sig Figs to GUM

The GUM approach begins with a measurement model y = f(x₁, x₂, …, x_n). Each input x_i has a standard uncertainty u(x_i). For independent inputs, the combined standard uncertainty is:

u_c(y) = √[ Σ ( ∂f/∂x_i )² u²(x_i) ]

This is given in GUM clause 5.1.2. If inputs are correlated, covariance terms must be added: u_c²(y) = Σ ( ∂f/∂x_i )² u²(x_i) + 2 Σ_i<j ( ∂f/∂x_i )( ∂f/∂x_j ) u(x_i, x_j). Expanded uncertainty is U = k u_c, with coverage factor k (GUM 5.2.2). When reporting, round u_c or U to one or two significant digits, round the result to the same decimal place, and state k (GUM 7.2.2). Significant figures are a coarse approximation: the last retained digit implies an uncertainty of roughly half a unit in that place. ISO 80000-1:2022 clause 7.3.3 advises against giving more digits than justified. ASTM E29-22 governs rounding test data for conformance decisions.

Worked Examples

Example 1: Density from mass and volume

Model: ρ = m/V. Let m = 12.345 g with u(m) = 0.002 g, and V = 5.00 mL with u(V) = 0.02 mL. Assume independence.

1. Relative uncertainty in mass: u(m)/m = 0.002/12.345 = 1.62 × 10^-4.
2. Relative uncertainty in volume: u(V)/V = 0.02/5.00 = 4.0 × 10^-3.
3. Combined relative uncertainty: √[(1.62 × 10^-4)² + (4.0 × 10^-3)²] = 4.003 × 10^-3.
4. Result: ρ = 12.345/5.00 = 2.469 g/mL.
5. Combined standard uncertainty: u_c = 0.004003 × 2.469 = 0.00988 g/mL.

Round u_c to 0.010 g/mL. Report ρ = 2.469 g/mL, u_c = 0.010 g/mL, or 2.469(10) g/mL. The last digit of the result is in the thousandths place, matching the uncertainty.

Example 2: Sum of lengths

Model: L = L₁ + L₂ + L₃. Let L₁ = 12.34 cm with u = 0.01 cm; L₂ = 5.6 cm with u = 0.1 cm; L₃ = 0.789 cm with u = 0.001 cm.

1. Sum: 12.34 + 5.6 + 0.789 = 18.729 cm.
2. Combined standard uncertainty: √(0.01² + 0.1² + 0.001²) = 0.1005 cm ≈ 0.10 cm.
3. Report: 18.73 cm ± 0.10 cm.

Note that a strict significant-figures addition rule would round to 18.7 cm because 5.6 has only one decimal place. The GUM result uses the actual input uncertainties and can retain more information when those uncertainties justify it.

Counter-Examples: Common Misapplications

• Adding standard uncertainties linearly: Two independent contributions of 0.1 each give 0.1 + 0.1 = 0.2, but quadrature gives √(0.1² + 0.1²) = 0.141. Linear addition overstates uncertainty.
• Rounding intermediate values: Rounding 2.469 to 2.47 before propagating changes u_c. Carry extra digits until the final report.
• Treating exact conversions as uncertain: 1 inch = 2.54 cm exactly. It contributes no uncertainty.
• Ignoring correlation: Using the same thermometer for two temperatures creates correlated inputs. The covariance term can increase or decrease u_c.
• Mismatched decimal places: Reporting 2.469 ± 0.01 g/mL when u_c = 0.00988 g/mL is inconsistent. Either report 2.47 ± 0.01 g/mL or 2.469 ± 0.010 g/mL.

Convention Comparison Table

Standards Citation

• JCGM 100:2008 (GUM): clauses 5.1.2 (combined standard uncertainty), 5.2.2 (expanded uncertainty), 7.2.2 (reporting uncertainty).
• NIST Technical Note 1297: Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, sections 2-5.
• ISO 80000-1:2022: Quantities and units, clause 7.3.3 on significant digits and rounding.
• ASTM E29-22: Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
• BIPM SI Brochure, 9th edition: section 5.4.1 on measurement uncertainty and significant figures.

Common Mistakes

1. Confusing significant figures with decimal places.
2. Using sig-fig propagation rules when actual standard uncertainties are available.
3. Rounding the uncertainty too aggressively to one significant digit when two are justified.
4. Forgetting the coverage factor k; u_c is not the same as expanded uncertainty U.
5. Mixing standard uncertainty and expanded uncertainty in a report.
6. Reporting an uncertainty budget with more digits than the inputs support.
7. Assuming all inputs are independent without checking correlation.

Quick Reference Table

Sources & Further Reading

• JCGM 100:2008. Evaluation of measurement data — Guide to the expression of uncertainty in measurement. BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, OIML.
• Taylor, B. N., & Kuyatt, C. E. (1994). Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. NIST Technical Note 1297.
• ISO 80000-1:2022. Quantities and units — Part 1: General. International Organization for Standardization.
• ASTM E29-22. Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. ASTM International.
• BIPM. (2019). The International System of Units (SI), 9th edition. Bureau International des Poids et Mesures.

Related Rules

• Error Propagation
• Measurement Uncertainty
• Rounding Rules
• Significant Figures Rules
• GUM Guide