Significant Figures vs. Uncertainty: When to Stop Counting Sig Figs and Start Propagating Error

Rule Statement: When Sig Figs Stop Being Enough

Significant figures are a compact convention for implied uncertainty. A reported value such as 2.74 g/mL does not mean the value is exactly 2.74; it means the last retained digit is uncertain by roughly one unit in that place. That convention is useful for quick communication, but it is not an uncertainty budget. Stop counting significant figures and start propagating error when any of the following conditions is true:

• An uncertainty budget, calibration certificate, tolerance, or repeatability estimate is available.
• Inputs have different uncertainties or different units.
• The calculation is nonlinear, such as squares, roots, exponentials, or logarithms.
• You need a coverage interval, conformance decision, or statement of confidence.
• A standard such as JCGM 100:2008 (GUM), ASTM E29-13a, or ISO 80000-1:2009 governs the report.

For a measurand y = f(x₁, …, x_n), the GUM law of propagation of uncertainty gives the combined standard uncertainty u_c(y) from the standard uncertainties u(x_i). In the independent case, the covariance terms vanish:

u_c²(y) = Σ (∂f/∂x_i)² u²(x_i)

The expanded uncertainty is U = k u_c(y), where k is the coverage factor. For many physical measurements, k = 2 corresponds to approximately 95% coverage under a normal distribution. Report the result as y ± U, with units, the coverage factor, and the definition of the measurand.

This site is a precision-and-rounding reference, not merely a calculator. Use our significant figures calculator to check sig-fig rules, and use the error propagation reference when uncertainty data exist.

Convention Comparison Table: Sig Figs vs. Uncertainty Propagation

Standards Citation: GUM, ASTM E29, ISO 80000, NIST

The distinction is not stylistic; it is standardized. The following sources define when significant-digit conventions are acceptable and when an uncertainty statement is required.

• JCGM 100:2008 (GUM), Evaluation of measurement data — Guide to the expression of uncertainty in measurement. Clause 5.1.2 gives the law of propagation of uncertainty; Clause 7.2.2 discusses reporting the expanded uncertainty and coverage factor. GUM Clause 7.2.6 recommends limiting uncertainty to one or two significant digits in many reports.
• NIST TN 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. This guide adapts GUM for NIST measurement results and emphasizes Type A and Type B evaluations.
• ASTM E29-13a, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. Section 6 addresses rounding; the standard is about conformance decisions, not a full uncertainty budget.
• ISO 80000-1:2009, Quantities and units — Part 1: General. Clause 6.2 and Annex B provide rounding and significant-digit conventions for quantities and units.
• BIPM SI Brochure, 9th edition. Defines SI units and quantity calculus; it reinforces that numerical values and units must be reported together.

Sig figs answer “how many digits should I write?” Uncertainty propagation answers “how well do I know the value?” They are related, but only the second supports a coverage interval.

Worked Examples: From Sig Figs to Uncertainty

Example 1: Density with Independent Inputs

Mass m = 12.345 g with u(m) = 0.002 g. Volume V = 4.50 mL with u(V) = 0.03 mL. Density is ρ = m/V.

1. Compute ρ = 12.345 / 4.50 = 2.743333… g/mL.
2. Relative uncertainties: u_m/m = 0.002/12.345 = 0.000162; u_V/V = 0.03/4.50 = 0.006667.
3. Combine in quadrature: u_c/ρ = √(0.000162² + 0.006667²) = 0.006669.
4. Combined standard uncertainty: u_c = 2.743333 × 0.006669 = 0.0183 g/mL.
5. Expanded uncertainty with k = 2: U = 0.0366 ≈ 0.04 g/mL.

Report: (2.74 ± 0.04) g/mL, k = 2. The sig-fig-only rule would also suggest three significant figures, 2.74, but it does not reveal the ±0.04 interval or the dominant volume uncertainty.

Example 2: Addition with Different Absolute Uncertainties

Lengths 1.23 m ± 0.01 m and 4.567 m ± 0.002 m. Sum = 5.797 m. Absolute uncertainties combine as √(0.01² + 0.002²) = 0.0102 m. Expanded uncertainty U = 0.0204 ≈ 0.02 m. Report (5.80 ± 0.02) m, k = 2. The addition rule for sig figs also rounds to the hundredths place, but the uncertainty statement is stronger.

Example 3: Nonlinear Function, Circle Area

Radius r = 2.50 cm with u(r) = 0.01 cm. Area A = πr² = 19.63495… cm². Relative radius uncertainty = 0.01/2.50 = 0.004. Because A ∝ r², relative area uncertainty = 2 × 0.004 = 0.008. Then u_c(A) = 19.63495 × 0.008 = 0.157 cm², and U = 0.314 ≈ 0.3 cm². Report (19.6 ± 0.3) cm², k = 2. The sig-fig result, 19.6 cm², matches the value but omits the interval.

Counter-Examples: What Not to Do

• Copying every calculator digit. A display reading 0.6666667 from 2.0/3.0 does not have seven significant figures. If 2.0 has uncertainty ±0.05, the result is approximately 0.67.
• Reporting uncertainty with too many digits. “25.000 ± 0.0001 g” is rarely defensible unless the calibration and repeatability support it.
• Averaging without reducing uncertainty. The mean of N independent readings can have u_mean = s/√N, not the same uncertainty as one reading. Ten readings each with u = 0.1 give u_mean ≈ 0.032, not 0.1.
• Adding relative uncertainties linearly for independent inputs. Use quadrature unless correlations are known; otherwise you overstate the combined uncertainty.
• Treating sig figs as a 95% interval. “2.74” implies a last-digit uncertainty convention, not a k = 2 coverage interval.

Common Mistakes

• Rounding intermediate values too early, causing double-rounding error.
• Mixing absolute and relative uncertainties in the same formula.
• Using the multiplication rule for addition or the addition rule for multiplication.
• Ignoring exact numbers, which have infinite significant figures and zero uncertainty.
• Claiming 95% coverage while using k = 1.
• Omitting units, the coverage factor, or the definition of the measurand.
• Using sig figs for a conformance decision where ASTM E29 or a tolerance specification applies.

For more on these pitfalls, see Common Mistakes in Significant Figures and Why Sig Figs Are an Approximation.

Quick Reference Table: Propagation Formulas

Practice Problems

1. Mass = 25.0 g ± 0.1 g; volume = 5.00 mL ± 0.02 mL. Find density and expanded uncertainty with k = 2.
2. Add 12.34 m ± 0.05 m and 0.567 m ± 0.001 m. Report the sum with U for k = 2.
3. Circle radius = 5.00 cm ± 0.05 cm. Find area and expanded uncertainty with k = 2.

Selected answers: (1) ρ = 5.00 g/mL; relative uncertainties 0.004 and 0.004; combined relative 0.00566; u_c ≈ 0.028 g/mL; U ≈ 0.06 g/mL; report (5.00 ± 0.06) g/mL. (2) Sum = 12.907 m; u_c = 0.05001 m; U ≈ 0.10 m; report (12.91 ± 0.10) m. (3) A = 78.5398 cm²; relative radius 0.01; relative area 0.02; u_c ≈ 1.57 cm²; U ≈ 3 cm²; report (78.5 ± 3) cm².

Sources & Further Reading

• JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM), Clause 5.1.2 and Clause 7.
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results.
• ASTM E29-13a, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications.
• ISO 80000-1:2009, Quantities and units — Part 1: General, Clause 6.2 and Annex B.
• BIPM, The International System of Units (SI), 9th edition.

Related reference pages: Significant Figures Rules, Rounding Rules, Measurement Uncertainty, GUM Guide, ASTM E29, and ISO 80000.