This page picks up directly where Accuracy, Precision, and Uncertainty left off. That page established that random error is what uncertainty describes, and that it behaves statistically rather than as a fixed offset. This page is about what happens when you calculate with several such uncertain values at once — add them, multiply them, raise them to a power — and need to know how uncertain the result is. It also closes a loop that’s been open since our sig figs guide: the addition and multiplication rules taught there (match decimal places; match sig fig count) are not arbitrary. They’re a simplified, digit-counting shortcut for exactly the calculation this page does properly, with real numbers. Example 4 below shows the two converging.

Two Methods, Not One

The simple (worst-case) method.

• Addition and subtraction: add the absolute uncertainties. If y = x₁ + x₂ (or x₁ − x₂), then u(y) = u(x₁) + u(x₂).
• Multiplication and division: add the relative (percentage) uncertainties. If y = x₁ × x₂ (or x₁ ÷ x₂), then u(y)/y = u(x₁)/x₁ + u(x₂)/x₂.

This method assumes the worst case: every input error pointed in the least favorable direction at once. It’s easy to compute by hand, which is why it’s what most intro science courses teach — but it’s deliberately pessimistic.

The quadrature method (the GUM’s actual method).

• Addition and subtraction: combine absolute uncertainties in quadrature. u(y) = √[u(x₁)² + u(x₂)²].
• Multiplication and division: combine relative uncertainties in quadrature. u(y)/y = √[(u(x₁)/x₁)² + (u(x₂)/x₂)²].

This is what the international measurement standard — the GUM (JCGM 100:2008) — actually specifies, under the name the law of propagation of uncertainty. It reflects a real statistical fact: if two error sources are genuinely independent, it’s very unlikely both land at their worst possible value on the same measurement, so the realistic combined uncertainty is smaller than the simple sum.

Quadrature is always smaller than or equal to simple addition. For any two non-negative numbers a and b, √(a² + b²) ≤ a + b, with equality only when one of them is zero. This isn’t a coincidence specific to uncertainty — it’s the same geometric fact that makes the diagonal of a rectangle shorter than the sum of its two sides. Every example below demonstrates it numerically.

Powers. If y = xⁿ, the relative uncertainty scales directly with the power: u(y)/y = |n| · u(x)/x. Squaring a value with 2% relative uncertainty gives a result with 4% relative uncertainty; cubing it gives 6%.

Where this formula actually comes from. The GUM derives both versions from one general formula, for y = f(x₁, x₂, …, xₙ) with independent inputs: uc²(y) = Σᵢ (∂f/∂xᵢ)² · u²(xᵢ) — the sum of each input’s uncertainty, weighted by how sensitively the result responds to that input. This is a first-order Taylor series approximation of the function f, which is why it works cleanly for the simple operations covered here but can break down for strongly nonlinear functions or unusually large uncertainties — in those harder cases, the GUM’s own supplementary guide (JCGM 101:2008) recommends a Monte Carlo simulation instead of the direct formula.

When inputs aren’t independent. Everything above assumes the input uncertainties come from unrelated sources. If two measurements share a common cause of error — the same miscalibrated instrument used for both, for instance — the quadrature formula on its own will underestimate the true combined uncertainty, and a covariance term (GUM Section 5.2) has to be added. This is a real caveat, not a footnote: shared error sources are common in practice, and this is exactly the setting where naive quadrature gives a falsely reassuring answer.

The connection back to significant figures. Our sig figs guide rounds a sum to the fewest decimal places among its terms, and a product to the fewest significant figures. That’s precisely a crude, discrete version of the simple method above — “the least precise input limits the result” — without ever calculating an actual numeric uncertainty. Example 4 makes this explicit.

Worked Examples

Example 1 — Addition: simple vs. quadrature

Values: x₁ = 12.5 ± 0.3, x₂ = 8.2 ± 0.2. Sum = 20.7.

Simple method: u(sum) = 0.3 + 0.2 = 0.5 → 20.7 ± 0.5

Quadrature method: u(sum) = √(0.3² + 0.2²) = √(0.09 + 0.04) = √0.13 ≈ 0.36 → 20.7 ± 0.36

The quadrature uncertainty is about 28% smaller — a real, meaningful difference, not rounding noise.

Example 2 — Multiplication: simple vs. quadrature

Values: x₁ = 5.0 ± 0.2 (4% relative), x₂ = 3.0 ± 0.1 (3.33% relative). Product = 15.0.

Simple method: relative u = 4% + 3.33% = 7.33% → absolute u = 15.0 × 0.0733 ≈ 1.10 → 15.0 ± 1.10

Quadrature method: relative u = √(4%² + 3.33%²) = √(0.16% + 0.111%)… expressed as decimals: √(0.04² + 0.0333²) = √(0.0016 + 0.00111) ≈ 0.0521 (5.21%) → absolute u = 15.0 × 0.0521 ≈ 0.78 → 15.0 ± 0.78

Again quadrature comes in noticeably tighter — roughly 29% smaller than the simple estimate.

Division works identically — it’s the same relative-uncertainty rule, just with the values divided instead of multiplied. For x₁ = 20.0 ± 0.4 (2% relative) divided by x₂ = 4.0 ± 0.2 (5% relative), quotient = 5.0:

Simple method: relative u = 2% + 5% = 7% → absolute u = 5.0 × 0.07 = 0.35 → 5.0 ± 0.35

Quadrature method: relative u = √(2%² + 5%²) = √29% ≈ 5.39% → absolute u = 5.0 × 0.0539 ≈ 0.27 → 5.0 ± 0.27

Example 3 — Powers

Value: x = 4.0 ± 0.1 (2.5% relative uncertainty). Calculate y = x².

Relative uncertainty of y: |2| × 2.5% = 5%. With y = 16.0, absolute uncertainty = 16.0 × 0.05 = 0.8.

Answer: y = 16.0 ± 0.8. Squaring doesn’t just square the value — it doubles the relative uncertainty.

Example 4 — Where sig figs and real propagation meet

Values, using only the implied precision of the written digits: 12.5 (written to 1 decimal place, implying roughly ±0.05) and 8.2 (also 1 decimal place, ±0.05).

Simple method: u(sum) = 0.05 + 0.05 = 0.10 → sum is trustworthy to about ±0.1 Quadrature method: u(sum) = √(0.05² + 0.05²) = √0.005 ≈ 0.07 → sum is trustworthy to about ±0.07

Both land in the same place: trust the first decimal place of the sum, not the second. Compare that to the sig-fig addition rule from our sig figs guide: round the sum to the fewest decimal places among the inputs — here, 1 decimal place either way. The sig-fig rule and the numeric propagation agree, because the sig-fig rule is a shortcut for exactly this reasoning. It just skips calculating an actual uncertainty value and jumps straight to “keep this many digits.” (This “implied uncertainty from the last written digit” is itself a rough convention, not a substitute for a real stated uncertainty — see our earlier gotcha on this — but it’s precisely the assumption sig fig rules are quietly built on.)

Example 5 — Propagating through a real calculation

Scenario: the area of a rectangle, with length L = 5.20 ± 0.05 cm and width W = 3.10 ± 0.04 cm. Area = L × W = 16.12 cm².

Relative uncertainties: u(L)/L = 0.05/5.20 ≈ 0.962%, u(W)/W = 0.04/3.10 ≈ 1.29%.

Simple method: relative u = 0.962% + 1.29% = 2.25% → absolute u = 16.12 × 0.0225 ≈ 0.36 → Area = 16.12 ± 0.36 cm²

Quadrature method: relative u = √(0.962%² + 1.29%²) ≈ 1.61% → absolute u = 16.12 × 0.0161 ≈ 0.26 → Area = 16.12 ± 0.26 cm²

This is what “uncertainty moves through a calculation” actually looks like end to end: two independently measured lengths, each with its own uncertainty, combining into one uncertainty on the derived area — smaller under quadrature, larger under the worst-case method, both defensible depending on what the number needs to guarantee.

Where This Still Trips People Up

• Uncertainties add for subtraction too. It’s tempting to think subtracting two values should partly cancel their uncertainties, since the central values are being subtracted. They don’t. u(x₁ − x₂) = u(x₁) + u(x₂) under the simple method, exactly like addition — the values subtract, but the uncertainties never do.
• Quadrature assumes independence. If two uncertainties share a root cause (the same uncalibrated instrument, the same flawed technique), quadrature alone understates the real combined uncertainty — see the correlated-inputs note above.
• This is a linear approximation. Both methods rest on a first-order Taylor expansion. For strongly nonlinear functions, or uncertainties that are large relative to the measured value, the approximation itself can break down, and a full propagation (Monte Carlo, per JCGM 101:2008) may be needed.
• “Just use the biggest uncertainty” isn’t propagation. Every genuine source of uncertainty contributes some real variance to the result, even the smaller ones — ignoring all but the largest source underestimates the true combined uncertainty, sometimes substantially, as Examples 1 and 2 show.
• Powers use the relative uncertainty, not the absolute one. A common error is applying u(y) = n · u(x) directly to absolute uncertainties; the power multiplies the relative uncertainty, and you convert back to an absolute value afterward — see Example 3.

Which Method Different Contexts Actually Use

Neither method is the “correct” one in general — they answer different questions. The simple method answers “what’s the largest this error could plausibly be if everything went wrong at once?” Quadrature answers “what’s the realistic, statistically expected uncertainty, assuming independent random errors?”

Where the Method Comes From

The quadrature approach isn’t a house convention — it’s the GUM’s (JCGM 100:2008) defined law of propagation of uncertainty, laid out in its Section 5, and it’s what our accuracy/precision pillar refers to as the formal companion to the VIM’s definitions. For the harder case of strongly nonlinear models or non-Gaussian input distributions, the GUM’s supplementary guide, JCGM 101:2008, describes a Monte Carlo alternative — propagating full probability distributions through the calculation by simulation rather than relying on the first-order approximation used here. That level of rigor is outside the scope of a sig figs calculator, but it’s worth knowing the name if a calculation ever demands it.

Common Mistakes

1. Adding absolute uncertainties for a multiplication problem, or relative uncertainties for an addition problem — these are reversed, exactly like the sig-fig arithmetic rules they mirror.
2. Assuming subtraction cancels uncertainty. It doesn’t — see the gotcha above.
3. Reporting the simple (worst-case) result as if it were a realistic statistical uncertainty, when quadrature is what scientific reporting convention actually expects.
4. Ignoring correlation between input uncertainties that share a common source.
5. Applying u(y) = n·u(x) to absolute uncertainty for a power calculation, instead of applying the multiplier to the relative uncertainty first — see Example 3.
6. Treating the biggest single uncertainty as the whole answer, discarding every other real contributing source.

Practice Problems

Concept: Simple method — addition

Q1. Using the simple method, x₁ = 5.0 ± 0.2 and x₂ = 3.0 ± 0.1 are added. What’s the propagated uncertainty? A) 0.1 B) 0.2 C) 0.3 D) 0.22 Answer: C) 0.3 (0.2 + 0.1).

Q2. Using the simple method, what’s the propagated uncertainty when subtracting x₂ from x₁ (same values as Q1)? A) 0.1 B) 0.2 C) 0.3 D) 0 Answer: C) 0.3 — uncertainties add regardless of whether the operation is addition or subtraction.

Concept: Quadrature — addition

Q3. Using quadrature, x₁ = 5.0 ± 0.2 and x₂ = 3.0 ± 0.1 are added. What’s the propagated uncertainty (2 sig figs)? A) 0.30 B) 0.22 C) 0.15 D) 0.10 Answer: B) 0.22 (√(0.2² + 0.1²) = √0.05 ≈ 0.22).

Q4. Why is the quadrature result in Q3 smaller than the simple result in Q1? A) It’s a rounding error B) Quadrature accounts for the statistical unlikelihood of both errors being at their worst simultaneously C) Quadrature only works for multiplication D) They should be identical Answer: B.

Concept: Relative uncertainty and multiplication

Q5. x = 5.0 ± 0.2. What is its relative uncertainty? A) 0.2% B) 2% C) 4% D) 20% Answer: C) 4% (0.2/5.0).

Q6. Using the simple method, x₁ has 4% relative uncertainty and x₂ has 3% relative uncertainty. What’s the propagated relative uncertainty of x₁ × x₂? A) 1% B) 5% C) 7% D) 12% Answer: C) 7% (4% + 3%).

Concept: Quadrature and multiplication

Q7. Using quadrature, x₁ has 4% relative uncertainty and x₂ has 3% relative uncertainty. What’s the propagated relative uncertainty? A) 3% B) 5% C) 7% D) 12% Answer: B) 5% (√(4² + 3²) = √25 = 5 — the same 3-4-5 relationship as a right triangle).

Q8. Why does multiplication propagate through relative uncertainties rather than absolute ones? A) It’s an arbitrary convention B) A product’s sensitivity to each factor scales with the size of the other factor, which normalizes naturally into a percentage C) Absolute uncertainty doesn’t apply to multiplication D) There’s no real difference Answer: B.

Concept: Powers

Q9. x = 4.0 ± 0.1 (2.5% relative uncertainty). What is the relative uncertainty of y = x³? A) 2.5% B) 5% C) 7.5% D) 25% Answer: C) 7.5% (3 × 2.5%).

Q10. Using Q9, with x = 4.0 so y = x³ = 64.0, what is the absolute uncertainty of y? A) 0.1 B) 1.6 C) 4.8 D) 16 Answer: C) 4.8 (64.0 × 0.075).

Simple vs. Quadrature, Same Inputs

Addition: 12.5 ± 0.3 + 8.2 ± 0.2 = 20.7 ± ?

• Simple method → ± 0.50 (wider bar)
• Quadrature method → ± 0.36 (narrower bar)

Multiplication: 5.0 ± 0.2 × 3.0 ± 0.1 = 15.0 ± ?

• Simple method → ± 1.10 (wider bar)
• Quadrature method → ± 0.78 (narrower bar)

Quick Reference

Rule of thumb: quadrature ≤ simple, always. Use simple for a guaranteed worst-case bound; use quadrature for a realistic statistical estimate.

Continue Learning

Related fundamentals:

• Accuracy, Precision, and Uncertainty: What Each One Actually Measures
• Significant Figures: The Complete Guide

Go deeper on one topic at a time:

• Absolute Error vs. Relative Error vs. Percent Error
• What Measurement Uncertainty Actually Is
• How to Read Uncertainty From a Ruler, Burette, or Digital Meter
• How Many Sig Figs Should an Uncertainty Have?
• Reporting Mean and Standard Deviation With Correct Precision
• Why Significant Figures Are a Simplification of Real Uncertainty

Tools:

• Error Propagation Calculator
• Absolute & Relative Uncertainty Calculator

Sources and Further Reading

• JCGM 100:2008, Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement (GUM), Section 5 — the primary source for the law of propagation of uncertainty and the general combined-uncertainty formula used throughout this page. (bipm.org)
• GUM Section 5, Determining Combined Standard Uncertainty (ISO-hosted full text) — confirms the formula’s basis in a first-order Taylor series approximation, cited above for the method’s known limits with nonlinear functions. (iso.org)
• NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results — reused from our accuracy/precision pillar, NIST’s own implementation guide of the same GUM methodology applied here. (emtoolbox.nist.gov)

Review and Methodology

Methodology: All formulas are cross-checked directly against the GUM (JCGM 100:2008) text, not a secondary paraphrase. Every worked example was independently recomputed during drafting. Calculator results referenced on this page use an arbitrary-precision decimal engine, not native floating-point math, validated against the site’s versioned regression fixture set.

Changelog

v1.0 — Initial draft completed, 2026-08-10.

The post Error Propagation: How Uncertainty Moves Through a Calculation appeared first on SignificantFiguresCalculator.

Rule Statement

The propagation of uncertainty follows two fundamental rules depending on the mathematical operation:

• Addition and Subtraction: The result should be rounded to the same number of decimal places as the measurement with the fewest decimal places (i.e., the least precise value). This is because the absolute uncertainty dominates.
• Multiplication and Division: The result should be rounded to the same number of significant figures as the measurement with the fewest significant figures. This is because the relative uncertainty dominates.

These rules are approximations that assume the uncertainties are independent and uniformly distributed. For rigorous work, the GUM method (JCGM 100:2008) is recommended, which uses standard deviations and partial derivatives.

Worked Examples

Addition and Subtraction

Suppose you measure lengths: 12.34 cm, 5.6 cm, and 0.789 cm. The least precise measurement is 5.6 cm (one decimal place). Adding them:

12.34 + 5.6 + 0.789 = 18.729 cm

Round to one decimal place: 18.7 cm. The result must not imply more precision than the least precise input.

Multiplication and Division

Calculate the area of a rectangle with length 3.45 m (3 sig figs) and width 2.0 m (2 sig figs). Multiply:

3.45 × 2.0 = 6.90 m²

Round to 2 significant figures: 6.9 m².

For a division example, compute density from mass 25.0 g (3 sig figs) and volume 5.0 cm³ (2 sig figs):

25.0 / 5.0 = 5.0 g/cm³

Round to 2 sig figs: 5.0 g/cm³.

Counter-Examples

Common errors often arise when the rules are misapplied. Consider the following pitfalls:

• Adding with different decimal places: 1.2 + 3.45 = 4.65. If you incorrectly round to the fewest significant figures (2 sig figs), you get 4.7, but the correct rule gives 4.6 (one decimal place). The result should be 4.6, not 4.7.
• Multiplying with too many sig figs: 2.5 × 3.14159 = 7.853975. If you keep all digits, you imply false precision. The correct answer is 7.9 (2 sig figs).
• Ignoring exact numbers: When a number is exact (e.g., a counted value or a conversion factor), it has infinite significant figures. For example, 2 × 3.14 = 6.28, not 6.3, because 2 is exact.

Convention Comparison Table

Standards Citation

Several standards define the proper handling of significant figures and uncertainty propagation:

• ASTM E29-22 – Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. This standard specifies how to round test results and how to interpret significant digits in specification limits.
• ISO 80000-1:2022 – Quantities and units – Part 1: General. Clause 7.3.4 addresses rounding and significant figures, recommending that the result of a calculation should not contain more significant digits than the least precise input.
• JCGM 100:2008 (GUM) – Evaluation of measurement data – Guide to the expression of uncertainty in measurement. Clause 5.1.2 provides the law of propagation of uncertainty, which is the rigorous method for combining standard uncertainties.
• NIST Technical Note 1297 – Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results (Taylor & Kuyatt, 1994). This document aligns with GUM and gives practical guidance for reporting uncertainties.

Common Mistakes

1. Rounding prematurely: Do not round intermediate values in a multi-step calculation. Carry extra digits until the final step, then round.
2. Confusing decimal places with significant figures: For addition, the rule depends on decimal places, not sig figs. For multiplication, it depends on sig figs, not decimal places.
3. Ignoring the uncertainty of exact numbers: Exact numbers (like 1000 in 1 L = 1000 mL) have infinite precision and do not limit the result.
4. Applying the rule to constants: Mathematical constants like π or e are exact if they are used as symbols; if approximated, their precision must be considered.
5. Forgetting to round the final answer: Always apply the rounding rule to the final reported value, not to intermediate steps.

Practice Problems

Test your understanding with these exercises:

1. Add: 2.34 + 5.6 + 0.789
2. Multiply: 4.56 × 1.2
3. Divide: 9.876 ÷ 3.2
4. Calculate: (12.34 + 5.6) × 2.0

Answers:

1. 8.7 (one decimal place)
2. 5.5 (two sig figs)
3. 3.1 (two sig figs)
4. 36 (two sig figs, because 2.0 has two sig figs and the sum 17.94 rounds to 17.9? Actually, stepwise: sum = 17.94, then multiply by 2.0 gives 35.88, round to 36 with two sig figs)

Software Behavior Note

Spreadsheet software (Excel, Google Sheets) and programming languages (Python, R, MATLAB) typically do not apply significant figure rounding automatically. They return full-precision results, which can mislead users. For example, =2.5*3.14159 in Excel yields 7.853975, not 7.9. To enforce significant figures, you must use rounding functions like ROUND() with the appropriate number of digits. In Python, the round() function uses banker’s rounding by default, which may not align with the half-up convention used in many standards. Always verify the rounding behavior of your tool against the relevant standard.

Quick Reference Table

Related Rules

For a deeper understanding, explore these related topics:

• Rounding Methods: A Comparison
• Significant Figures: The Complete Guide
• Understanding Measurement Uncertainty
• Error Propagation in Complex Calculations

The post Propagating Uncertainty Through Addition and Multiplication: Rules, Examples, and Standards appeared first on SignificantFiguresCalculator.