Error Propagation: How Uncertainty Moves Through a Calculation

Error propagation is how the uncertainty in your measured inputs combines into the uncertainty of a calculated result — and there are two genuinely different, both legitimate, ways to combine them. The simple method (taught in most intro classes) adds uncertainties directly: absolute uncertainties for addition/subtraction, relative (percentage) uncertainties for multiplication/division. The rigorous method — the one the international measurement standard actually specifies — combines them in quadrature (square root of the sum of squares) instead of simple addition, and it always gives a smaller, more realistic number. Neither is wrong; they answer slightly different questions.

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.

<!– BLOCK: B02 – Inline Mini-Calculator –> <!– DEV NOTE: Embed the Error Propagation Calculator tool here (Tier 2 tool #10 in the site blueprint). Shortcode: [sfc_error_propagation_calculator show_both_methods=”true”]. Should accept multiple values with uncertainties, the operation (add/subtract/multiply/divide/power), and display both the simple and quadrature results side by side so the difference is visible on every calculation, not just in worked examples. –>

[Live Error Propagation Calculator embeds here] — Enter values with their uncertainties and an operation to see the propagated result under both the simple and quadrature methods, side by side.

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

Reviewed by: [Pending — reviewer assignment required before publication] Last reviewed: [Pending] 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.