Propagation of uncertainty

The formula

The general law, for a result y = f(x₁, x₂, …, xₙ) computed from measured inputs ^[1]^[6]:

u_c²(y) = Σᵢ (∂f/∂xᵢ)²·u²(xᵢ) + 2·Σᵢ Σⱼ>ᵢ (∂f/∂xᵢ)(∂f/∂xⱼ)·u(xᵢ, xⱼ)

Reading: each input's variance gets scaled by the square of its sensitivity coefficient, the scaled variances add, and correlated inputs pick up a cross term that can push the total up or down.

With independent inputs — the everyday case, and the reason engineers say "RSS" — the cross terms vanish:

u_c(y) = √( (c₁·u₁)² + (c₂·u₂)² + … + (cₙ·uₙ)² ) where cᵢ = ∂f/∂xᵢ

Reading: root-sum-square of the sensitivity-weighted standard uncertainties.

Two special cases cover most bench math ^[6]:

y = a·x₁ ± b·x₂ → u_c = √( (a·u₁)² + (b·u₂)² ) — for sums and differences, absolute uncertainties RSS.

y = x₁^m · x₂^n → u_c/|y| = √( (m·u₁/x₁)² + (n·u₂/x₂)² ) — for products, quotients, and powers, relative uncertainties RSS, and each exponent multiplies its input's contribution.

Two support formulas from the GUM you use alongside it ^[1]:

u = a/√3 ≈ 0.577·a — converts a hard tolerance limit ±a (rectangular distribution) into a standard uncertainty.

U = k·u_c — the expanded uncertainty on the cal cert; k = 2 covers roughly 95% for near-normal results.

Where you meet it

The bench. You measure power as P = V²/R. The voltmeter is good to 0.5%, the resistor to 0.2%. The exponent on V doubles its contribution: u_P/P = √((2·0.5)² + 0.2²) = √1.04 ≈ 1.02%. The voltage term is 96% of the budget — a better resistor buys you nothing here.
The test stand. Thrust comes off a load cell through excitation supply, bridge, amplifier gain, and ADC. Every calibration certificate on that chain quotes an expanded uncertainty U with k = 2, because ISO/IEC 17025 labs report per the GUM ^[1]^[3]. Combining those into one number for the thrust channel is this formula, executed as an uncertainty budget table.
The design review. Three parts stack: ±0.5, ±0.3, ±0.2 mm. Worst case is ±1.0 mm; RSS is √(0.25 + 0.09 + 0.04) ≈ ±0.62 mm. Which one goes on the drawing is a judgment call about independence and volume — and someone at the table should be able to defend it.
The review board. When a customer asks whether your measurement is good enough to verify a requirement, the answer is a test uncertainty ratio, and the numerator comes out of this calculation. "The DMM is 6½ digits" is not an uncertainty statement.

How it works

The law is a first-order Taylor expansion: replace f with its tangent plane at the operating point, and the variance of a linear combination of independent variables is the sum of scaled variances. That origin defines its limits. It holds when the uncertainties are small relative to the curvature of f ^[1]^[6]^[7]. Where the linearization breaks — dividing by a quantity whose uncertainty is a decent fraction of its value, a function near a stationary point where the first derivative is zero, a hard nonlinearity like a threshold — the formula quietly reports garbage. The GUM's own escape hatch for those cases is Monte Carlo propagation (JCGM 101): sample the inputs, push each draw through the real f, read the spread of the outputs ^[9].

Variances add, not sigmas. That single fact drives most of the useful behavior:

Independent errors partially cancel. Adding sigmas linearly (the pre-RSS habit) says three 1% errors make 3%; RSS says 1.73%. The linear sum answers a different question — "what if every error conspires against me at once" — which is a worst-case bound, not a standard uncertainty.
The biggest term owns the budget. Combine 1.0 and 0.3 and you get √1.09 ≈ 1.04 — the smaller term moved the answer 4%. Corollary for spending money: improving anything but the dominant term is decoration. Build the budget table first, then shop.
Independence is the load-bearing assumption. Two 1% resistors in a divider, drawn independently: the ratio is uncertain by about 0.71%. Same two resistors from the same reel, tracking together: the fractional errors largely cancel in the ratio and the RSS answer is badly pessimistic. Flip side: channels sharing a reference, a supply, or a temperature environment have positively correlated errors, and assuming independence understates the combined uncertainty ^[7]. The covariance term is not a footnote; it is the difference between the two failure modes.
Random and systematic are different animals. A bias does not RSS away with averaging, and RSSing a known offset in with random scatter — instead of correcting for it — is the most common way an uncertainty budget lies. The GUM's discipline: correct what you can, propagate the uncertainty of the correction ^[1].
Type A and Type B feed the same formula. Uncertainties you computed from repeated readings (Type A) and ones you took from a spec sheet or cert (Type B, tolerance over √3 for rectangular) combine identically once they're standard uncertainties ^[1]^[8]. The classification records where the number came from, not how much it counts.

The mistake people make on reports: quoting u_c and U interchangeably. One is a one-sigma-equivalent, the other has a coverage factor baked in; mixing them silently doubles or halves the claimed quality of a measurement.

History

The mathematics is Gauss's. His Theoria combinationis observationum erroribus minimis obnoxiae (1823, supplement 1828) — the definitive statement of his least-squares theory — worked out how the mean errors of observations carry through to combinations of them, the machinery that becomes the modern law when you linearize f ^[4]^[5]. For the next century and a half every lab propagated errors in Gauss's spirit but in its own notation, with its own conventions about what "the error" even meant — probable error, standard deviation, limit of error — which made uncertainty statements from two labs genuinely incomparable.

The standardization is a metrology story. In 1977 the CIPM, recognizing that there was no international agreement on how to state measurement uncertainty, asked the BIPM to sort it out; the working group of national-lab experts it convened produced Recommendation INC-1 in 1980, which introduced the Type A/Type B classification, and the CIPM approved it in 1981 and reaffirmed it in 1986 ^[1]^[2]. ISO's Technical Advisory Group 4 turned the one-page recommendation into the full Guide to the Expression of Uncertainty in Measurement — the GUM — published in 1993 and reprinted with corrections in 1995 on behalf of seven international organizations ^[2]^[3]. NIST adopted the approach for its own measurement results and condensed it into Technical Note 1297 by Barry Taylor and Chris Kuyatt (1994 edition), the version most working US engineers actually read ^[3]^[8]. The Joint Committee for Guides in Metrology now maintains the GUM as JCGM 100, free to download, along with the Monte Carlo supplement for the cases the first-order law can't handle ^[1]^[9].

Related tools

/tools/strain-gauge-bridge — excitation, gauge factor, and bridge nonlinearity each carry an uncertainty that propagates through to the strain number.
/tools/voltage-divider — the two-resistor ratio is the textbook case for sensitivity coefficients and for correlation between inputs.
/tools/shunt-resistor — shunt tolerance and drift RSS with the voltmeter spec to set the current-measurement uncertainty.
/tools/adc-resolution — quantization is a Type B rectangular component: one LSB wide, LSB/√12 standard uncertainty as a centered version of the a/√3 rule.
/tools/loop-4-20ma — transmitter, loop resistor, and readout errors combine down the chain exactly this way.

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →