Measurement uncertainty (GUM framework)

The formula

Everything starts from a measurement model — the equation that turns what you actually read into what you report:

y = f(x₁, x₂, …, x_N)

Your result y is computed from N input quantities: readings, calibration corrections, temperature, lead resistance, whatever touches the number.

Type A — uncertainty you get by taking repeated readings and doing statistics:

u(x̄) = s/√n ,   where  s = √( Σ(x_k − x̄)² / (n−1) )

The standard uncertainty of the average of n readings is the sample standard deviation divided by √n — scatter you measured, shrunk by averaging.

Type B — uncertainty you get any other way: spec sheets, cal certificates, handbooks, physics. The two workhorses:

u = a/√3        limits ±a, nothing else known (rectangular distribution)
u = U_cert/k    certificate quotes expanded uncertainty U_cert at coverage factor k

A spec that says "±a, period" is treated as a flat distribution between the limits; its standard deviation is a/√3 ≈ 0.577·a. A certificate value quoted at k = 2 gets divided by 2 before it enters your budget. If values near the middle are more likely, the GUM offers a/√6 (triangular) ^[1].

Combined standard uncertainty — the law of propagation of uncertainty, first-order Taylor expansion of the model:

u_c²(y) = Σ ( ∂f/∂xᵢ )² · u²(xᵢ)          (independent inputs)
        + 2·Σ Σ (∂f/∂xᵢ)(∂f/∂xⱼ)·u(xᵢ,xⱼ)  (add this when inputs are correlated)

Each input's standard uncertainty gets scaled by its sensitivity coefficient cᵢ = ∂f/∂xᵢ — how hard the output moves per unit wiggle of that input — then everything adds in quadrature. This is RSS with the physics attached.

Expanded uncertainty — what goes on the certificate:

U = k · u_c(y)

k is the coverage factor. k = 2 covers about 95% of a normal distribution (95.45%, exactly), k = 3 about 99.7% ^[1]. When a small number of readings dominates the budget, k comes from Student's t at the effective degrees of freedom given by the Welch–Satterthwaite formula:

ν_eff = u_c⁴(y) / Σ ( uᵢ⁴(y) / νᵢ )

Type A components carry ν = n − 1; spec-sheet components are conventionally taken as ν → ∞, so their term vanishes and they never drag ν_eff down ^[1].

Where you meet it

Reading a calibration certificate. Every accredited cal lab reports y ± U with the k value stated, almost always k = 2. Before that number goes in your own budget, divide by k. Feeding a k = 2 value into an RSS as if it were a standard uncertainty doubles that component's weight — the single most common budget error in circulation.
Qualifying a measurement chain at the test stand. Load cell, signal conditioner, ADC, cal source: the data package for a thrust or pressure measurement carries an uncertainty budget — one spreadsheet row per source, each converted to a standard uncertainty, RSS'd, expanded. Repeatability from the checkout runs is your Type A row; everything off a datasheet is Type B.
At the review board, arguing pass/fail near a limit. A unit measures 99.8 against a 100.0 limit with U = 0.5. Whether that's a pass, a fail, or a guard-banded retest is a measurement decision risk conversation, and it runs entirely on the GUM numbers. Same math sits under the 4:1 test uncertainty ratio your quality system asks for.
During an ISO/IEC 17025 audit. Accreditation requires documented uncertainty budgets for every calibration on the scope. The auditor will pull one and walk the rows. NIST's own policy for reporting measurement results follows the same framework, published as Technical Note 1297 ^[3]^[5].

How it works

The recipe is five steps: write the model, evaluate each input's standard uncertainty (Type A by statistics, Type B by judgment from what you know), compute sensitivity coefficients, combine in quadrature, expand by k ^[1]^[8]. The output is one defensible number instead of a shrug.

The framework's central move is what it deleted. Before 1980, labs split errors into "random" and "systematic" and fought about how to combine them — linear addition, RSS, or not at all. Recommendation INC-1 declared the categories describe how you evaluated a component, not what kind of error it is, warned that the term "systematic uncertainty" is misleading, and ordered everything expressed as a standard deviation so it all combines by the ordinary rules for variances ^[1]. Once every source speaks the same unit, a resistor tolerance, a thermocouple table, and thirty repeat readings can live in one budget. That bookkeeping trick is the whole invention.

A Type B evaluation is not a second-class guess. With ten readings, your estimate of s itself carries about 24% relative uncertainty; a well-founded datasheet number can beat it ^[1]. The sin isn't using Type B — it's using it lazily, like calling a ±0.1% spec "2 sigma" when the manufacturer said no such thing. Bounds with nothing else known are rectangular: 0.1%/√3 = 0.058%.

Where it bites:

k = 2 is a habit, not a law. It delivers ~95% only when the combined distribution is near-normal with healthy degrees of freedom. If your budget is dominated by a Type A estimate from ten readings, ν_eff ≈ 9 and the honest 95% factor is t₉₅(9) = 2.26, not 2 — a 13% understatement of U. The Welch–Satterthwaite formula exists to catch exactly this, and it's the step most spreadsheets skip ^[1].

Correlation breaks the RSS. Two channels calibrated against the same reference share its error; two dimensions measured with the same micrometer share its bias. Quadrature addition assumes independence. For fully correlated inputs the uncertainties add linearly, and skipping the covariance term makes the budget optimistic — the expensive direction.

The Taylor expansion is first-order. Strongly nonlinear models — anything with squares, quotients near zero, or a minimum — can make the sensitivity-coefficient math meaningless. The official fix is Supplement 1 (JCGM 101:2008): push actual probability distributions through the model with Monte Carlo and read the coverage interval off the output histogram ^[4]^[6].

Uncertainty is not error. If you know the bias, correct for it; uncertainty is your doubt about what's left after every known correction is applied ^[1]. Folding a known, uncorrected offset into U to avoid paperwork is how a budget turns into fiction. Related failure: counting the same physics twice, once as the instrument's "accuracy" spec and again as your measured repeatability, when the spec already includes it.

History

The GUM exists because the world's cal labs couldn't agree on arithmetic. In 1977 the CIPM — the international committee that oversees the SI — asked the BIPM to find a consensus on stating uncertainties. The BIPM sent a questionnaire to 32 national metrology labs; the 21 replies received by early 1979 agreed the problem mattered and agreed on nothing else ^[1]^[2]. So in 1980 the BIPM convened experts from eleven national labs, and that Working Group on the Statement of Uncertainties produced Recommendation INC-1 — five short clauses that created the Type A/Type B split and ruled that everything be carried as a standard deviation. The CIPM adopted it in 1981 and reaffirmed it in 1986 ^[1]^[2].

Turning a one-page recommendation into a usable manual fell to ISO, whose Technical Advisory Group on Metrology (TAG 4) formed Working Group 3 with experts from the BIPM, IEC, ISO, and OIML ^[1]^[3]. The 100-page result — the Guide to the Expression of Uncertainty in Measurement — appeared in 1993 in the name of seven international organizations, with a corrected reprint in 1995 ^[1]^[3]. NIST issued its plain-language adaptation, Technical Note 1297 by Barry Taylor and Chris Kuyatt, in 1994 ^[3]^[5]. The effective-degrees-of-freedom machinery inside Annex G was borrowed, not invented: B. L. Welch published it across 1936–1947 and F. E. Satterthwaite independently in 1941 and 1946, decades before anyone called it metrology ^[1].

In 1997 the sponsoring organizations formed the Joint Committee for Guides in Metrology (JCGM), chaired by the BIPM director, which took the GUM over from ISO ^[4]^[7]. The JCGM reissued it — free, at last — as JCGM 100:2008, added the Monte Carlo supplement JCGM 101:2008 and a multivariate extension JCGM 102:2011, and has since begun recasting the Guide as a series of parts (GUM-6 in 2020, GUM-1 in 2023) ^[4]^[6].

Related tools

/tools/adc-resolution — quantization is the textbook Type B rectangular source: u = q/√12 per code width
/tools/snr-enob — converting noise to effective bits is an uncertainty statement about the digitizer
/tools/strain-gauge-bridge — the bench measurement whose budget has Type A scatter riding on Type B excitation and gauge-factor specs
/tools/wheatstone-bridge — the classic cal-lab resistance measurement the GUM's own worked examples are built around
/tools/resistor-e-series — tolerance limits with nothing else known: rectangular, divide by √3
/tools/loop-4-20ma — transmitter accuracy specs are the Type B rows in every process-measurement budget

Sources

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