Foundations

Detectable change and response scale are different instrument properties.

Response depends on timescale, gain and phase.

A measurand is the quantity intended to be measured, including the relevant system state, components and conditions.

Accuracy describes closeness of a measured value to a true measurand value; trueness concerns agreement of the limiting replicate mean with a reference value; precision concerns agreement among repeated values under specified conditions.

Measurement error is measured value minus reference value.

An influence quantity changes the relation between instrument indication and measurement result without changing the quantity actually measured in a direct measurement.

Verification provides objective evidence that specified requirements are fulfilled.

An uncertainty budget states the measurement model, input estimates, standard uncertainties, distributions, covariances, sensitivities, evaluation methods and degrees of freedom, together with how they combine and any coverage factor.

For n independent identically distributed observations of a stable quantity with finite variance, the replicate-mean standard uncertainty is estimated by s/√n, where s²=Σ_j(x_j−x̄)²/(n−1).

An assigned rectangular distribution on estimate±a gives u=a/√3, while a symmetric triangular distribution on the same interval gives u=a/√6.

For independent input standard-uncertainty estimates in a linearized model, Welch–Satterthwaite gives ν_eff=u_c⁴/Σ_i[(c_i u_i)⁴/ν_i].

A reusable measurement report identifies the measurand, value and unit, relevant conditions and corrections, and whether uncertainty is standard u_c or expanded U.

An illustrative temperature model y=x̄+C+δ has x̄=20.00 °C from four independent readings with s=0.20 K, C=+0.50 K with u_C=0.20 K, and independent zero-centered δ uniform on ±0.05 K.

An instrument calibration curve models indication y as a function of reference input x, such as y=a+bx+ε.

For the valid linear calibration y=a+bx, an unknown input is estimated as x̂=(y₀−â)/b̂.

Calibration-model validation examines residuals for curvature, unequal variance, dependence and outliers, and compares independent or check-standard results across the working range.

A measurement standard realizes a specified quantity with a stated value and uncertainty for reference use.

A reference material is sufficiently homogeneous and stable for specified properties and an intended measurement or examination use.

A detection limit depends on the measurement procedure's false-presence probability α and false-absence probability β.

Results for quantities of the same kind are metrologically comparable when traceable to the same reference.

An ideal baseband signal containing no frequencies above B can be reconstructed from exact uniform samples at f_s>2B.

For a uniform rounding quantizer with step Δ and no overload, the error magnitude is at most Δ/2.

For repeated indications, the mean has variance Var(x̄)=(1/n²)Σ_iΣ_j Cov(x_i,x_j).

Instrumental drift is a continuous or incremental change in indication caused by changes in an instrument's metrological properties.

Fixing a defining constant's numerical value makes that SI definition exact; it does not make an experiment or transferred standard exact.

A primary caesium frequency standard relates its oscillator to the defining unperturbed Cs-133 hyperfine transition by estimating and correcting design-dependent shifts, including atomic-motion Doppler and thermal blackbody effects.

With calibrated light frequency f, vacuum wavelength is λ₀=c/f.

A Kibble balance combines weighing mg=I Bℓ with velocity-mode U=vBℓ to eliminate the common geometry-field factor: mgv=IU.

For a differentiable real model Y=f(X), first-order output covariance is Σ_Y≈JΣ_XJᵀ, J_ai=∂f_a/∂x_i at the input estimate; the rule is exact for an affine model with finite input covariance.

In a constructed scalar model Y=X² with X uniformly distributed on [−a,a], a>0, exact distribution propagation gives E(Y)=a²/3 and Var(Y)=4a⁴/45, hence u(Y)=2a²/√45.