Foundations

Distinguish exact model values, rounded approximations and estimates from measurements. Declare the rounding rule, preserve information through calculations and unit changes, and interpret resolution, precision, accuracy and the stated uncertainty type without turning a report’s digits into guaranteed bounds.

Choose the physical quantity you want to investigate, specify the point, event or location involved, and explain how a reading relates to that quantity.

Connect a result to a reference through documented calibrations.

Combine input uncertainty through a measurement model.

Relate indications to standards, then infer measurement results.

The classification describes how a component is evaluated.

A coverage factor scales standard uncertainty to a reported interval.

Shared input variation belongs in the joint uncertainty description.

Specify how input quantities determine the measurand.

Sample the joint input model to approximate the output distribution.

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.

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.

For a validated linear first-order sensor model τẏ+y=Kx with τ>0, a step to x_f gives y(t)=Kx_f+[y(0)−Kx_f]e^(−t/τ).

For a vector measurand Y=f(X), propagating a joint input distribution retains output dependence and non-Gaussian shape.

For a nonsingular m-variate Gaussian assigned distribution with mean y and known covariance U, (Y−y)ᵀU⁻¹(Y−y)≤χ²_(m,p) gives probability p.

Monte Carlo draws from the joint input model and evaluates all outputs from each common draw.

More repetitions do not resolve an underspecified measurand.

Relative uncertainty uses a nonzero measured magnitude.

A zero or blank reading can still have uncertainty.

An intended-use target differs from achieved uncertainty.

Report useful digits without treating extra digits as proof of accuracy.