Calibration and measurement uncertainty

A measurand is the specified quantity intended to be measured, including the conditions needed to make its value sufficiently well-defined for the stated accuracy. A measurement model for a scalar measurand $q$ is a specified function $q=f(x_1,\dots ,x_d)$ where $f:U\to \mathbb{R}$ is defined on an open set $U\subseteq {\mathbb{R}}^d$ and the $x_i$ are the input quantities that contribute to the reported result. For the uncertainty model below, assume $\widehat{x}\in U$. To treat the nonlinear output $f(X)$ as a random output, either require $X\in U$ almost surely and require $\omega \mapsto f(X(\omega ))$ to be measurable on the event $\{X\in U\}$, defining it measurably on the null complement, or specify a measurable extension of the model to the input values outside $U$. The almost-sure domain condition alone does not ensure measurability. If the variance of $f(X)$ is used, also require the resulting output to be square-integrable. The input list includes each significant correction, correction factor, calibration quantity, reference value, and other influence quantity used by the model. The estimate based on input estimates $\widehat{x}=({\widehat{x}}_1,\dots ,{\widehat{x}}_d)$ is $\widehat{q}=f(\widehat{x})$.

For uncertainty evaluation, represent the possible input values by an ${\mathbb{R}}^d$-valued random element $X=(X_1,\dots ,X_d)$ on one probability space, with square-integrable real coordinate variables. Their means are the input estimates, $\mathbb{E}[X_i]={\widehat{x}}_i$, and their input covariance matrix is ${\mathrm{\Sigma }}_{ij}=\mathrm{Cov}(X_i,X_j),1\le i,j\le d\mathrm{.}$ This probability model represents the evaluated uncertainty of the inputs; it does not assert that every input varies physically at random. No independence of the inputs is assumed.

A systematic effect is an identified influence whose estimated contribution can be corrected. A random effect is an influence that appears as variation across repeated observations under the declared conditions. These descriptions concern how an effect enters the measurement model; they are separate from the Type A/Type B evaluation labels. A correction is an estimated signed value added algebraically to an uncorrected result to compensate for a recognized systematic effect. A correction factor changes a result multiplicatively according to the stated model. Applying a correction does not remove uncertainty: uncertainty in the correction and in its reference values remains an input contribution unless its contribution is shown negligible for the stated result. A reference standard is a stated reference artifact or realization used to compare or calibrate a measurement system; its assigned value and uncertainty, including covariance with other shared reference inputs, belong in the measurement model. A calibration curve is the specified relation between a reference quantity and an instrument response used to convert a later response into an estimate of the measurand.

Type A evaluation means evaluating an uncertainty component by statistical analysis of a series of observations. Type B evaluation means evaluating it by other information or methods. These labels describe how a component is evaluated, not whether its cause is random or systematic. A repeatability component describes variation in repeated readings under stated, comparable conditions. A later reading of a calibrated instrument can therefore add a new repeatability component; include its covariance with other inputs when the model does not justify treating them as uncorrelated.

A standard uncertainty is an uncertainty measure expressed as a standard deviation. The combined standard uncertainty below is the standard deviation of a linearized output, not an interval statement.

When $f$ is differentiable at $\widehat{x}$, write $g=\mathrm{\nabla }f(\widehat{x})$ and define the first-order linearized output by $Q_{\mathrm{l}\mathrm{i}\mathrm{n}}=f(\widehat{x})+g^T(X-\widehat{x})\mathrm{.}$ The first-order combined standard uncertainty is $u_c=\sqrt{\mathrm{Var}(Q_{\mathrm{l}\mathrm{i}\mathrm{n}})}$, the standard deviation of this Taylor-linearized output. The next theorem gives its covariance formula. Its use as an approximation to the uncertainty of the nonlinear output requires the first-order model to be adequate for the stated measurement. This standard uncertainty is not itself an interval and implies no coverage probability or confidence level.

A measurement batch in the example on the companion page is a declared measurement occasion or configuration, such as one day of operation. It is a label for a process condition; it is not a random-effect assumption by itself. Whether a calibration relation remains stable across batches is an empirical question about the stated process and study.

The broader Scientific statistical model contracts requires the observation law, units and index roles, target, observation mechanism, support (a measurable set assigned probability one), conditioning convention, and status of model-adequacy assumptions; it does not prove that a proposed model applies physically. Here Random elements and real random variables supplies the measurable-map meaning of each input variable, Moments, variance, and covariance on a probability space defines the finite means and covariances, and The real Frechet derivative on real and complex matrix spaces with the Frobenius norm supplies the first-order derivative remainder (identify ${\mathbb{R}}^d$ with real column matrices and use the Euclidean/Frobenius norm).

Let $d\ge 1$, let $\widehat{x}\in U$ for an open set $U\subseteq {\mathbb{R}}^d$, and let $f:U\to \mathbb{R}$ be differentiable at $\widehat{x}$. Let $X$ be an ${\mathbb{R}}^d$-valued random element on one probability space with square-integrable coordinate variables $X_i$, means $\mathbb{E}[X_i]={\widehat{x}}_i$, and $X\in U$ almost surely. Let ${\mathrm{\Sigma }}_{ij}=\mathrm{Cov}(X_i,X_j)$, set $g=\mathrm{\nabla }f(\widehat{x})$, and define $Q_{\mathrm{l}\mathrm{i}\mathrm{n}}=f(\widehat{x})+g^T(X-\widehat{x}),u_c^2=\mathrm{Var}(Q_{\mathrm{l}\mathrm{i}\mathrm{n}})\mathrm{.}$ Then $u_c^2=g^T\mathrm{\Sigma }g={\sum }_{i=1}^d{g}_i^2{\mathrm{\Sigma }}_{ii}+2{\sum }_{1\le i<j\le d}{g}_i{g}_j{\mathrm{\Sigma }}_{ij}\mathrm{.}$ The equality is exact for the first-order linearized output. To interpret it as an approximation to uncertainty in the nonlinear output $f(X)$, require $f(X)$ to be a measurable, square-integrable random output (including a measurable definition on any null event where $X\mathrm{\notin }U$) and require the first-order model to be adequate. This interpretation is not an assertion of exact finite-sample variance or interval coverage.

No independence assumption is used. The formula includes covariance between calibration coefficients, reference standards, corrections, and any later reading included in $X$. The first-order Taylor rule is appropriate only when the nonlinear remainder is negligible for the measurement at hand. JCGM 100:2008/Amd.1:2026 calls for higher-order terms when nonlinearity is significant; Monte Carlo methods are outside this result. Neither $u_c$ nor $u_c^2$ alone supplies a confidence interval or coverage probability.