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A physicist associated with quantitative measurement and the evaluation of measurement uncertainty. With Chris E. Kuyatt, he wrote NIST guidance that separates uncertainty evaluated statistically from uncertainty evaluated using other information, then expresses both on a common standard uncertainty scale. This allows repeated readings, instrument specifications and other contributions to be combined into an explicit measurement uncertainty budget; correcting an estimated systematic error does not eliminate uncertainty in that correction.^[ref]

Measurement calibration establishes a relation between an instrument's indications and reference values, with the relevant measurement uncertainty. It is distinct from changing the instrument until it reads correctly: an instrument can be calibrated by determining a response curve and applying a correction afterward.

A practical voltage experiment connects a meter to several reference values across its working range and repeats the sequence in both directions. Fitting $V_{\mathrm{indicated}}=a{V}_{\mathrm{reference}}+b$ estimates a scale factor and measurement zero offset; departures from that relation test the response model. Repetition at later times checks instrumental drift. Uncertainty in the reference, the fitted parameters, and unresolved departures belongs in the measurement uncertainty budget, even if the displayed readings appear very stable.^[ref]

Measurement uncertainty describes the spread of values reasonably attributable to a specified measurand, using the available information. It differs from measurement error, which is a difference from a reference value: the uncertainty can be quantified even when that difference is unknown. A useful result therefore states what was measured, the unit of measurement, the value, and how its uncertainty was evaluated.^[ref]

For a wire's electrical resistance obtained from $R=V/I$, uncertainty in the voltage and electric current propagates through that measurement model. Shared measurement calibration effects can make the inputs correlated, so adding percentages or treating every contribution as independent can be wrong. The measurement uncertainty budget records those assumptions. Report a standard uncertainty when a standard-deviation scale is intended, or an expanded uncertainty with its stated coverage factor; a bare plus-or-minus number does not specify which was used.

A systematic error is a reproducible displacement caused by the apparatus, procedure or measurement model. Repeating the same experiment can determine its mean very precisely while retaining that displacement. Reducing random measurement error therefore does not establish accuracy.

Start with a measurement model listing the response, zero offset, calibration scale and relevant environmental corrections. Compare the instrument with traceable standards at several points across its working range; a single zero check cannot reveal a gain error or nonlinearity. Measure blanks and backgrounds using the same acquisition chain as the signal. Repeat calibration before and after the run to detect drift, and retain the uncertainty of every applied correction.

systematic-error cancellation by reversal separates a wanted effect from an offset with a different symmetry. For example, a resistance measurement with thermoelectric offset $V_0$ gives $V_{+}=RI+V_0$ and $V_{-}=-RI+V_0$. The difference givescancelling a stable offset, while $(V_{+}+V_{-})/2$ diagnoses it. Alternate the current direction quickly enough that the offset is approximately unchanged. This does not cancel a current-scale calibration error or resistance changes from Joule heating; vary the current magnitude and monitor temperature to test those effects. Control measurements must target the actual proposed bias.

Randomizing the order of settings prevents slow drift from consistently imitating a physical trend. Interleaving a reference measurement helps track that drift. Reversing geometry, exchanging matched detectors and changing alignment can expose effects tied to one part of the apparatus. Repeating the measurement with an independent method is particularly valuable when its dominant biases differ; agreement between two instruments sharing the same calibration does not test a common calibration error.

Apply justified corrections, propagate their uncertainties and include correlations when the same calibration enters several observations. A systematic component shared by all observations does not shrink as $1/\sqrt{N}$. A measurement uncertainty budget should distinguish the physical origin of each effect from the method used to evaluate its uncertainty: statistical Type A and other Type B evaluations do not simply mean random and systematic, respectively.^[ref]

A sensitivity coefficient describes how an estimated output of a measurement model changes when one input changes slightly. For $y=f(x_1,\dots ,x_n)$, the first-order coefficients, evaluated at the input estimates, areA sensitivity coefficient can have a sign and a unit of measurement: for an output in ohms and an input in volts, its unit is ohms per volt. The GUM also describes estimating sensitivity coefficients experimentally by varying inputs.^[ref]

In a measurement uncertainty budget, propagation of uncertainty uses these coefficients to transform input variances and covariances:Here $u_i$ denotes an input standard uncertainty. The derivative's sign disappears from its independent variance term but still matters in a covariance term.^[ref]

For an original electrical resistance example, $R=V/I$, $V=10\mathrm{V}$ and $I=2\mathrm{A}$ give $R=5\Omega$ andIndependent input standard uncertainties of0.1volt and0.02ampere each contribute0.05ohm on the output scale, giving $u_R=\sqrt{0.05^2+0.05^2}\Omega \approx 0.0707\Omega$. A larger sensitivity coefficient alone does not identify the dominant uncertainty: the input standard uncertainty and any correlations also matter.

The derivative is a local approximation. If the input spread spans a strongly nonlinear part of the measurement model, first-order propagation of uncertainty can be inadequate even when every derivative has been computed correctly.

A shared calibration error is a measurement error contribution inherited by several results from the same uncertain measurement calibration information. A common reference offset, scale factor or fitted coefficient can couple readings that would otherwise have independent noise. The GUM identifies common measuring instruments, standards and reference data as sources of correlation.^[ref]

Consider an original additive model for repeated readings of a fixed quantity $\theta$:Here $B$ represents the remaining uncertainty in a common correction, and the centered ${\epsilon }_i$ are independent of one another and of $B$. These assumptions implyThis is covariance induced by a shared latent variable: the same unknown contribution appears in every reading. A fixed but unknown physical offset need not fluctuate between readings for uncertainty about it to be shared.

With100readings, individual noise standard deviation1unit and common-offset standard uncertainty0.2unit, the mean has standard uncertainty $\sqrt{1/100+0.2^2}\approx 0.224$unit. Treating all100readings as independent would instead give $\sqrt{(1+0.2^2)/100}\approx 0.102$unit. Repetition reduces the individual noise contribution but not the common-offset floor.

A difference $Y_i-Y_j={\epsilon }_i-{\epsilon }_j$ cancels this additive common offset. It does not cancel an arbitrary common gain error. Reference changes and deliberate systematic-error cancellation by reversal test different effects; the measurement uncertainty budget should identify which quantities actually share the uncertain correction.

A systematic error is a component of measurement error that remains constant or changes predictably when measurements are repeated under specified conditions. A ruler with an incorrect scale, a thermometer with a measurement zero offset, and an unaccounted instrumental drift can all produce such effects. Closely agreeing readings can therefore have poor measurement accuracy: repetition tests their scatter, not the correctness of the shared reference.

An estimated systematic error should be corrected when possible, with the correction's measurement uncertainty included in the measurement uncertainty budget. Suppose a balance reads $m+b$ and a reference gives an estimate $\hat{b}$ of its offset. The corrected result is $m_{\mathrm{read}}-\hat{b}$; uncertainty in $\hat{b}$ remains even if the same unknown offset affected every weighing. Changing the reference, using a blank, or testing systematic-error cancellation by reversal can reveal effects that a long sequence of identical readings misses. The Type A evaluation of uncertainty and Type B evaluation of uncertainty distinction concerns how uncertainty is evaluated, so neither label is synonymous with systematic error.^[ref]

Vibroscope area uncertainty quantifies uncertainty in an area inferred from a vibration model and its measured inputs. For the flexible-string vibroscope area estimator, length and frequency enter with inverse-square dependence, whereas tension and density enter to the first power. The parent article gives the full first-order propagation formula. A measurement uncertainty budget records the input standard uncertainties, their sensitivities and any correlations.^[ref]

For an original independent-input example, stipulate relative standard uncertainties of $1\%$ in tension, $2\%$ in density, $0.5\%$ in length and $0.1\%$ in frequency. After applying the sensitivity coefficients, their relative area contributions are $1\%,2\%,1\%,0.2\%$. Combining their squares givesFor a stipulated estimate $\hat{A}=80\mu {\mathrm{m}}^2$, this is $u_A\approx 1.97\mu {\mathrm{m}}^2$. Density contributes $4/6.04\approx 66.2\%$ of the combined variance. Improving the frequency readout alone would therefore make little difference to this budget. Reducing the density contribution from $2\%$ to $1\%$ instead gives $\sqrt{3.04}\%\approx 1.74\%$ overall.

These input uncertainties are assigned model values, not the readout resolutions quoted for the Favimat+ fibre testing machine. A resolution increment does not determine a standard uncertainty without a model for the associated error. Correlated inputs require covariance terms; their signed sensitivities can either reinforce or cancel contributions.

The budget also presumes an adequate measurement model. Bending-stiffness bias in the vibroscope area estimator, uncertain supports or deformation conventions may require a correction and additional uncertainty terms. Increasing the numerical precision of the frequency does not repair a wrong physical model.