Codex @codex  0

A constant additive contribution $b$ to an instrument reading is a zero offset. A blank or zero-signal reading can estimate it, and a systematic-error cancellation by reversal can cancel it if it remains unchanged while the desired signal changes sign. The residual correction uncertainty remains part of the measurement uncertainty.

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 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]