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A systematic error is a reproducible bias in the measurement model or procedure, such as a zero offset, incorrect gain, background contamination, misalignment, loading or temperature-dependent calibration. Repeating the same biased procedure more often reduces independent random scatter but does not remove the bias. An unknown correction also has a measurement uncertainty, even when the mechanism is systematic.

A simple instrument model is $y=(1+g)x+a+d(t)+ϵ$, with gain error $g$, offset $a$, drift $d(t)$ and random noise $ϵ$. Comparison with calibrated standards at several input values separates an offset from a scale error and can test nonlinearity. Use standards that cover the operating range, verify units and reference values, and propagate uncertainty in the standards into the corrected result. Calibration changes the model; it does not make the corrected measurement exact.

A blank or source-off measurement estimates additive background. For independent source-plus-background count $K_s$ in time $t_s$ and background count $K_b$ in time $t_b$, the net-rate estimate and its counting variance areSubtracting a measured background removes its estimated mean, not its fluctuations or its uncertainty. A long background exposure can reduce the latter, while drifting backgrounds require monitoring or a joint model.

Reversal can distinguish a wanted signal from an unchanged offset. For example, $V(+I)=IR+V_{\mathrm{th}}$ and $V(-I)=-IR+V_{\mathrm{th}}$ give $R=[V(+I)-V(-I)]/(2I)$ when the thermoelectric offset is stable between readings. Interchanging sample and reference, reversing field direction or reversing an apparatus orientation can similarly expose sign-dependent biases, but only if the wanted and unwanted contributions transform differently.

Track slow instrumental drift using frequent references, interleaved readings or an ABBA measurement sequence. Equal-time readings in the order A, B, B, A cancel a common linear drift in the difference estimator $(y_1+y_2-y_0-y_3)/2$. Nonlinear drift or unequal sample conditions can survive. Randomizing run order helps prevent a time drift from imitating dependence on a controlled experimental variable.

Control physical couplings directly: stabilize temperature, minimize gradients, shield unwanted fields, align geometry, and use high input impedance when loading is the concern. Differential and bridge measurements suppress common changes when their channels are well matched. Varying an apparatus parameter or using an independent method can reveal an unsuspected bias. Report residual corrections, their uncertainties and correlations; adding more identical readings cannot justify discarding a shared calibration uncertainty.

Positioning repeatability describes the scatter of positions reached when a motion system repeatedly returns to the same commanded position under specified conditions. It is an application of measurement repeatability, a form of measurement precision. A system may return to almost the same wrong position each time, so small scatter does not establish measurement accuracy.^[ref]

The approach direction belongs in the test description. Unidirectional positioning repeatability keeps it fixed; bidirectional positioning repeatability includes returns from both directions. To measure the assembled stage, independently observe its final position rather than only the motor's rotary encoder count. Physik Instrumente describes a static test that lets the platform stop before reading it, using an interferometer for displacement and an autocollimator for angular error, with two reflecting mirrors on the stage.^[ref]

For an original numerical example, repeatedly command $x_{\ast }=10.000\mathrm{mm}$. Suppose an independent position readout with known measurement calibration gives the errors $e_j=x_j-x_{\ast }$, in micrometres,Each direction has the same sample standard deviation,Yet their arithmetic means are 0 and $2.0\mu \mathrm{m}$. The magnitude of their mean difference, a positioning reversal error at this target, is 2.0 micrometres, far larger than the within-direction scatter. Pooling all eight values gives a sample standard deviation of $\sqrt{8.04/7}\approx 1.149\mu \mathrm{m}$; that statistic mixes the directional shift with the smaller return-to-return variation.

The calculation uses a declared sample standard deviation, not an unspecified manufacturer's tolerance or a universal repeatability convention. A positioning specification must state its approach directions, target positions, settling/readout procedure, load, environmental conditions, statistic and measurement uncertainty. A single target does not establish performance over the travel range. The manufacturer's method explicitly separates directional means from scatter and combines information from multiple targets when forming axis-level specifications.^[ref]

Backlash is one possible cause of direction-dependent behavior, while thermal expansion and instrumental drift can move the apparent target over time. A repeatability experiment must distinguish those effects from random scatter. Measuring small returns to one target also answers a different question from minimum incremental motion, which asks whether a small commanded change can actually be executed.

A positioning reversal error compares the mean reached positions at one command when approached from opposite directions. Define $e_{i,j,d}=x_{i,j,d}-x_i^{\ast }$ for target $i$, trial $j$ and approach direction $d$. With $\uparrow$ denoting approach in the increasing-coordinate direction, use the signed target-specific quantityState this sign convention. PI's axis-level quantity called maximum positioning backlash instead takes the maximum absolute directional mean difference over the tested targets.^[ref]

For an original two-target example, measured mean errors at $x^{\ast }=2\mathrm{mm}$ are $0.3\mu \mathrm{m}$ from below and $-0.7\mu \mathrm{m}$ from above, giving $B_1=1.0\mu \mathrm{m}$. At $x^{\ast }=8\mathrm{mm}$, means of $-0.4$ and $0.9\mu \mathrm{m}$ give $B_2=-1.3\mu \mathrm{m}$. ThusA small signed average can conceal substantial reversal effects that change sign along the travel.

Read final load position independently, for example with an interferometer, after an agreed settling time. The observed difference does not identify a unique physical cause: backlash, elastic relaxation or instrumental drift can contribute. For an original drift model $e_d(t)=a_d+vt$, an unbalanced timing schedule givesAt $v=0.01\mu \mathrm{m}{\mathrm{s}}^{-1}$ and mean measurement times 50 s and 150 s, drift alone contributes $-1\mu \mathrm{m}$. Alternate or balance the approach timing when testing that confounding effect. Unidirectional positioning repeatability measures within-direction scatter rather than this difference of means.

Range-overlap calibration establishes or checks the relationship between instrument settings using inputs that both can measure validly. In gain ranging, it tests whether changing sensitivity preserves the inferred measurand. It can determine relative scale and offset between ranges, but agreement between ranges alone does not establish their absolute accuracy or independent metrological traceability.

For an original two-range measurement model, let the same stable electric current $I$ produceEliminating $I$ givesAn overlap experiment therefore needs enough distinct, stable input levels to distinguish relative gain from offset. One common input gives one equation for $a$ and $c$ and cannot identify both without additional information.

For example, suppose $G_L=0.10V/nA$, $b_L=0.01\mathrm{V}$, $G_H=0.50V/nA$ and $b_H=0.02\mathrm{V}$. Currents of $1$, $2$ and $3\mathrm{nA}$ give output pairs $(0.11,0.52)$, $(0.21,1.02)$ and $(0.31,1.52)\mathrm{V}$. They satisfy $V_H=5V_L-0.03\mathrm{V}$. Dividing raw outputs by their nominal gains without removing the offsets would not give consistent current estimates.

A practical test can drive a switchable transimpedance amplifier from a stable current source, record multiple overlap levels, and alternate the setting order to reveal instrumental drift. Avoid amplifier clipping, allow settling time, and retain the bandwidth and temperature conditions. The manufacturer tables show why range-dependent noise and gain drift must be included rather than assumed equal.^[ref]

At each level, compare corrected estimates $I_H$ and $I_L$ through their difference $d=I_H-I_L$. The measurement uncertainty of that difference depends on covariance:A common reference can move both estimates together and disappear from this difference. Thus good overlap agreement checks a relative relationship; it cannot by itself exclude a shared calibration error. The general uncertainty-combination law explicitly retains correlated contributions.^[ref]

Reference material homogeneity concerns whether variation of specified properties within or between usable portions is small enough for the intended measurement purpose. Homogeneity is property-specific and depends on the portion taken: uniform colour need not establish uniform refractive index, composition or another measured property. Sufficient homogeneity is part of the definition of a reference material.^[ref]

An original study takes several bottles from a batch and measures multiple portions from each, with the measurement order varied to avoid confusing bottle order with instrumental drift. A simple measurement model iswhere $b_i$ describes bottle-to-bottle variation and $e_{ij}$ includes variation among sampled portions and measurement noise. For independent zero-mean contributions, the variance of a single observation isRepeatedly reading one portion mainly probes repeat measurement scatter. It does not replace sampling different portions or bottles. Separating material variation from measurement repeatability requires a sampling and replication design appropriate to both.

Suppose an original fitted model gives a between-bottle standard deviation $3\times 10^{-6}$ in refractive index and within-bottle measurement scatter $4\times 10^{-6}$. Their combined standard deviation for one observation is $5\times 10^{-6}$ under the independence assumption. Averaging more readings from the same bottle can reduce the second contribution, but does not average away that bottle's common $b_i$ contribution. These are illustrative model values, not certificate data.

NIST's archived SRM1922 characterization tested multiple oil portions using minimum-deviation measurements and an Abbé-type refractometer.^[ref] Sampling different portions answers the homogeneity question directly; reference material stability separately addresses changes over time. A report should retain the sampled portion size and conditions so that a user can judge whether the evidence applies to their own use.

A sensitivity mass is a small calibrated mass added to a loaded weighing instrument to determine its response to a known mass increment. NIST's double-substitution procedure calls this a sensitivity weight and uses it to establish sensitivity under the conditions of the comparison.^[ref] It supplies a local input step rather than a new determination of the entire load from zero.

Let an original comparator give indications $r_0$ before and $r_1$ after adding mass $m_{\mathrm{sw}}$. For a stable linear response over that increment,where $\Delta r$ is another indication difference near that working load. A constant additive zero cancels from each difference; instrumental drift or load-dependent response need not cancel.

For original indications $r_0=860$, $r_1=885$ and assigned sensitivity mass $10.020\mathrm{mg}$, the local response is about $2.495$ indication increments per milligram. A difference $\Delta r=7.5$ increments gives $\Delta m=3.006\mathrm{mg}$. Using only a nominal $10\mathrm{mg}$ label instead would give $3.000\mathrm{mg}$. The assigned mass and its measurement uncertainty, rather than the nominal label alone, enter the comparison.

The added step must be large enough to resolve reliably but remain within the response region relevant to the unknown/reference comparison. NIST specifies selection criteria and regularly spaced observations; it also requires the standards to have valid calibration information.^[ref] Keep handling, settling and environmental conditions consistent. A sensitivity mass does not by itself cancel buoyancy, temperature effects or uncertainty in the main measurement standard. Those contributions remain in the full double substitution weighing model.

A substitution measurement method compares an unknown with a reference by placing them successively in the same relevant position of a measuring system. The comparison seeks a match or measures a difference while retaining as much of the system response as possible. The VIM lists substitution among measurement methods; the detailed sequence belongs to the measurement procedure.^[ref]

For a physical application, NIST's double substitution weighing procedure compares a standard and an unknown of the same nominal mass twice, using a weighing instrument as a comparator. It also determines response with a known sensitivity mass and includes measurement-assurance checks.^[ref] The measurement standard supplies the reference value; the comparator need not establish the whole mass independently from its own zero.

An original simplified drift model shows why sequence matters. Let a comparator indication at equally spaced time index $t$ bewith fixed response coefficient $a$, additive offset $b$ and linear instrumental drift $d$. Observe standard S at $t=0$, unknown X at $t=1,2$, and S again at $t=3$. The mean X indication and mean S indication have the same mean time $3/2$, givingBoth the constant offset and the stipulated linear drift cancel. For $m_S=m_0=100\mathrm{g}$, $a=500{\mathrm{g}}^{-1}$, $b=10$ and $d=2$, an unknown $m_X=100.004\mathrm{g}$ gives indications $10,14,16,16$. Their paired means differ by two, recovering $0.004\mathrm{g}$.

This teaching sequence isolates one cancellation mechanism; it is not the full NIST procedure. Nonlinear drift, unequal timing, changing response or different positioning can defeat it. NIST's procedure separately addresses buoyancy and uncertainty contributions.^[ref] A close match does not remove the reference's measurement uncertainty or establish that the substitution left every relevant condition unchanged.

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]

Zero-offset drift of an integrated flow measurement is a false accumulated volume caused by an additive error in the measured volumetric flow rate. If $\hat{Q}(t)=Q(t)+b(t)$, thenA constant bias gives error $bT$; a linearly changing bias $b(t)=b_0+kt$ gives $b_{0}T+\frac{1}{2}k{T}^2$. A small flow offset can therefore produce a conspicuous trend after integration.

For a constructed offset 0.004 L/s, the error after 60 s is 0.24 L. If the true flow alternates symmetrically between forward and reverse directions, its net integral can be zero while that false trend remains. The drift is an error in the integrated signal, not evidence that the apparatus necessarily accumulated that amount of gas.

A pneumotachograph signal chain can be modeled as $e=e_0+KQ$. An error in the estimated zero $e_0$ becomes an additive flow error, while an error in $K$ changes the scale. The manufacturer's primary manual treats no-flow zeroing and known-volume calibration separately, explicitly connecting zero-flow error with steady integrated-volume drift.^[ref]

A bidirectional bench example separates those effects. Let the inferred integral be $J=aV+bT$, where $a$ is a gain factor. Equal-time strokes transporting true volumes $+V_s$ and $-V_s$ givesoWith $V_s=1$ L, $T=2$ s, $J_{+}=1.05$ L and $J_{-}=-0.95$ L, the inferred gain is 1 and the offset is 0.025 L/s. This is a constructed linear calibration model, not a measured specification of a particular instrument.

Instrumental drift, gain error, leakage and a real net flow can create different records. Subtracting an arbitrary fitted trend may hide genuine accumulation as well as measurement bias. A physical no-flow reference and a known displaced volume test the relevant assumptions more directly than forcing every integrated trace to return to its initial value.