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Independent observations with known errors add a positive-semidefinite Fisher information matrix $G$ to the existing information $F$. Writing $F+G=F^{1/2}(I+F^{-1/2}G{F}^{-1/2})F^{1/2}$ shows its inverse covariance cannot increase in any parameter direction. Improvement can be zero along an uninformative direction. This statement does not assume that data-dependent residual variance estimates or model systematic errors must decrease.

If a controlled reversal changes the desired signal's sign while leaving an additive offset unchanged, readings $y_{+}=S+b$ and $y_{-}=-S+b$ separate them by half-difference and half-sum. Offset drift, gain changes, hysteresis or a background that also reverses violate the simple cancellation. Interleaving polarities and independent blank measurements test those assumptions; signal reversal does not cancel an arbitrary systematic error.

An instrument for comparing two lengths through their difference. Suppose a test rod and reference both have length $L_0$ at temperature $T_0$, and have constant linear thermal expansion coefficients ${\alpha }_s$ and ${\alpha }_r$ over a small interval. Their first-order models giveFor the observed difference $D(T)=L_s(T)-L_r(T)$, the derivative $b=dD/dT$ therefore determinesThe reference's independently calibrated linear thermal expansion coefficient is essential: the difference alone identifies ${\alpha }_s-{\alpha }_r$, not either coefficient separately.

For an original numerical example, stipulate $L_0=1.00\mathrm{m}$, ${\alpha }_s=1.0\times 10^{-6}{\mathrm{K}}^{-1}$ and ${\alpha }_r=9.0\times 10^{-6}{\mathrm{K}}^{-1}$. Heating both by $10\mathrm{K}$ gives sample and reference increases of $10\mu \mathrm{m}$ and $90\mu \mathrm{m}$. Thus the sample expands even though its measured difference decreases by $80\mu \mathrm{m}$. These are chosen model parameters, not measured values for a historical Invar specimen or reference.

If ${\alpha }_r$ and $b$ have independent standard uncertainties $0.2\times 10^{-6}{\mathrm{K}}^{-1}$ and $0.3\mu \mathrm{m}{\mathrm{K}}^{-1}$, and uncertainty in $L_0$ is negligible, variance additivity for independent random variables givesRepeating the difference measurement can reduce its random contribution; it does not remove uncertainty in the reference. A temperature gradient or unequal rod temperatures also breaks the common-temperature model and can introduce a systematic error.

Even a shared field does not cancel if it is sampled at different times. Write $f_A(t)=g_{A}C(t)$ and $f_B(t)=g_{B}C(t)$, where $C(t)={\mu }_{B}B(t)/h$. As an original proposed diagnostic, bracket the $B$-state reading at $t=0$ with $A$-state readings at $t=\pm \tau$:If $C(t)=C_0(1+bt)$, the denominator is exactly $g_A{C}_0$, so $R_{\mathrm{bracket}}=g_B/g_A$. A single earlier $A$ reading instead gives $R_{\mathrm{unpaired}}=(g_B/g_A)/(1-b\tau )$. For the preceding numerical model and $b\tau =0.001$, this unpaired comparison falsely gives approximately $a_e=0.00186649$; interpolation recovers $0.0012$. The sequence illustrates cancellation and is not asserted to be Kusch and Foley's exact historical timing protocol.

For a curved field $C(t)=C_0(1+bt+c{t}^2)$, the same calculation givesThe linear drift cancels but the curvature remains. Different fields sampled by the two beams, additive frequency offsets and incorrect atomic corrections also survive. For a small ratio error $\delta R$, direct differentiation of the inverse relation givesRepeated readings can reduce random measurement uncertainty; agreement among genuinely different state comparisons is what tests shared systematic errors in their interpretation.

In 1915, Albert Einstein and Wander Johannes de Haas suspended a soft-iron cylinder by glass fibre between coaxial coils. Alternating magnetization drove torsional resonance; a mirror reflected lamplight onto a scale. They compensated Earth's magnetic field, tilted the coils, and rotated the suspension attachment to test unwanted magnetic dipole torques. Added known moment of inertia calibrated the mechanical response.^[ref]

Their paper acknowledges a failed phase determination and claims only ten-percent precision for the revised measurement. Its agreement with the orbital gyromagnetic ratio prediction was not a direct discovery of spin.^[ref] Comparing that orbital gyromagnetic ratio prediction with the electron spin g-factor explains why a successful qualitative demonstration of angular momentum transfer can coexist with an inadequate quantitative interpretation. The stated precision is the authors' estimate; it does not establish that all systematic errors were bounded by that amount.

The general control problem follows directly from the model. Unwanted magnetic dipole torque can drive the same mechanical resonance as the desired reaction torque. In the ideal geometry $m_z$ and $B_z$ alone produce no axial cross product, but transverse magnetic dipole moments, stray transverse magnetic fields and imperfect alignment can. Nulling stray magnetic fields and deliberately varying alignment test this contribution. Merely observing a large resonant twist cannot identify its cause.

For a nearly instantaneous magnetic response, a sinusoidal magnetization makes $m_z$ follow the drive while $-{\dot{m}}_z/\gamma$ is in phase quadrature. An ordinary magnetic contribution proportional to $m_z$ is then distinguishable by phase, after removing the mechanical frequency response. Hysteresis or delayed magnetic response can spoil this simple separation. The measured magnetization, rather than the coil electric current alone, must set the reference. Mechanical vibration, magnetostriction coupled through imperfect geometry, and changing suspension properties require separate checks; none is eliminated simply by reversing the coil electric current.

Improve a circuit's signal-to-noise ratio by reducing unwanted coupling and physical noise sources while retaining the signal bandwidth. Important methods include lowering appropriate resistances or temperature to reduce thermal electrical noise, controlling leakage and bias currents relevant to shot noise, minimizing pickup with shielding and small loop area, and selecting an operational amplifier for the source electrical impedance. Filtering limits integrated noise only if it preserves the information-bearing spectrum; a lock-in amplifier can move detection above a flicker noise region and select a narrow band. Uncorrelated noise variances add, whereas coherent pickup requires its phases and correlations to be considered. Repeated averaging reduces independent random error, but does not remove an arbitrary systematic error or slow drift.

A laboratory determination of the gravitational constant must connect a sensor's reading to a gravitational signal from independently characterized masses. For an ideal two-point mass force measurement, invert the model:These are signed small changes obtained by taking the logarithmic derivative, not an instruction to add standard uncertainties with signs. For independent small input uncertainties, linear propagation of uncertainty instead givesCorrelated inputs require their covariances too. In particular, common geometric calibration errors must not be treated as independent simply because several distances were recorded.

For a concrete model check, hold the actual force and masses fixed but overestimate a true $r=0.100\mathrm{m}$ by $1\mu \mathrm{m}$. The exact inferred ratio isThe resulting bias is $20.0001$ parts per million, already comparable with the relative standard uncertainty of the recommended gravitational constant. A sensor can resolve very small changes while its inferred answer remains limited by source geometry. The example concerns one idealized separation; it is not an uncertainty estimate for any actual experiment.

Making the source mass larger or bringing it closer strengthens the signal, but also makes detailed mass density and geometry important. Nearby apparatus supplies its own Newtonian gravitational field; electromagnetic shielding does not remove that attraction. Sensor offsets, changing temperature, vibration and source motion can imitate or alter a signal. The Cavendish experiment illustrates calibration of a torsion constant; a different sensor requires its own calibrated response and a full source model.

Repeating readings can reduce independent measurement noise, but a fixed scale error survives averaging. A useful experiment deliberately changes the source configuration, checks whether the signal follows the predicted change, and characterizes what moves with the source. A systematic error that changes with that configuration also survives background subtraction. Comparing methods with different sensors and different dominant errors is therefore valuable.

Institutional guidance on measurement uncertainty. Sections 2–5 distinguish uncertainty evaluation from the physical classification of effects and combine uncertainty contributions; Appendix D.1 discusses systematic error and correction.

Charles-Édouard Guillaume sought affordable, stable length standards. His 1920 lecture describes two micrometer microscopes on stone supports sighting marks on a rod immersed in water. Their separation was not fixed: one method alternated the sample with a reference in a second trough at nearly constant temperature. His relative method compared sample and platinum–iridium reference in the same bath, adding the reference's known expansion to the measured difference. Composition and treatment affected the nickel–iron alloys; low expansion held over a limited temperature range.^[ref]

The differential length comparator calculation explains why a calibrated reference is needed even when its expansion is larger than the sample's. Keeping both rods in the same bath targets one source of systematic error; allowing for changes in the instrument baseline targets another. Discovering a low-expansion alloy and demonstrating its dimensional stability are connected experimental tasks.

Measurement error is the difference between a recorded variable and the quantity it is intended to measure. It can attenuate associations, create misclassification, and bias nonlinear estimators even when the error has mean zero.

Measurement resolution is the smallest change in an instrument indication that its reading method can distinguish. It contributes to measurement uncertainty, but does not by itself establish accuracy or remove systematic error. The manometer height-to-pressure calibration shows how a height scale affects the inferred pressure.

A record of uncertainty contributions in a specified measurement model, their evaluation methods, sensitivity coefficients and relevant correlations. For $y=g(x_1,\dots ,x_n)$, first-order propagation of uncertainty gives $u_y^2={\sum }_{ij}({\partial }_{i}g)({\partial }_{j}g)\mathrm{Cov}(x_i,x_j)$. A correction for a known systematic error can itself have uncertainty; applying the correction does not remove that contribution.^[ref]

Albert Abraham Michelson's 1907 lecture, Recent Advances in Spectroscopy, pp. 171–172, connects counting interference fringes with a reproducible length standard. Broad or multiple spectral lines obscured fringes at large path differences; red cadmium emission supplied a suitable source. His 1907 Nobel Prize in Physics recognized optical precision instruments and their spectroscopy and metrology applications.^[ref]

His authored account of the metre comparison at Sèvres describes a heated, evacuated cadmium discharge tube and a spectrometer selecting one spectral line. A movable reference mirror was compared with pairs of silvered parallel surfaces defining intermediate standards. Counted interference fringes established a short interval; repeated comparisons extended it to a decimetre and then a metre. White-light fringes located surface coincidence, while monochromatic fringes supplied counts and fractional orders. A thermally shielding box, firm pier, calibrated translation screw and tested thermometers controlled drafts, vibration, travel and temperature. Uniform temperature mattered in comparisons; the final standard required an absolute temperature reading. Optical microscopes made the final comparison to the marked metre bar, limiting the whole result.^[ref]

This is a measurement chain, rather than one enormous unchecked fringe count: short intervals establish whole orders; fractional-order readings refine each longer comparison. The optical coherence envelope helps locate white-light coincidence, whereas the carrier phase difference measures small departures from it. Repetition checks counting mistakes, but agreement alone cannot exclude shared systematic errors. The general lesson of metrology is to preserve the link from the optical ruler through each transfer and control to the final length.

Background subtraction must account for uncertainty in the background measurement itself. In an original idealized experiment, an on-source exposure contains source and background, while an independent off-source exposure contains background alone. Assume stationary rates and identical background acceptance per unit live time. With $\alpha =t_{\mathrm{on}}/t_{\mathrm{off}}$, let independent counts obey Poisson distributions:Then linearity of expectation and addition of variances for independent random variables giveThe result concerns accepted source counts $S$, after all efficiencies. For large counts the normal approximation gives the descriptive signal-to-noise ratio in photon countingThis plug-in ratio is not an exact null-hypothesis significance. Small counts require the discrete Poisson model; searching many sky directions also changes a detection claim's p-value.

Choose $t_{\mathrm{on}}=100\mathrm{s}$ and $t_{\mathrm{off}}=200\mathrm{s}$, with observed counts $N_{\mathrm{on}}=1000$ and $N_{\mathrm{off}}=800$. Thus $\alpha =1/2$, $\hat{S}=600$, and the estimated standard deviation is $\sqrt{1200}=34.64$ counts. The inferred $F=\hat{S}/(A_{\mathrm{eff}}{t}_{\mathrm{on}})=0.300\pm 0.0173\mathrm{photons}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$ uses the preceding $A_{\mathrm{eff}}=20\mathrm{c}{\mathrm{m}}^2$ and includes counting uncertainty only. The descriptive ratio is $17.32$. Treating the scaled off-source background as exact would omit 200 counts squared from the estimated variance.

For a selection-design comparison, suppose before a veto the expected source contribution is $S_0=750$, the on-source background is $B_0=2000$, and $\alpha =1/2$. Let source acceptance be ${ϵ}_s=0.8$ and background acceptance be ${ϵ}_b=0.2$. Then $S=600$, $B=400$, and $S/\sqrt{S+(1+\alpha )B}=17.32$, whereas omitting the veto gives $750/\sqrt{750+3000}=12.25$. Rejection helps because the background is reduced enough to compensate for lost source events. A different veto can make matters worse; source efficiency and background rejection are separate probabilities. This comparison is an independent teaching example, not the efficiency calibration used above.

If both exposure times grow with fixed $\alpha$ and stationary rates, $S=s{t}_{\mathrm{on}}$ and $B=b{t}_{\mathrm{on}}$ imply $Z\simeq s\sqrt{t_{\mathrm{on}}}/\sqrt{s+(1+\alpha )b}$. Four times the live time doubles this ideal ratio. Misestimated background rates, uncertain X-ray effective area, and source variability can defeat that scaling; longer exposure does not remove systematic error.

Volumetric analysis determines an amount of substance from the measured volume of a reagent of known strength that reacts with it in a known proportion. A measured aliquot is placed in a suitable vessel, the standard solution is delivered from a burette, and the volume required to reach the chosen titration endpoint is read. A pipette measures the aliquot; a volumetric flask prepares or dilutes the standard solution.

For a reaction $aA+bB⟶\text{products}$, if the concentration of the titrant $B$ is $c_B$ and the corrected volume delivered is $V_B$, thenWith volume in litres and concentration in moles per litre, the answer is an amount in moles. The corresponding mass is $m_A=M_A{n}_A$, using the appropriate molar mass. For an unknown solution of aliquot volume $V_A$,These are modern symbols for the same proportional reasoning that a nineteenth-century analyst could express with equivalent weights and a specified standard solution.

A suitable reaction is sufficiently rapid, complete and selective, with a recognisable endpoint. In an acid-base titration, an indicator can change colour near neutralization; other reactions use precipitation, redox or a colour intrinsic to a reagent. The observed titration endpoint must represent the required equivalence point closely enough. The reagent's nominal preparation alone is insufficient if its concentration changes in storage; standardisation of a solution, a reagent blank where needed, and correct reading of the meniscus control systematic error. A titration measures the intended analyte only insofar as interfering substances have been excluded or accounted for.

Assuming independent Gaussian measurement errors with the quoted standard deviations, use the inverse-variance weighted mean:Here ${\sum }_i{w}_i=0.654514$ and ${\sum }_i{w}_i{x}_i=65.805556$, givingThe consistency statistic is ${\chi }^2=4.45$ for four degrees of freedom, so there is no evident excess scatter requiring error rescaling. A shared systematic error would require a covariance matrix and would not be reduced by this averaging rule.

Measure an unwanted effect separately or design a comparison in which it cancels. Calibration against reference standards at several settings tests zero offset, gain and nonlinearity. Blanks reveal background. Reversal is useful when the wanted response is odd and the nuisance is even: $D_{\pm }=\pm S+b$ gives $S=(D_{+}-D_{-})/2$ and $b=(D_{+}+D_{-})/2$. For example reversing current in a resistance measurement cancels a stable thermoelectric voltage, but does not cancel the current-scale error or even-in-current Joule heating.

Interleave reversals to detect drift; interchange sensors or leads to expose asymmetry. Null or bridge methods reduce dependence on detector gain. Alignment checks, shielding and temperature control target specified physical couplings. Independent measurement methods provide a stronger test than repetition of the same setup. Repeated readings reduce independent random errors, not a shared bias. Correct significant identified effects and propagate the uncertainty of the correction and standards; complete removal of an unknown systematic error cannot generally be certified.^[ref]

Identify the physical coupling that causes a systematic error and change the apparatus so that it can be measured or cancelled. measurement calibration against known reference standards at several points reveals zero and gain errors; blanks and background runs measure additive contributions. Carry uncertainties of the standards and fitted corrections into the final result.

Reversal provides a particularly transparent test. If $D_{+}=S+b$ and $D_{-}=-S+b$, the combination $(D_{+}-D_{-})/2$ isolates $S$ while $(D_{+}+D_{-})/2$ measures the measurement zero offset. Interleave and repeat the reversals so drift can be detected. The cancellation assumes a stable measurement zero offset and a genuinely odd signal; a common gain error remains. null measurements and bridge comparisons can reduce dependence on detector gain. Swapping sensors or leads exposes asymmetry; checking alignment, shielding, grounding and temperature control tests specific unwanted couplings. Varying signal amplitude or measurement frequency can reveal heating, nonlinear response or pickup.

Use independent measurement principles where possible, and test the predicted response to each deliberate change. Repeated readings alone do not remove a fixed systematic error. Correct identified significant effects and retain uncertainty for imperfectly known corrections; complete elimination cannot generally be demonstrated. Type A and Type B classify how uncertainty is evaluated, rather than simply meaning random measurement errors and systematic errors.^[ref]

A systematic error shifts results in a reproducible or correlated way relative to the measurand. Examples include zero offsets, gain miscalibration, nonlinearity, drift, geometrical misalignment, loading, background contamination and an inadequate response model. Repeated measurements under unchanged conditions can reduce independent noise without exposing such a shift; a small standard error is not proof of measurement accuracy.

Use traceable standards and independent measurement methods, test zero and span, reverse the physical signal where possible, and vary the conditions that could cause the bias. For common-offset cancellation by signal reversal, take measurements $y_{+}=S+b$ and $y_{-}=-S+b$. The half-difference estimates $S$ while the half-sum diagnoses an unchanged offset $b$. Drift between the two readings would violate that cancellation, so alternate or interleave measurements. Randomizing acquisition order separates slow drift from controlled parameter changes; blanks and calibration samples test backgrounds and instrument gain. Blinding helps reduce analysis choices that follow an expected answer.

Corrections have uncertainties and may be correlated across data. Include residual offsets and gains as nuisance parameters or propagate their calibration covariance, rather than treating a common calibration error as fresh independent noise for each datum. Document assumptions and check stability with controlled perturbations. These methods suppress or quantify specified systematic effects; they cannot guarantee elimination of every unknown bias.