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Gaussian pulse-width deconvolution uncertainty is the uncertainty in an inferred pulse duration after subtracting a characterized Gaussian instrument contribution. For the Gaussian instrument broadening in third-order pulse correlation model, let $M$ be measured correlation full width at half maximum, $R$ the additional response width and $P$ the inferred pulse width:These widths must use the same convention. The relation already includes the ideal nonlinear gate; $R$ represents an additional response.

First-order propagation uses the sensitivity coefficientsThus the standard uncertainty obeysThe negative covariance term follows from the subtraction, not a rule that uncertainties should simply be subtracted. The GUM's general law retains covariance and requires considering higher-order terms when nonlinearity is significant.^[ref]

For an original example, $M=120\mathrm{fs}$ and $R=60\mathrm{fs}$ give $P\approx 84.85\mathrm{fs}$. With independent $u(M)=u(R)=2\mathrm{fs}$, the formula gives $u(P)\approx 2.11\mathrm{fs}$. If instead the two width estimates have correlation coefficient 0.8 with those same uncertainties, the covariance is $3.2\mathrm{f}{\mathrm{s}}^2$ and $u(P)\approx 1.26\mathrm{fs}$. The assumed relationship between the estimates changes the result.

Near $M=R$, both derivatives grow because $P$ approaches zero. With $M=61\mathrm{fs}$, $R=60\mathrm{fs}$ and independent 2 fs uncertainties, the central inferred value is only about $8.98\mathrm{fs}$ while the first-order uncertainty is about $12.7\mathrm{fs}$. This is an unresolved Gaussian pulse-width estimate under these measurement conditions, rather than a reliably resolved 9 fs pulse.

The quantity being subtracted is $Q=M^2-R^2$, with approximate variance $4[M^2{u}^2(M)+R^2{u}^2(R)-2MR\mathrm{cov}(M,R)]$. If plausible width values include $Q\le 0$, linear symmetric error bars on $P$ can be misleading. Report the response assumptions and consider an appropriate nonlinear inference or bound; simply discarding negative radicands would silently change the statistical model. Even a precise inversion remains conditional on the Gaussian response model being physically appropriate.

Gravimetric dilution determines a dilution from measured masses. A measured amount of a stock solution is combined with material containing none of the quantity being diluted. For a closed transfer and a homogeneous mixture, the conserved amount divided by the final mass determines the new mass-based concentration.

Let $m_s$ be the stock's net mass, $m_d$ the added net mass, and $M=m_s+m_d$. Define the stock fraction $f$ and the gravimetric dilution factor $F$ byIf the stock has solute mass fraction $w_s$, thenFor stipulated $m_s=2.000\mathrm{g}$, $m_d=18.000\mathrm{g}$ and $w_s=0.2000$, the final mass is $20.000\mathrm{g}$, $F=10$, and $\boxed{w_f=0.02000}$: 2% by mass. The solute mass is $0.4000\mathrm{g}$ before and after dilution. The net masses exclude their containers. This mass basis does not require an assumption that liquid volumes add.

The uncertainty of a gravimetric dilution depends on which readings share errors. For independently evaluated inputs $w_s,m_s,m_d$, first-order propagation of uncertainty givesHere $u$ denotes standard uncertainty. The sensitivity factors follow from the partial derivatives $\partial f/\partial m_s=m_d/M^2$ and $\partial f/\partial m_d=-m_s/M^2$. With a chosen independent $u(m_s)=u(m_d)=0.001\mathrm{g}$ at the example's nominal masses, the relative mass-ratio contribution is approximately 0.0453%. Adding an independent 0.5% relative stock-value contribution in quadrature gives approximately 0.502%. These are illustrative inputs, rather than a balance specification.^[ref]

If input errors have covariance, their cross terms must be retained. A useful exact check isA common multiplicative mass-scale factor therefore cancels from $f$. Counting it as two independent contributions would lose that correlation. Container subtraction, transfer losses and changes in composition require their own assessment; this cancellation establishes only the common-scale property.

For massic activity, the same conservation argument applies after expressing all activities at a common reference time. A 2000 NIST study compared massic activity ratios inferred from gravimetric dilution with independent liquid scintillation counting. Their agreement within reported uncertainties provided an experimental check on transferring standards through mass ratios.^[ref]

The gravitational constant $G$ sets the strength of Newton's law of universal gravitation. Two point masses $M$ and $m$ a center-to-center distance $r$ apart attract with force magnitudeFor fixed masses and separation, doubling $G$ would double the force. For bodies with spherical symmetry, the same exterior formula follows from the spherical shell theorem; for arbitrary extended bodies, add the attractions of their mass density elements instead. The gravitational constant connects a laboratory attraction to the Newtonian gravitational field controlling a Kepler orbit.

The gravitational constant $G$ and local gravitational acceleration $g$ are different quantities. Applying Newton's second law to a test particle gives $g=F/m=GM/r^2$ for one spherical source. The source's mass and the observer's position change $g$; $G$ remains the same parameter of the law. A local falling-body experiment therefore measures a field, not $G$ alone. On rotating Earth, an instrument fixed to the surface also responds to centrifugal acceleration, while surrounding mass density and altitude change the gravitational contribution.

Dimensional analysis explains the units. Denote the base physical dimensions of length, mass and time by $\mathsf{L},\mathsf{M},\mathsf{T}$. Since force has dimension $\mathsf{ML}{\mathsf{T}}^{-2}$,By contrast, $[g]=\mathsf{L}{\mathsf{T}}^{-2}$ and the standard gravitational parameter $\mu =GM$ has dimension ${\mathsf{L}}^3{\mathsf{T}}^{-2}$. A small numerical value of a dimensional constant depends on the units chosen: it is the resulting force for specified physical masses and separation that conveys the laboratory scale.

NIST's current complete listing givesfor the 2022 CODATA adjustment, the latest adjustment listed when checked on 14 October 2026. Parentheses indicate a standard uncertainty $u(G)=0.00015\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, approximately $22$ parts per million relative to $G$. The adjustment year is 2022; NIST's May 2024 data update is a different date.^{[ref] [ref]}

For an original scale calculation, stipulate two $1\mathrm{kg}$ point masses separated by $1\mathrm{m}$ and insert this recommended $G$: $F=6.67430\times 10^{-11}\mathrm{N}$. The corresponding acceleration of either mass is $6.67430\times 10^{-11}\mathrm{m}{\mathrm{s}}^{-2}$ at that instant. Reducing the separation to $0.10\mathrm{m}$ multiplies both by $100$. These are ideal model predictions, not observations; their tiny scale explains why stray forces matter.

Being a constant of a physical law does not make the numerical value exact. The SI fixes the numerical values of seven defining constants, including the speed of light and Planck constant; $G$ is determined experimentally. Exact unit definitions still require physical realizations with measurement uncertainty: calibrating a length or a mass does not become error-free.^[ref]

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.

An orbit around an isolated source constrains its standard gravitational parameter $\mu =GM$ rather than independently measuring the source's mass and the gravitational constant. In the test particle approximation, a circular orbit of radius $a$ and orbital period $T$ satisfiesThe left equation equates the required centripetal acceleration to the gravitational acceleration. More generally, a Kepler orbit depends on the same $\mu$. In the exact relative two-body problem, replace $GM$ by $G(M+m)$.

For any positive dimensionless number $q$, replacing $G$ by $qG$ and $M$ by $M/q$ leaves $GM$ and every predicted test particle trajectory unchanged. This is the gravitational mass degeneracy: more observations of the same orbit can refine $GM$ without separating its factors. A laboratory source whose mass is calibrated independently supplies the missing information.

Consequently an astronomical mass expressed in kilograms as $M=\mu /G$ inherits measurement uncertainty from $G$, even if $\mu$ is known much more precisely. To first order,When the relative standard uncertainty of $\mu$ is negligible, that of $M$ is approximately $u(G)/G$. Conversely, the ratio of two source masses inferred using the same $G$ is $M_1/M_2={\mu }_1/{\mu }_2$, so this common factor cancels. That distinction explains how excellent orbit determination can coexist with less precise absolute astronomical masses.

Ian Mann is a metrology and laboratory-quality specialist credited with Brian Brookman as co-editor of the 2021 third edition of Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann. The editorial page identifies him with SAS in Switzerland. This identifies the proficiency-testing author without attributing the careers of other people named Ian Mann to him.^[ref]

The guide connects proficiency testing with practical decisions about laboratory measurements. Its selection criteria ask whether a scheme's material, measurand, concentration range and assessment approach fit the laboratory's actual work. A comparison using an unsuitable material can fail to test the procedure for which competence is being assessed.^[ref]

An original example illustrates why interpretation needs the assessment criterion. For a participant result $x=102$, assigned value $x_{\mathrm{pt}}=100$ and standard deviation for proficiency assessment ${\sigma }_{\mathrm{pt}}=1$ in compatible units, the proficiency-test z-score isA scheme using ${\sigma }_{\mathrm{pt}}=2$ would give $z=1$ for the same result. The result and its difference from the reference did not improve; the assessment scale changed. The chosen standard deviation for proficiency assessment must therefore be read with the score.

A participant's reported standard uncertainty answers another question: how uncertain is that participant's measurement result? It is not automatically the scheme's performance criterion. The guide separates evaluation scores, assessment spread and laboratory investigation of unexpected results.^[ref] Mann's documented editorial contribution is to this shared measurement-quality guidance; no separate invention or personal experimental result is implied.

Identifying flow-sensor gain and offset with timed transfers uses a known gas volume delivered over different durations to distinguish multiplicative response from accumulated zero error. In the model $Q_m=gQ+b$, a transfer of volume $V_0$ lasting $T_i$ producesThe observations form a linear regression with design matrix rows $(V_0,T_i)$. If $V_0\ne 0$, gain and offset are identifiable only when the durations are not all equal. Repeating one duration measures the same combination of $g$ and $b$ more precisely, without separating them.

For two noise-free transfers,Stipulated 3 L transfers lasting 1 and 6 s, measured as 3.020 and 3.120 L, give $\hat{b}=0.020\mathrm{L}{\mathrm{s}}^{-1}$ and $\hat{g}=1.000$. The different durations expose an offset that one transfer could misinterpret as a gain error.

Timing spread and gain-offset estimator covariance quantifies the design. Suppose $n$ observations have independent equal volume-error variance $s^2$, and define $S_{TT}={\sum }_i(T_i-\overline{T}{)}^2$. Ordinary least squares givesUnder the stipulated model, wider timing separation improves offset precision. For the two durations above and $s=0.010$ L, the offset's standard uncertainty is about $0.00283\mathrm{L}{\mathrm{s}}^{-1}$ and the gain's about 0.00406.

The reference syringe volume was treated as known in those formulas. If it is uncertain, repeated sensor tests share that reference uncertainty. Indeed the observations identify only the product $g{V}_0$ when both factors are unknown. Additional repetitions cannot establish an absolute gain without an independently characterized volume. The GUM treats fit uncertainties and common-reference correlations as distinct contributions.^[ref]

The primary spirometry statement's multiple-flow syringe checks test the practical response over different delivery rates.^[ref] Actual leaks, flow-dependent nonlinearities and uncertain timing require a richer model than a constant gain and offset.

Illumination-initiation interaction in dye-loss controls tests whether changing illumination has the same response in the absence and presence of chemical initiation. A crossed factorial design separates the four conditions. For mean fractional losses $L_{ij}$, with $i$ indicating initiation and $j$ illumination, the contrastmeasures statistical interaction on the fractional-loss scale. In the STY-BODIPY assay context, the optical observable is carrier loss during co-autoxidation, not a direct label of its chemical cause.^[ref]

The crucial qualification is response-scale dependence of dye-loss interactions. Consider an original first-order survival model with fully additive rate constants,ThenFor positive $\alpha$, $\beta$ and $t$, this is negative even though the rates have no interaction term. With stipulated $k_0=0$, $\alpha =\beta =0.1$ in reciprocal time units and $t=1$, the losses are $0$, $0.09516$, $0.09516$ and $0.18127$, giving $I_L\approx -0.00906$. Saturation of fractional loss produces the contrast without a new chemical coupling.

On the transformed scale $Y_{ij}=-\log (1-L_{ij})/t$, the same noiseless model gives $Y_{ij}=k_{ij}$ and zero interaction contrast. This transformation is appropriate only when the measured signal genuinely follows the stated survival model; product absorption or changing backgrounds can invalidate it. A transformed zero is therefore a conditional model check, not proof of a reaction mechanism.

For independent replicate measurements, the variance of the estimated loss contrast is ${\sum }_{ij}{\sigma }_{ij}^2/n_{ij}$. Four replicates per condition with common standard deviation 0.01 give a contrast standard uncertainty of 0.01. The modeled contrast above is smaller than that scale. Shared references instead introduce covariances that must be retained.^[ref]

Match temperature, oxygen access, timing and readout exposure across conditions. Specify the response scale before interpreting a nonzero contrast as evidence about photobleaching or initiated oxidation.

An interlaboratory consensus value is a value derived from results supplied by a specified group of laboratories to represent their common location for a stated measurand. It can serve as an assigned value in proficiency testing, but consensus and independent reference assignment are different ways of establishing that benchmark. The procedure must specify the participating results, estimator and associated measurement uncertainty.^[ref]

For an original example, three laboratories report $9.90$, $10.00$ and $10.10$ in the same units. Their arithmetic mean is $10.00$ and their sample standard deviation is $0.10$. Under a model of independent, equally distributed, unbiased laboratory errors, the estimated standard uncertainty of this mean is $0.10/\sqrt{3}\approx 0.0577$. This calculation does not include an unrecognized systematic error common to all three laboratories.

Choosing a location estimator is a substantive decision. For stipulated results $100.0$, $100.2$, $100.4$ and $105.0$, the arithmetic mean is $101.4$, whereas the median is $100.3$. Robust statistics can reduce the influence of an isolated extreme result, but cannot decide from these four numbers whether that result is erroneous or represents a different measurement method or material condition. The guide discusses robust approaches while distinguishing consensus assessment from fitness-for-purpose criteria.^[ref]

When laboratories supply valid, mutually uncorrelated standard uncertainties $u_i$ for unbiased estimates of the same quantity, an inverse-variance weighted mean usesThe last equality belongs to that model. Shared references, method-specific effects or underestimated $u_i$ can invalidate its interpretation. A laboratory reporting a small uncertainty should not receive overwhelming influence merely because that number was submitted.

An interlaboratory consensus value therefore summarizes a specified collection of evidence. It is not automatically the true value, a certified reference material value, or an independent test of those same laboratories. Shared bias in an interlaboratory consensus and the influence of participants on their own benchmark must be considered before interpreting a small deviation as evidence of accuracy.

Inverse-variance weighting combines estimates of the same quantity by giving a more precise estimate a larger weight. Suppose $X_i$ are independent, unbiased estimators of $\theta$, with known variances $0<{\sigma }_i^2<\infty$. For normalized weights,The inverse is taken after squaring the standard uncertainty: twice the standard uncertainty gives one quarter as much weight. The corresponding weighted least squares criterion is ${\sum }_i(X_i-\theta {)}^2/{\sigma }_i^2$; differentiating it with respect to $\theta$ gives the same weighted arithmetic mean.^[ref]

The optimality statement is precise. Any linear unbiased estimator ${\sum }_i{a}_i{X}_i$ requires ${\sum }_i{a}_i=1$. By the Cauchy-Schwarz inequality,Equality holds for the displayed weights. Thus they minimize variance among linear unbiased combinations under these assumptions; a normal distribution is not needed for that proof.

For an original example, combine observed values 10 and 14 with standard uncertainties 1 and 2 in the same units. The normalized weights are $0.8$ and $0.2$, giving estimate $10.8$, variance $0.8$ and standard uncertainty approximately $0.894$. Equal weights give 12 and standard uncertainty $\sqrt{1.25}\approx 1.118$.

Correlated errors change the problem. A contribution common to every reading survives every weighting whose coefficients sum to one; precision limits of weighting correlated repeats derives the resulting floor. More generally, use the full covariance matrix and generalized least squares rather than treating correlated readings as independent.^[ref] Uncertainty in estimated inverse-variance weights is another limitation: variances estimated from the same data are not fixed known quantities, and an accidentally tiny estimated variance does not justify arbitrary confidence.

John S. Beers is a NIST dimensional metrology researcher and coauthor, with Ted Doiron, of The Gauge Block Handbook by Ted Doiron and John S. Beers. Its preface explicitly attributes the handbook's single-wavelength interferometry discussion to Beers's work. The official publication record retains the name John S. Beers.^[ref]

A gauge block provides a physical length standard, but its calibrated length depends on a stated convention. The handbook defines length using a wrung auxiliary plane and discusses mechanical comparison as well as interferometry.^[ref] This makes contact geometry and gauge block wringing part of the measurement, rather than treating the artifact as an abstract number printed on a block.

An original comparison example uses a reference and unknown of nominal length $50\mathrm{mm}$. Let the reference's certified deviation from nominal be $+0.120\mu \mathrm{m}$, and let the corrected unknown-minus-reference difference be $-0.035\mu \mathrm{m}$. The inferred unknown deviation is then $+0.085\mu \mathrm{m}$.

For independent reference and comparison standard uncertainties $u_r=0.025\mu \mathrm{m}$ and $u_d=0.015\mu \mathrm{m}$,If five independent repeat comparisons reduce only the second contribution, the result becomesThe common reference contribution does not disappear by repeating the comparison. This gauge-block reference uncertainty floor illustrates why high measurement repeatability alone cannot establish the same numerical measurement uncertainty as a reference-free measurement. Correlated inputs require the corresponding covariance terms.^[ref]

These chosen numbers explain a calibration principle; they are not reported performance values for Beers's apparatus. His documented contribution concerns the measurement methods and their controlled realization, including the link between an optical observation and a defined physical length.

Consider an original illustrative resonance model, with all numerical settings chosen here. On one isolated branch approximate the transition frequency by $\nu (B)={\nu }_0+aB$, with known slope $a=20MHz/mT$ and drive frequency ${\nu }_d=1300\mathrm{MHz}$. Choose ${\nu }_0=1100\mathrm{MHz}$, hence resonance field $B_r=10\mathrm{mT}$. This local linear model omits hyperfine structure and nonlinear level mixing; it is not the historical hydrogen Zeeman splitting calculation.

Use a Lorentzian line shape for a weak-drive current dip,Choose baseline $C=1$, depth $A=0.25$ in arbitrary detector signal units, and full width at half maximum $w_{\nu }=100\mathrm{MHz}$, so $w_B=5\mathrm{mT}$. The half-depth points are $B_r\pm w_B/2$; this checks the factor of four. If an exponential decay with lifetime $\tau$ alone set this width, natural linewidth would give $w_{\nu }=1/(2\pi \tau )$ and $\tau \simeq 1.59\mathrm{ns}$. The chosen width is an explanatory input, not a measurement of a historical lifetime.

Generate 65 equally spaced fields from $2$ to $18\mathrm{mT}$ and add independent Gaussian noise with known standard deviation ${\sigma }_I=0.004$ signal units. A simultaneous nonlinear regression fit of $C,A,B_r,w_B$ gives $B_r\simeq 10.0007\mathrm{mT}$, $w_B\simeq 4.9305\mathrm{mT}$ and ${\nu }_0\simeq 1099.986\mathrm{MHz}$. With residual derivatives $J_{ik}=\partial I(B_i)/\partial {\theta }_k$ and $\theta =(C,A,B_r,w_B)$, the local covariance matrix estimate isIt gives centre standard deviation $0.01427\mathrm{mT}$, or $0.285\mathrm{MHz}$ in the inferred intercept. This uses the stipulated noise variance and a local linear approximation, rather than treating the width as the centre error. Repeated noisy samples can locate a broad line's centre much more closely than its width.

Include a common magnetic calibration offset with standard deviation $u_b=0.020\mathrm{mT}$, a fractional gain error $u_g=0.001$, and independent drive-frequency error $u_d=0.100\mathrm{MHz}$. To first order the inferred variance isHere $u$ denotes a standard uncertainty, not a frequency. The four contributions in $\mathrm{MHz}$ are $0.285$, $0.400$, $0.200$, and $0.100$; adding their squares gives $u({\nu }_0)\simeq 0.540\mathrm{MHz}$. The calibration offset shifts all sampled fields together, so increasing the sample count cannot average it away. Uncertainty in $a$, unresolved components, asymmetric line distortion, and errors in the assumed field dependence would require additional terms or a richer model. The historical inference requires the appropriate Zeeman splitting branches, rather than simply drawing a straight line through one resonance.

A leave-one-out interlaboratory consensus estimates a benchmark for laboratory $i$ from the other participants. For a fixed weighted mean with positive coefficients $w_j$ and at least two results, defineThe omitted result does not enter its own benchmark. This removes the direct participant influence on a consensus assigned value for the stated fixed-weight estimator. It does not establish independence when laboratories share a systematic error. The proficiency-testing guide distinguishes participant consensus from expert or independent reference assignment; the leave-one-out construction here is a mathematical comparison, not a claim that the guide requires this scoring procedure.^[ref]

With equal weights and original results $(9,10,11)$, the omitted-result arithmetic means are $(10.5,10,9.5)$, giving residuals $(-1.5,0,1.5)$. Residuals from the full arithmetic mean are $(-1,0,1)$. In general,The larger magnitudes reflect a changed comparator rather than a changed laboratory result.

The joint dependence of leave-one-out consensus residuals matters when interpreting the collection of scores. Suppose the $X_i$ are independent random variables with a common mean and finite common variance ${\sigma }^2$. A result is independent of the arithmetic mean of the other results, soFor different participants, however, their leave-one-out benchmarks overlap. Linearity of covariance givesThus independent laboratory observations do not give independent residual scores. The zero-sum identity makes their covariance matrix a singular matrix. For $n=4$ and $\sigma =2$ units, each residual has standard deviation $\sqrt{16/3}\approx 2.31$ units, and each distinct pair has correlation coefficient $-1/3$. No normal distribution assumption is needed for these moment identities.

A leave-one-out inverse-variance weighted consensus applies an inverse-variance weighted mean to the remaining participants when their standard uncertainties are known. Under independent unbiased observations with weights $w_j=u_j^{-2}$, put $W={\sum }_j{w}_j$. ThenThese are uncertainties under that observation model; the resulting quantity should not automatically be called a proficiency-test z-score with a fixed assessment scale. Computing $({\sum }_j{w}_j{x}_j-w_i{x}_i)/(W-w_i)$ is algebraically convenient, but direct summation of the remaining terms can avoid cancellation when one participant dominates the total weight.

If a robust estimator has weights depending on the full data set, merely deleting one weighted numerator term need not reproduce the estimator fitted without that participant. The estimator must be recomputed according to its specified rule. Finally, a common additive calibration error remains undetectable in these residuals: replacing every $X_i$ by $X_i+B$ leaves all $D_i$ unchanged. Shared bias in an interlaboratory consensus therefore remains a separate question from self-influence and the variance of the residuals.

A leave-one-out inverse-variance weighted consensus applies an inverse-variance weighted mean to all laboratories except the one being assessed. Suppose their results $X_j$ are independent random variables with common expected value $\theta$ and known positive variances $u_j^2$. For laboratory $i$, defineUnder this model it is the best linear unbiased estimator of $\theta$ from the remaining results. The omitted observation does not determine its own leave-one-out interlaboratory consensus value.

The minimum-variance property follows directly from the Cauchy-Schwarz inequality. For any fixed coefficients $c_j$ satisfying ${\sum }_{j\ne i}{c}_j=1$, the linear unbiased estimator $T={\sum }_{j\ne i}{c}_j{X}_j$ hasEquality holds precisely for $c_j=u_j^{-2}/W_{-i}$. ThereforeThis is a comparison among linear unbiased estimators under the stated independent-error model; it requires no normal distribution assumption.

For an original example, assess a laboratory reporting $x_i=12$ with standard uncertainty $u_i=0.5$. The other laboratories report $(10,12,8)$ with respective standard uncertainties $(1,2,4)$, all in the same units. Their total precision is $W_{-i}=1+1/4+1/16=21/16$, givingThe held-out difference is $D_i=12/7\simeq 1.714$, withThe denominator includes the benchmark's uncertainty even though $X_i$ was excluded. This uncertainty-normalized difference and a proficiency-test z-score based on a predetermined assessment scale answer different questions.

Shared bias in an interlaboratory consensus changes the uncertainty model. Suppose instead $X_j=\theta +B+{\epsilon }_j$, where the common reference contribution $B$ has variance $u_B^2$ and the mutually independent residuals, independent also of $B$, have variances $s_j^2$. Coefficients summing to one preserve $B$ in the benchmark. Its variance is $u_B^2+\sum c_j^2{s}_j^2$, so the minimizing coefficients use $s_j^{-2}$, andUsing the full marginal uncertainties $\sqrt{u_B^2+s_j^2}$ as if they were independent changes the coefficients and loses the shared-reference structure. The propagation of uncertainty retains the covariance of dependent inputs.^[ref]

If the held-out laboratory shares the same $B$, that contribution cancels in $D_i$, while it remains in the benchmark's absolute measurement uncertainty. Precision of a difference and knowledge of an absolute value are distinct properties. Estimated, result-dependent uncertainties or additional correlations require analysis of the actual estimator; the fixed known-variance proof does not establish optimality for those cases.

Low-temperature calorimetry measures a small sample's heat capacity by resolving a small temperature change after a known electrical energy input. The experimental challenge is that the support, heater and thermometer also warm up. Their calorimeter addenda must be removed before interpreting the sample's phonons or electronic heat capacity. The Debye T-cubed test turns the corrected response into a test of the low-temperature vibrational spectrum.

Manuel Brando's 2009 experiment used a sapphire platform suspended by four alloyed platinum wires from a copper ring fixed to a cryostat stage. Grease attached the sample; a chip heater and ruthenium-oxide resistance thermometer served platform PLS-2. A controller stabilized the bath; a lock-in amplifier supplied heating pulses and read the calibrated sensor. The steady rise calibrated thermal conductance as $G=P/\Delta T$. A separate $59.1\mathrm{mg}$, $99.99\%$ pure gold sample, compared with reference heat capacities, established the calorimeter addenda. Small pulses limited property variation; imperfect sample contact required a second relaxation time. The platform contribution rose below $1.5\mathrm{K}$, making simple extrapolation unsafe.^[ref]

For an original idealization, assume sample and platform share one temperature $T$, the bath remains at $T_b$, and both the total heat capacity and the weak-link thermal conductance $G$ are approximately constant across one small pulse. Ignore other heat transfer and any phase transition. The first law of thermodynamics becomesHere $C_s$ is sample heat capacity, $C_a$ is calorimeter addenda, $P$ is deposited electrical power from Joule heating, and $\tau$ is a time constant. This is the same balance as calorimeter heat-loss correction, used here with a deliberately weak cryogenic link and a separately subtracted platform. For a constant pulse $P_0$ beginning at bath temperature, then switched off at $t_h$,The cooling exponential decay determines $\tau$, while the heating plateau determines $G=P_0/(T_{\infty }-T_b)$. Together with calorimeter addenda, they give $C_s=G\tau -C_a$. Cooling alone determines a ratio, not an absolute heat capacity. With finite $t_h$, the observed end rise is smaller than the plateau; fit the full response or apply the displayed correction rather than identifying the two.

For the stipulated figure parameters, $G=5.00\times 10^{-8}\mathrm{W}{\mathrm{K}}^{-1}$, $\tau =20.0\mathrm{s}$, and $C_a=2.00\times 10^{-7}\mathrm{J}{\mathrm{K}}^{-1}$ give $C_s=8.00\times 10^{-7}\mathrm{J}{\mathrm{K}}^{-1}$. The asymptotic rise is $0.0400\mathrm{K}$, only $2\%$ of bath temperature. At $100\mathrm{s}$, the rise is $0.0397305\mathrm{K}$. Of $2.00\times 10^{-7}\mathrm{J}$ input, only $3.97305\times 10^{-8}\mathrm{J}$ remains stored: the other $1.60270\times 10^{-7}\mathrm{J}$ has flowed to the bath. Treating this long pulse as an isolated $Q/C$ experiment would overestimate total heat capacity by a factor of $5.034$.

Suppose independent standard uncertainties are $u(G)=0.05\times 10^{-8}\mathrm{W}{\mathrm{K}}^{-1}$, $u(\tau )=0.2\mathrm{s}$ and $u(C_a)=0.10\times 10^{-7}\mathrm{J}{\mathrm{K}}^{-1}$. First-order propagation of uncertainty givesThe uncertainty is one standard deviation under those assumptions, about $2.2\%$ of sample heat capacity. Shared thermometer calibration errors can correlate fitted $G$ and $\tau$; then their covariance contributes too. Smaller samples amplify the relative influence of calorimeter addenda. Proposed checks include varying pulse amplitude, repeating with a reference sample, checking sensor self-heating, and comparing single- and two-time-constant fits. A drifting bath, slow internal equilibration or a transition invalidates this simple reduction even if a cooling curve looks smooth.

For $z=\sqrt{R^2+X^2}$ with reactance $X=\omega L$, linear differentiation gives $dz=(RdR+XdX)/z$. Independent standard uncertainties give the title formula; correlated inputs add $2RX\mathrm{Cov}(R,X)/z^2$. Hard error bounds instead add absolute linear contributions. The meaning of an error bar is an additional measurement assumption, not a consequence of the impedance formula.

The ideal measurement calibration is a straight line through the equal-pressure zero. For the stipulated water mass density and gravitational acceleration above, its slope is $9.81\mathrm{Pa}$ per vertical millimetre. A lower liquid mass density gives a larger length change for the same pressure, improving the height-reading contribution to measurement resolution but requiring more vertical space for a given pressure range. The pressure scale depends on the actual mass density, which depends on temperature; $1000\mathrm{kg}{\mathrm{m}}^{-3}$ here is a chosen model value, not an exact property of all water samples.

Suppose each liquid meniscus height is rounded to the nearest scale division $q=1\mathrm{mm}$. Each height measurement error lies within $q/2$, so the error in their difference lies within $q$. The resulting pressure error is bounded by $\rho gq=9.81\mathrm{Pa}$. To quantify a standard uncertainty, make the additional model assumption that the two rounding errors are independent random variables, each having a continuous uniform distribution on $[-q/2,q/2]$. Each then has variance $q^2/12$; adding the two variances givesHere $u$ denotes standard uncertainty. The $9.81\mathrm{Pa}$ worst-case bound and $4.0049\mathrm{Pa}$ standard uncertainty answer different questions. At $588.6\mathrm{Pa}$ they are respectively $1.67\%$ and $0.680\%$ of the reading. Shared reading bias or a different reading method can violate the assumed independence.

If the mass density, gravitational acceleration and height difference also have small independent standard uncertainties, a first-order estimate follows from the partial derivatives of $\Delta p=\rho g\Delta h$:This displays why a finer scale does not remove uncertainty in mass density or gravitational acceleration. A pressure transducer compared against the column needs its own differential-pressure calibration to establish an electrical response.

For the same massless spring obeying Hooke's law, the frequency is $\nu =(2\pi {)}^{-1}\sqrt{k/m}$. The displayed mass ratio follows. For small independent frequency errors with common fractional standard uncertainty $ϵ$, monomial uncertainty propagation gives ${\sigma }_R/R\simeq \sqrt{8}ϵ$. If $ϵ$ instead describes individual measurements and each sample mean uses $n$ independent trials, its standard error is smaller by $\sqrt{n}$, giving ${\sigma }_R/R\simeq \sqrt{8/n}ϵ$. Four times as many trials halve this statistical uncertainty; a common calibration error needs its covariance included.

A measured quantity value is a value used to represent a measurement result. It may be obtained from one observation or calculated from replicate observations, for example as an arithmetic mean. The VIM distinguishes this value from a raw measurement indication and from the fuller result with its measurement uncertainty.^[ref]

For an original electrical resistance example, suppose three instrument indications are $100.04$, $100.06$ and $100.05\Omega$. A measurement calibration gives additive indication offset $b=0.02\Omega$, so the corrected values are $100.02$, $100.04$ and $100.03\Omega$. Their arithmetic mean is the representative measured quantity valueThis value is a numerical estimate of the specified measurand, such as the resistor's resistance at a stated temperature. It is not established as an exact true value merely by correcting the indications.

The three values have sample standard deviation $s=0.01\Omega$. If repeat deviations are independent, their contribution to the mean's standard uncertainty is $s/\sqrt{3}\simeq 0.0058\Omega$. Suppose the common correction has independent standard uncertainty $u(b)=0.02\Omega$. First-order propagation of uncertainty givesRepeating the readings reduces the first term under these assumptions; it does not repeatedly remeasure and average away the same common correction. These are illustrative uncertainty inputs, not observed equipment data.

Reporting the measured quantity value therefore requires its unit and measurand definition. Reporting the complete measurement result also identifies the associated measurement uncertainty, its convention and relevant conditions. A change from ohms to milliohms changes the numerical expression, while leaving the represented physical value unchanged.

Measurement precision describes agreement among repeated indications or measured values obtained under stated conditions. It is commonly characterized using a standard deviation, variance or coefficient of variation, which describe the observed spread. Measurement repeatability is precision under a particular restricted set of conditions; intermediate measurement precision and measurement reproducibility concern other specified conditions.^[ref]

For an original balance example, take three repeat readings of a reference object: $100.100$, $100.080$ and $100.120\mathrm{g}$. Their arithmetic mean and sample standard deviation areSuppose the reference object's assigned mass is $100.000\mathrm{g}$ with standard uncertainty $0.002\mathrm{g}$. The observed mean differs from that reference by $0.100\mathrm{g}$, much more than the within-series scatter. These chosen readings therefore illustrate why close repeat readings do not establish agreement with a reference. They do not determine the true systematic offset from only three trials.

Measurement accuracy and measurement trueness answer different questions from precision. A common additive offset leaves the deviations $x_j-\bar{x}$, and hence $s$, unchanged. Averaging more independent readings can reduce random uncertainty in the mean but does not by itself remove that common offset.^[ref]

A useful physical study repeatedly removes and replaces the reference object while recording load procedure, environmental conditions and elapsed time. Specify which parts of the measurement were repeated: repeated display readings alone may omit placement variation or instrumental drift. The statistic describes the stated experiment, not an unconditional property of the instrument.