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An assigned value in proficiency testing $x_{\mathrm{pt}}$ is the benchmark attributed to the proficiency-test item for evaluation of participant results. Its origin, measurement uncertainty and relationship to the specified measurand are part of its meaning. It is not automatically a true value known without uncertainty.

The guide distinguishes assignment by formulation, a certified reference material, results from one laboratory, an expert-laboratory consensus or participant consensus. These methods provide different evidence and dependencies; a simple interlaboratory consensus value is only one possibility.^[ref]

For an original formulation example, suppose the relevant mass concentration is analyte mass divided by final solution volume:The assigned value in this stipulated complete-transfer model is $c=2.00mg/L$. If $u(m)=0.02\mathrm{mg}$ and $u(V)=0.005\mathrm{L}$ are independent standard uncertainties, first-order propagation givesso $u(c)\approx 0.0224mg/L$. Purity, incomplete transfer, stability or nonuniform mixing would require additional model terms. Formulation alone may not reproduce extraction or preparation difficulties encountered in routine samples.^[ref]

Participant consensus introduces a different dependency. If four laboratories report 100 and a fifth reports 110, their arithmetic mean is 102. The fifth laboratory's deviation from this benchmark is 8, whereas its deviation from the other four laboratories' mean is 10. In general,where ${\bar{x}}_{-i}$ is the arithmetic mean excluding participant $i$. This original identity shows participant influence on a consensus assigned value; the same result being assessed helps determine its comparator. It does not imply that every scheme should use a leave-one-out rule, but the dependence must not be mistaken for an independent reference comparison.

The provider should record how the assigned value was obtained and how its uncertainty was evaluated. A precise participant consensus can retain shared bias in an interlaboratory consensus, while an independent reference can itself have substantial uncertainty. Those are distinct limitations requiring different evidence.

An interlaboratory comparison organizes measurements of the same or similar items by at least two laboratories under agreed conditions. Its purpose may be to compare measurement reproducibility, evaluate a measurement procedure, or assess participants through proficiency testing. The last purpose evaluates performance against criteria specified beforehand; it is one use of interlaboratory comparison, not a synonym for every such study.^[ref]

A comparison begins with its experimental design. When the aim is to compare laboratory analysis, the organizer needs evidence of reference material homogeneity and reference material stability: unequal or changing test items would otherwise be confounded with differences between laboratories. If collection is part of the measurement procedure, the design must instead include analytical sampling explicitly. In a 2002 collaborative study, several samplers collected wheat and green coffee from bulk lots. The reported between-sampler variation was significant for two wheat analytes; for coffee, a significant repeatability contribution was detected without a significant sampling-reproducibility contribution. Thus even the importance of collection differences depended on the material and analyte.^[ref]

For a common proficiency-test z-score, let laboratory $i$ report $x_i$, let the assigned value in proficiency testing be $x_{\mathrm{pt}}$, and let the standard deviation for proficiency assessment be ${\sigma }_{\mathrm{pt}}>0$. ThenThe denominator describes the scheme's assessment scale. It is not automatically the laboratory's own measurement uncertainty or the observed standard deviation of participants. The guide distinguishes this proficiency-test z-score from scores incorporating uncertainty, and discusses choosing the assessment scale to match the intended use.^[ref]

An original hypothetical example shows why the assigned value matters. Suppose three laboratories report $9.8$, $10.0$ and $10.2$ in the same units, with a stipulated assessment scale ${\sigma }_{\mathrm{pt}}=0.2$. Using their arithmetic mean as an interlaboratory consensus value gives $x_{\mathrm{pt}}=10.0$ and scores $(-1,0,1)$. An independently established assigned value of $11.0$, with the same assessment scale, instead gives $(-6,-5,-4)$. Close agreement between participants is compatible with a shared displacement from an independent reference.

The identifiability issue is visible in a simple model,where $\theta$ is the reference quantity value, $b$ is a common systematic error, and ${\epsilon }_i$ describes laboratory-specific deviations. Pairwise differences eliminate both $\theta$ and $b$. Moreover, replacing $(\theta ,b)$ by $(\theta +c,b-c)$ leaves every predicted $x_i$ unchanged. Consequently agreement alone cannot identify a common bias. Evidence from an independently characterized certified reference material or an independent measurement calibration can help resolve this ambiguity. Its own measurement uncertainty and any shared dependencies must then enter the comparison model; one should not silently interpret the simple proficiency-test z-score as a full uncertainty calculation. This is the practical importance of distinguishing measurement precision, measurement trueness and shared bias in an interlaboratory consensus.

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.

Participant influence on a consensus assigned value is the dependence created when a result being assessed also contributes to its benchmark. The deviation from that benchmark is then generally smaller than the deviation from a corresponding consensus formed solely from the other participants. This is a property of the estimator, not evidence that the laboratory improved.

For an original equal-weight calculation with $n>1$ results, writewhere ${\bar{x}}_{-i}$ is the leave-one-out interlaboratory consensus computed as the arithmetic mean of the other results. With results $(8,10,10,12)$, the overall mean is 10. The participant reporting 12 differs from it by 2, but differs from the other three laboratories' mean $28/3$ by $8/3$. The factor $3/4$ accounts for the difference exactly.

A fixed-weight version makes the effect more general. If normalized weights $a_j$ sum to one, $0\le a_i<1$, andthen $x_i-x_c=(1-a_i)(x_i-x_{c,-i})$. A participant with weight 0.8 retains only one fifth of its difference from the remaining weighted consensus. These are fixed-weight identities; robust statistics with data-dependent weights requires analysis of the actual estimator, for example through an influence function.

Dependence also enters uncertainty. For independent results $X_i$ having common mean $\theta$ and variance ${\sigma }^2$,Treating $X_i$ and $\bar{X}$ as independent would instead add their variances and give the wrong result. This is a stated sampling model, not a universal uncertainty formula for every interlaboratory comparison.

The primary guide distinguishes assigned values based on participant consensus, expert laboratories, formulation and certified reference materials. It also requires the assigned value's uncertainty to be evaluated and reported.^[ref] Using an independent reference or a documented leave-one-out method changes the dependence, but neither is automatically appropriate for every scheme.

Finally, reducing self-influence does not resolve shared bias in an interlaboratory consensus. All remaining laboratories can still share the same calibration error. Benchmark independence, robustness to an extreme result and absolute accuracy are separate questions.

A reference material is established as suitable for a specified measurement use because the relevant properties are sufficiently homogeneous and stable. The requirement concerns particular properties and an intended use, rather than every possible attribute of the material.^[ref] Reference material homogeneity concerns variation within or between the portions used; reference material stability concerns maintaining the relevant properties over time and under stated conditions.

Reference materials can support measurement precision control even without an assigned quantity value. An assigned value is needed when the material serves in measurement calibration or a check of measurement trueness.^[ref] A certified reference material adds authoritative documentation specifying property values with associated uncertainties and traceabilities obtained through valid procedures.^[ref]

A concrete physical example is the archived 2009 certificate for National Institute of Standards and Technology SRM 1922, a mineral oil characterized for refractive index. Its characterization used a precision spectrometer and goniometer arrangement with the minimum deviation method, oil in a hollow optical prism, and water-bath temperature control. Measurements covered several wavelengths and temperatures; multiple portions were examined to assess homogeneity.^[ref]

The certificate's Table 1 reports, at wavelength 589.3 nm and temperature $20{}^{\circ }\mathrm{C}$, refractive index 1.46945 with expanded uncertainty $6\times 10^{-5}$ and coverage factor 2. This wavelength's value was obtained through a dispersion fit. The listed temperature coefficient is $-3.68\times 10^{-4}{\mathrm{K}}^{-1}$.^[ref]

An original calculation using that linear coefficient predictsThe predicted change, $0.000736$ in magnitude, is larger than the quoted expanded uncertainty at the reference conditions. Temperature therefore belongs with the assigned optical value; it is not an incidental label. This calculation illustrates the certificate's model, rather than supplying a new certified value or uncertainty.

Record the material identity, certificate version, property, wavelength, temperature and handling history with a measurement result. The archived certificate supplies a documented example of those links, rather than a claim about a currently available product or the validity of a particular bottle today.

Shared bias in an interlaboratory consensus occurs when a common systematic error shifts several laboratories' results together. Their interlaboratory consensus value can then be highly reproducible while displaced from an independent reference. Shared calibration standards, a common correction or the same inadequately modeled measurement step can create such dependence. The GUM explicitly warns that using the same instrument, standard or reference datum can correlate input quantities.^[ref]

A simple original model iswhere $\theta$ is the target quantity value, $b$ a common fixed offset and ${\epsilon }_i$ a laboratory-specific deviation. The arithmetic mean is $\bar{x}=\theta +b+\bar{\epsilon }$. Increasing the number of independent laboratories reduces the fluctuations in $\bar{\epsilon }$ but leaves $b$ in place.

The obstacle is also one of identification. For any constant $c$, replacing $(\theta ,b)$ with $(\theta +c,b-c)$ leaves all predicted results unchanged. Pairwise differences $x_i-x_j$ cancel both terms. Thus the comparison's internal agreement cannot determine the absolute common offset without additional information.

To represent residual common-reference uncertainty, use an uncertain zero-centered quantity $B$ with standard uncertainty $u_B$, independent of mutually independent laboratory errors of variance ${\sigma }^2$. ThenThis is a stated uncertainty model, distinct from asserting that an unknown fixed bias has been observed to vary randomly. The common contribution is counted once in the mean's uncertainty; it is not divided by the number of laboratories.

For stipulated $u_B=0.50$, $\sigma =0.20$ and $n=25$, treating everything as independent would retain only $0.20/5=0.04$. The stated correlated model instead gives $u(\bar{x})=\sqrt{0.25+0.0016}\approx 0.502$, in the same units as the measurement. The appropriate covariance terms prevent the implausible conclusion that enough copies of a shared reference can remove its uncertainty.^[ref]

An independently characterized certified reference material, a different measurement principle or an independently calibrated reference can help test the suspected common shift. Independence must itself be checked: a nominally separate reference derived from the same calibration chain may retain the same dependency. Recognized biases should be corrected where appropriate, with the correction's uncertainty retained.