Interlaboratory consensus value ($x_c$)

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.

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.