= Shared bias in an interlaboratory consensus
\b[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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.4 and AnnexF.1.2.3, common references and correlated inputs]{ref}
A simple original model is
$$
x_i=\theta+b+\varepsilon_i,
$$
where $\theta$ is the target quantity value, $b$ a common fixed offset and $\varepsilon_i$ a laboratory-specific deviation. The <arithmetic mean> is $\bar x=\theta+b+\bar\varepsilon$. Increasing the number of independent laboratories reduces the fluctuations in $\bar\varepsilon$ 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 $\sigma^2$. Then
$$
\operatorname{cov}(x_i,x_j)=u_B^2\quad(i\ne j),\qquad
u^2(\bar x)=u_B^2+\frac{\sigma^2}{n}.
$$
This 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.
\Image[/shared-calibration-uncertainty-floor.png]
{title=An original interlaboratory uncertainty model: a common calibration contribution remains as independent laboratory variation averages down}
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 $u(\bar x)=\sqrt{0.25+0.0016}\approx0.502$, 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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2 and §5.2.2 note1, covariance combination and shared-resistor example]{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.
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