= Assigned value in proficiency testing
{title2=$x_{\mathrm{pt}}$}
An \b[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.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[§7.2.2, printed p. 15; Annex D.2, assigned value and uncertainty]{ref}
For an original formulation example, suppose the relevant <mass concentration> is analyte mass divided by final solution volume:
$$
c=\frac mV,\qquad m=2.00\,\mathrm{mg},\quad V=1.000\,\mathrm L.
$$
The assigned value in this stipulated complete-transfer model is $c=2.00\,\mathrm{mg/L}$. If $u(m)=0.02\,\mathrm{mg}$ and $u(V)=0.005\,\mathrm L$ are independent standard uncertainties, first-order propagation gives
$$
\frac{u(c)}c=\sqrt{\left(\frac{0.02}{2.00}\right)^2+
\left(\frac{0.005}{1.000}\right)^2}\approx0.01118,
$$
so $u(c)\approx0.0224\,\mathrm{mg/L}$. 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.<Selection, Use and Interpretation of Proficiency Testing (PT) Schemes by Brian Brookman and Ian Mann>[Annex D.2, formulation and the need for a preparation uncertainty model]{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,
$$
x_i-\bar x=\frac{n-1}{n}(x_i-\bar x_{-i}),
$$
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.
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