= Analytical sampling uncertainty
<Analytical sampling uncertainty> is the contribution to <measurement uncertainty> arising when collected material is used to infer a property of a specified sampling target. The target might be a bulk lot or a defined location and time interval. Its composition need not equal that of one convenient portion. Sampling and physical preparation therefore belong in the <measurement model> when the <measurand> concerns the larger target.<Measurement uncertainty arising from sampling by Michael H. Ramsey and others>[Sections 7–9, target definition and empirical uncertainty]{ref}
A duplicated experiment can separate two random components. Within each target, collect two independent samples by the stated protocol, prepare them separately, and analyse each twice. For a simple model,
$$
Y_{ij}=\mu+S_i+\varepsilon_{ij},\qquad
\operatorname{Var}(S_i)=\sigma_s^2,\quad
\operatorname{Var}(\varepsilon_{ij})=\sigma_a^2,
$$
where $i=1,2$ labels collection and $j=1,2$ labels analysis; all random terms are centered and independent except for the shared $S_i$ within a sample. The within-sample difference has <variance> $2\sigma_a^2$. The difference between the two sample means has <variance> $2\sigma_s^2+\sigma_a^2$. Repeating this design across targets supports estimation of both components; different target means cancel in each within-target difference.<Measurement uncertainty arising from sampling by Michael H. Ramsey and others>[Sections 9.3–9.5; §9.4.2 and Figure 2, balanced duplicate design]{ref}
For original stipulated difference <standard deviations>2and5milligrams per kilogram, respectively,
$$
\sigma_a^2=2^2/2=2,\qquad
\sigma_s^2=(5^2-2)/2=11.5
$$
in squared concentration units. One analysis on one sample has combined <standard deviation> $\sqrt{13.5}\approx3.67$milligrams per kilogram. Averaging two analyses of that same sample gives $\sqrt{11.5+2/2}\approx3.54$; averaging one analysis from each of two independent samples gives $\sqrt{(11.5+2)/2}\approx2.60$. More laboratory replication and more independent collection address different contributions.
A collaborative wheat-and-coffee trial varied the sampler as well as the collected material. Its contrasting reproducibility and repeatability findings show why the uncertainty depends on which collection conditions are allowed to vary.<Collaborative trials of the sampling of two foodstuffs, wheat and green coffee by Michael Thompson and others>[Abstract, multiple-sampler design and contrasting analyte results]{ref}
Duplicate sampling with one sampler and protocol does not establish absence of a shared sampling bias. Reference comparisons, alternative protocols or inter-organisational trials can test additional effects. Common preparation losses or <measurement calibration> errors also require <covariance> terms beyond the independent model.
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