= Analytical sampling
<Analytical sampling> selects and collects material from a larger target for chemical measurement. It connects the material actually measured with the batch, location or time period about which a conclusion is wanted. A small, well-measured portion need not represent the larger material merely because its instrument reading is precise.
An original two-region example has90% of a batch's mass at $1\,\mathrm{mg\,kg^{-1}}$ and10% at $100\,\mathrm{mg\,kg^{-1}}$ of an <analyte>. The batch-average <mass fraction> is
$$
0.9(1)+0.1(100)=10.9\,\mathrm{mg\,kg^{-1}}.
$$
Taking equal masses from the two regions instead produces $50.5\,\mathrm{mg\,kg^{-1}}$. That sample answers a different weighted question. <Probability sampling> or a justified <stratified sampling> design can connect selection probabilities and weights to the intended target.
A <sampling increment> is an individual collected portion. Combining increments and reducing material to a laboratory portion can introduce further variation, so the protocol should state where, when, how much and how material was handled. Repeated readings of one prepared solution chiefly test later measurement stages.
For an original nested model, let independent collected portions have sampling variance $\sigma_s^2$, and repeated analyses within a portion have variance $\sigma_a^2$. Averaging $m$ analyses on each of $n$ independent portions gives
$$
\operatorname{Var}(\overline y)=\frac{\sigma_s^2}{n}+\frac{\sigma_a^2}{nm}.
$$
With stipulated $\sigma_s=4$, $\sigma_a=1$, $n=4$ and $m=3$ in compatible units, the mean's <standard uncertainty> is approximately2.02. Increasing laboratory replication alone cannot eliminate the sampling component. Correlated portions or shared biases require an expanded model.
A primary collaborative trial had multiple samplers collect wheat and green coffee from bulk lots. Its contrasting findings for between-sampler and within-condition variation demonstrate why <analytical sampling uncertainty> needs a design that actually varies the collection stage.<Collaborative trials of the sampling of two foodstuffs, wheat and green coffee by Michael Thompson and others>[Abstract, multiple-sampler design and sampling reproducibility/repeatability findings.]{ref}
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