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.^[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,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.^[ref]

For original stipulated difference standard deviations2and5milligrams per kilogram, respectively,in squared concentration units. One analysis on one sample has combined standard deviation $\sqrt{13.5}\approx 3.67$milligrams per kilogram. Averaging two analyses of that same sample gives $\sqrt{11.5+2/2}\approx 3.54$; averaging one analysis from each of two independent samples gives $\sqrt{(11.5+2)/2}\approx 2.60$. 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.^[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.

An unbalanced sampling-uncertainty design estimates collection and analytical contributions while performing different numbers of analyses on the collected samples. One useful duplicate design collects two samples independently at each target, analyses one twice and the other once. The Eurachem guide presents this three-result arrangement as a less costly alternative to analysing both samples twice.^[ref]

An original variance calculation shows why both components remain identifiable. At target $i$, letHere $j=1,2$ labels independently collected samples; only $j=1$ has analytical replicates $r=1,2$. Assume centered independent random effects, common variance components and the same analytical precision for each result. Define within-target contrastsThe target mean cancels. The first contrast also cancels the sample effect; the second compares independent collections with unequal analytical averaging. Consequently,The coefficient $3/2$ matters: one sample mean contains half the analytical variance, while the other result contains all of it. Applying the balanced-design coefficient would misallocate uncertainty.

For stipulated contrast variances $s_D^2=4$ and $s_Q^2=25$ in squared concentration units, the corresponding method-of-moments estimates areA single analysis of a single collected sample then has modeled standard uncertainty $\sqrt{11+2}\approx 3.61$ in concentration units. These are invented values illustrating the model, not measurements from the guide.

Across eight targets, this design requires sixteen collections and twenty-four analyses, compared with thirty-two analyses for two analytical replicates on each collection. The saving is in analytical runs; equal cost does not imply equal statistical precision. Repeat the nested design across targets to estimate the contrast variances. One triple of results does not reliably estimate the collection variance, and duplicate collection by one unchanged protocol does not reveal a bias shared by every sample. A negative estimated variance component signals finite-data or model issues requiring an appropriate estimation method, rather than a physically negative uncertainty.