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 $1\mathrm{mg}\mathrm{k}{\mathrm{g}}^{-1}$ and10% at $100\mathrm{mg}\mathrm{k}{\mathrm{g}}^{-1}$ of an analyte. The batch-average mass fraction isTaking equal masses from the two regions instead produces $50.5\mathrm{mg}\mathrm{k}{\mathrm{g}}^{-1}$. 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 givesWith 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.^[ref]

A composite sample is formed by combining material from several collected portions into one mixture, from which material for laboratory measurement is taken. It can represent an aggregate composition when the collection and mixing weights match the intended target. IUPAC describes compositing as preparation of a mixture from several samples; combining them does not by itself prove representativeness.^[ref]

Suppose sampling increments have masses $m_i$ and analyte mass fractions $c_i$. With complete transfer, no chemical change and adequate mixing, conservation of analyte givesEqual increment masses produce an arithmetic mean; unequal masses produce a mass-weighted mean. Weights chosen to represent different parts of the sampling target can differ from the fractions accidentally collected.

An original example shows what is lost. Four equal-mass portions with concentrations $(0,0,0,40)$ milligrams per kilogram give the same composite concentration, 10 milligrams per kilogram, as portions with concentrations $(10,10,10,10)$. A single composite measurement cannot distinguish a localized high value from uniform composition. It addresses the weighted average, not the maximum or the spatial distribution.

The uncertainty of that average depends on dependence between portions. For a random concentration vector with covariance matrix $\Sigma$ and fixed mixing weights,For four equal-weight portions each with standard deviation 2 concentration units and common pairwise correlation coefficient 0.25, this givesand standard deviation about 1.32 units. Assuming independence would instead give 1 unit. This stipulated model concerns collected-material variation; common losses or measurement errors can add other contributions.^[ref]

A practical protocol specifies increment selection, mixing proportions, mixing procedure and reduction to a test portion. Loss of volatile material, segregation of particles or incomplete mixing can make the analysed portion differ from the intended composite. Replicate readings of one final solution test later measurement stages; independent composites are needed to investigate variability introduced during collection and pooling.

A sampling increment is an individual portion collected by one operation of a sampling device. One scoop, one probe withdrawal or one defined cut through a moving stream can produce an increment, provided the collection operation and its extent are specified. IUPAC uses this definition for bulk materials and large units.^[ref]

Several sampling increments may be combined into a composite sample. Their number describes collection operations; repeated instrument readings of the same prepared material do not create more sampling increments. Nor is an increment necessarily representative of the entire lot on its own: its location, time, amount and selection mechanism connect it to the analytical sampling target.

An original mass-balance example combines a0.10kilogram increment containing2milligrams per kilogram of an analyte with a0.30kilogram increment containing10milligrams per kilogram. With complete transfer and no chemical change, the composite mass fraction isThe unweighted average of the two concentrations would be6milligrams per kilogram. Equal weighting of reported increment concentrations is therefore different from physically combining unequal increment masses. Whether either result estimates the intended lot mean also depends on the collection design.

Sparse contamination supplies a second distinction. In a stipulated model where each independent increment has probability0.01of intercepting a contaminated region, $k$ increments all miss it with probability $0.99^k$. Even100such increments have a miss probability of about0.366. This is an illustrative detection model, not a universal sample-count rule; spatial dependence and changing increment size alter it.

A collection record should identify the sampling device, each increment's position or collection interval, its amount and any combining or reduction step. Different increment plans can change analytical sampling uncertainty even when the same analytical instrument is used. The collection stage must be varied to assess that uncertainty, rather than only remeasuring one final extract.

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.