Representative sampling, preparation and contamination

The bottle is not automatically the intended target

A sampling target may be a batch, a defined location, a time-dependent stream or a population of objects. A primary sample is taken to represent that target, then preparation and analysis produce an indication. Repeating analysis of one bottle probes a different part of this chain from independently sampling the target.

Representativeness depends on the target, heterogeneity and design. Convenient access can systematically miss a region or fraction. A liquid is not automatically homogeneous: gradients, settling or separate phases can matter. Sampling plans specify which portions, places or times the result is intended to represent and how potential variation is assessed.

Use weights that belong to the intended bulk quantity

In an invented target, three internally uniform strata have masses 200, 300 and 500 g and analyte contents 1.00, 3.00 and 7.00 mg g⁻¹. Their analyte masses are 200, 900 and 3500 mg. Bulk mass is 1000 g and bulk analyte mass is 4600 mg, giving 4.60 mg g⁻¹.

The unweighted mean of the three content values is 11/3≈3.66667 mg g⁻¹. It represents equal weighting of the three strata values, rather than their unequal masses. The mass-weighted fractions 0.200, 0.300 and 0.500 reconstruct the bulk value. This fully specified teaching model does not prove how a real unknown heterogeneous target should be sampled.

The dominant uncertainty component determines what improvement helps

For an optional deeper random-effects toy, let independent zero-mean sampling and analytical effects have standard deviations 0.600 and 0.200 mg L⁻¹. Variances add: 0.360+0.0400=0.400 (mg L⁻¹)², giving total standard deviation approximately 0.632456 mg L⁻¹. Analytical variation supplies only 10% of this total variance.

Halving the analytical standard deviation gives total variance 0.360+0.0100=0.370. Halving the sampling standard deviation instead gives 0.0900+0.0400=0.130. The larger improvement follows from reducing the dominant component. These random-effect calculations omit sampling bias and require the same defined quantity and valid independence assumptions.

Repeated readings can reduce an analytical contribution to the uncertainty of a mean. They cannot make an unrepresentative sampling design representative or independently reproduce the sampling stage. Likewise, variation between different targets is not automatically uncertainty of sampling within one target.

The same contamination mass has a different concentration effect

Suppose a measurement model assigns a fixed additive background analyte mass of 0.0200 μg. In a 0.00500 L final mixture its apparent contribution is 4.00 μg L⁻¹. In 0.0500 L it is 0.400 μg L⁻¹. A tenfold volume difference changes the concentration contribution tenfold, despite the same background mass.

Preparation can also lose analyte, change forms or introduce matrix-dependent responses. A suitable blank or control can investigate a defined stage, but no single blank automatically reproduces every effect of the target matrix. This toy supplies hypothetical background and volume values, without a sample-handling recipe.

A design can distinguish readings from independently prepared samples

Consider four independent primary samples from one hypothetical target, each independently prepared and read twice. Invented pairs in mg L⁻¹ are (9.9, 10.1), (10.3, 10.5), (9.5, 9.7) and (10.1, 10.3). Their means are 10.0, 10.4, 9.6 and 10.2; grand mean is 10.05. Sampling and preparation share one group effect in this design.

Under a balanced independent random-intercept model with two readings per group, within-group sum of squares is 0.0800 with four degrees of freedom, giving analytical variance estimate 0.0200 (mg L⁻¹)². Between-group sum of squares is 0.700 with three degrees of freedom, giving mean square 0.233333. The sample/preparation component estimate is (0.233333−0.0200)/2≈0.106667 (mg L⁻¹)².

For the grand mean of four independent groups with two readings each, the model variance is 0.106667/4+0.0200/8≈0.0291667 (mg L⁻¹)². Standard uncertainty is approximately 0.170783 mg L⁻¹. This small invented dataset illustrates the calculation, rather than a validated uncertainty study. The design does not separately identify preparation versus sampling, or establish absence of shared bias.

Worked examples

Try it yourself

1. Two fully specified strata have equal masses and contents 2.00 and 6.00 mg g⁻¹. Find the bulk content.

   Show the answer and reasoning

   4.00 mg g⁻¹.

   Equal masses give equal weights; this equality did not hold in the chapter’s unequal-mass model.

2. The same invented 0.0200 μg background appears in 0.0100 L final mixture. Find its concentration contribution.

   Show the answer and reasoning

   2.00 μg L⁻¹.

   Divide the supplied mass by the final mixture volume.

3. Do ten readings from one prepared bottle independently replicate target sampling?

   Show the answer and reasoning

   No.

   They replicate readings under the specified conditions; the primary-sampling stage was not repeated.

4. Can the stated nested design separate sampling variance from preparation variance?

   Show the answer and reasoning

   No.

   Each independently prepared primary sample supplies one shared group effect, combining those two stages in this toy.

What to carry forward

• Define the target before taking a sample.
• Weights belong to the intended target quantity.
• Reading precision does not establish representativeness.
• Preparation and background effects need their own evidence.
• Replication designs determine which variance components can be identified.