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Batch and measurement-process variation — Examples

1 · Prerequisites

2 · Summary

Three stable reference standards are measured on two days under stipulated linear response curves. The worked calculations recover the day-specific slopes, fit the equally weighted pooled line, and compare its check-standard residuals and inverse readings with the separate curves. The example is deterministic and illustrates how a shared curve can conceal day-specific behavior; it makes no empirical, inferential, or general stability claim.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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5 · Examples, counterexamples and false statements

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Day-specific calibration slope drift and check standards

Example

Suppose the three stable reference standards have assigned quantities $q \in {0, 5, 10}$ . Remeasure these same standards on two days, obtaining the stipulated exact response table

Reference quantity $q$	Day 1 response	Day 2 response
$0$	$1$	$1$
$5$	$11$	$12$
$10$	$21$	$23$

Take $[0, 10]$ as the declared calibration range. The standards are check standards here: fixed reference artifacts measured again over time to reveal changes in the measurement process. The numbers are a deterministic illustration, not observed data. Fit one equally weighted straight line with an intercept to all six response pairs and compare it with the separate day-specific calibration curves.

Facts & Assumptions

Given: The assigned reference values and stipulated responses in the Example, with the pooled line defined to minimize the sum of squared vertical residuals over all six pairs.

[F1]

Measurands, calibration, and measurement uncertainty: a calibration curve is a specified relation between a reference quantity and instrument response used to convert a later response into a measurand estimate.

[F2]

Batch and measurement-process variation: day is a batch/process condition, and calibration curves are pooled only under an explicit shared curve or equivalence model.

[A1]

The three reference standards remain stable with assigned values $0, 5, 10$ on both days, and the response table is stipulated exactly as shown.

[A2]

The pooled line is $a + b q$ with $a, b \in R$ , fitted by minimizing $\sum_{d = 1}^{2} \sum_{q \in {0, 5, 10}} (r_{d q} - a - b q)^{2}$ with equal weight for each of the six readings.

Verification

1.1givenA1algebra

The day-specific lines are $r_{1} (q) = 1 + 2 q$ and $r_{2} (q) = 1 + 2.2 q$ : each intercept is the response at $q = 0$ , and the slopes are $(21 - 1) / 10 = 2$ and $(23 - 1) / 10 = 2.2$ . In particular the known standard at $q = 5$ has responses $11$ and $12$ .

1.2givenA1algebra

The six reference values have mean $\overset{ˉ}{q} = 5$ and the six responses have mean $\overset{ˉ}{r} = 69 / 6 = 11.5$ . Their centered sums are $S_{q q} = \sum (q - \overset{ˉ}{q})^{2} = 100$ and $S_{q r} = \sum (q - \overset{ˉ}{q}) (r - \overset{ˉ}{r}) = 210$ : the two $q = 0$ readings contribute $105$ , the $q = 5$ readings contribute $0$ , and the two $q = 10$ readings contribute $105$ .

2.1givenA2step 1.2algebra

Write each residual as $(r - \overset{ˉ}{r}) - b (q - \overset{ˉ}{q}) + (\overset{ˉ}{r} - a - b \overset{ˉ}{q})$ . The first two terms sum to zero over the six pairs, so expanding the sum of squared residuals gives $\sum [(r - \overset{ˉ}{r}) - b (q - \overset{ˉ}{q})]^{2} + 6 (\overset{ˉ}{r} - a - b \overset{ˉ}{q})^{2}$ . For fixed $b$ this is minimized at $a = \overset{ˉ}{r} - b \overset{ˉ}{q}$ ; the remaining slope-dependent terms are $S_{q q} b^{2} - 2 S_{q r} b$ , whose completed square is $S_{q q} (b - S_{q r} / S_{q q})^{2}$ plus a constant. Since $S_{q q} = 100 > 0$ , the unique minimizing slope is $b = 210 / 100 = 21 / 10$ and $a = 11.5 - (21 / 10) (5) = 1$ , giving $\hat{r} (q) = 1 + (21 / 10) q$ .

3.1F1F2A1step 1.1step 2.1algebra∎

At the $q = 5$ check standard, the pooled line predicts $11.5$ , so its day-specific residuals are $11 - 11.5 = - 0.5$ and $12 - 11.5 = 0.5$ . Inverting the pooled line maps the two observed responses to $10 / (21 / 10) = 100 / 21$ and $11 / (21 / 10) = 110 / 21$ , whereas the separate day lines each map their own response back to $q = 5$ . The three-standard table exposes the differing slopes in this stipulated example; it supplies no empirical evidence, statistical test, or general justification for pooling calibration curves.

Remarks

There is one stipulated reading per standard per day, so the table does not estimate repeatability, a variance component, a control limit, or uncertainty. It only demonstrates the finite arithmetic of comparing three standard responses under two specified lines.