Calibration standards made from one stock solution share part of their uncertainty. Preparing more dilutions can improve knowledge of repeatability, yet the common uncertainty in the stock concentration does not shrink simply because more bottles or more calibration points were measured.
This guide follows that shared component through absolute concentrations, a concentration ratio and an average. It also explains why a very straight calibration plot can retain an uncertain concentration scale. The calculations are original small-error teaching models, with every independence assumption stated.
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Trace the shared input · Chapters 1–2. Build the concentration model and an absolute budget.
Compare quantities · Chapters 3–5. Work ratios, dilution trees and calibration limits.
Practise and check · Chapters 6–8. Test averaging, interpretation and evidence.
Full chapter index · Practice and evidence checks · Return to Chemistry
Chapter index
Trace the shared input · Chapters 1–2
Compare quantities · Chapters 3–5
Practise and check · Chapters 6–8
Begin with a stock solution assigned concentration c0. Dilution A has concentration cA = dA c0, and dilution B has concentration cB = dB c0. Each d is a dimensionless final-to-stock concentration factor. A factor of 0.10 means one tenth the stock concentration.
Even if A and B are prepared separately, they inherit the same assigned c0. A change in the stock value moves both assigned concentrations in the same direction. Separate bottles therefore do not establish independent concentration uncertainties.
Draw one parent labelled c0 with two branches labelled dA and dB. Keep a separate branch for each genuinely independent preparation contribution. If the standards form a serial ladder, draw every intermediate as well. The drawing identifies which inputs reappear in more than one result.
Eurachem’s uncertainty guide discusses both covariance and shared effects from calibration stock solutions. The central practical step is to model the shared input once and carry it through the quantities being compared.
Let c0 = 100 mg L⁻¹ with standard uncertainty u(c0) = 2 mg L⁻¹, a relative standard uncertainty of 2%. First take the dilution factors as exact: dA = 0.10 and dB = 0.020. Then cA = 10 mg L⁻¹ with u(cA) = 0.20 mg L⁻¹, and cB = 2 mg L⁻¹ with u(cB) = 0.04 mg L⁻¹.
Now add a 1% relative standard uncertainty for each dilution factor. For this example, the two preparation contributions are independent of one another and of the stock value. First-order product propagation gives the same relative standard uncertainty for each concentration: √(2² + 1²)% ≈ 2.24%.
| Result | Assigned concentration | Stock component | Independent dilution component | Combined standard uncertainty |
|---|---|---|---|---|
| A | 10 mg L⁻¹ | 2% | 1% | 0.224 mg L⁻¹ |
| B | 2 mg L⁻¹ | 2% | 1% | 0.0447 mg L⁻¹ |
These uncertainties describe knowledge of the assigned concentrations under the model. They do not mean both solutions physically fluctuate in concentration by a new random 2% each time an instrument reads them. The common stock component concerns a shared concentration scale.
Take R = cA/cB. Substitute the preparation relationships before propagating uncertainty: R = (dA c0)/(dB c0) = dA/dB = 5. The common stock factor cancels exactly in this model.
With exact dilution factors, R has no uncertainty contribution from c0. With the independent 1% dilution uncertainties, its relative standard uncertainty is √(1² + 1²)% ≈ 1.41%. Thus u(R) ≈ 0.0707. This statement does not claim that all physical effects cancel; it identifies the cancellation built into the stated model.
If you treated cA and cB as independent values with relative uncertainties of 2.24% each, you would obtain √(2.24² + 2.24²)% ≈ 3.16%. That wrongly counts the shared stock scale as if it could independently push one concentration up and the other down.
The covariance form reaches the same answer. Each relative variance is 0.02² + 0.01²; their shared relative covariance contribution is 0.02². For the ratio, subtract twice that shared contribution: 0.0005 + 0.0005 − 2 × 0.0004 = 0.0002. Its square root is about 0.0141.
The sign comes from the question. A common upward scale change increases both the numerator and denominator and leaves their ratio unchanged. For an absolute concentration, that same scale change remains important.
Suppose B is made from A instead of directly from the stock. Write cA = c0 d1 and cB = c0 d1 d2. Then cB/cA = d2. Both the stock and the first dilution factor cancel from this ratio.
For the absolute value of cB, those factors have not vanished. Its uncertainty still depends on c0, d1 and d2, with their dependence structure. A precise ratio between adjacent stages can coexist with a substantially uncertain absolute concentration.
Cancellation also requires the same input. If A and B come from independently characterised stock preparations, their stock uncertainties need not be the same shared contribution. If the stocks are separate bottles from one certified batch, some components may still be common. Bottle count alone cannot settle the question.
Similarly, using the same pipette can introduce a shared calibration component alongside independent dispensing variation. Treating each transfer as wholly independent may miss that relationship. Do not assume complete correlation either: separate the documented components and model their effects.
Imagine all assigned standard concentrations share an unknown multiplicative scale factor. Their points can still lie almost perfectly on a straight signal-versus-concentration line. A high coefficient of determination describes closeness to that fitted pattern; it does not independently establish the correctness of the common concentration scale.
A fitted sample concentration can inherit uncertainty from the standards used to establish the scale. Adding more points from the same stock can improve some fitted quantities while leaving that shared component. The measurement model must connect the sample estimate to the calibration inputs.
A separate check material can test a different part of the chain only if its relevant uncertainty is sufficiently independent and its behaviour is suitable for the method. A check made from the same stock may confirm preparation or instrument consistency while sharing the very scale error you hoped to detect.
This is distinct from extraction recovery or matrix response. Those effects concern what reaches or influences the measurement. The present question is which assigned concentrations share a common source of uncertainty. A complete method may need to account for all of them without double-counting.
Changed-context check: average four preparations
Prepare four nominally equal standards from one stock. Suppose the stock contributes 2% relative standard uncertainty, and each independent preparation contributes 1%. Assume equal weighting, no additional shared effects and small errors. The average has relative standard uncertainty √[2² + (1/√4)²]% ≈ 2.06%.
Dividing the whole single-standard uncertainty, 2.24%, by √4 would give about 1.12%. That is incorrect here because only the independent preparation contribution averages down. The common stock component remains.
Changed-context check: change the question
If the only question is the concentration ratio of two dilutions with exact factors from the same stock, does the stock’s 2% uncertainty limit that ratio? Under the stated relationship, no. If the question is the absolute concentration of either dilution, yes. Name the reported quantity before choosing an uncertainty calculation.
For an evidence check, retain stock identity and certificate basis, the dilution tree, calibration information, independent preparation variation, any common temperature or volume effects, and the equation for the final result. Distinguish standard uncertainty from an expanded uncertainty stated with a coverage factor.
Does correlation mean the measurements were manipulated?
No. Here it means that uncertainty components are related because results share an input. It is a property of the measurement model, not an accusation about the data.
Can I simply remove every shared uncertainty?
No. Shared components may cancel, reinforce or partly cancel depending on the function being calculated. The stock scale cancels from the stated ratio but remains in an absolute concentration or the average of concentrations.
Does a stock certificate report standard uncertainty?
Check the certificate. An expanded uncertainty may be accompanied by a coverage factor. Convert and interpret it using the stated basis before combining components; do not assume every printed ± value is one standard deviation.
Does a perfect calibration line establish traceability?
No. Traceability concerns the documented relationship to reference values and their uncertainties. Straightness of the response plot answers a different question.
Start with a family tree of assigned concentrations. Then ask whether the result is an absolute concentration, a ratio, a difference or an average. The same shared input can behave differently in each calculation.
Try rebuilding the ratio example with another stock concentration. The answer stays at five because the stock factor cancels. Then recalculate the two absolute concentrations and watch that common scale return. That contrast is the understanding a memorised square-root formula cannot supply on its own.
Subject and learning map
The Gold Standard of Chemistry — the connected subject guide.
Education: learning that transfers into life — explain the relationship again when the setting changes.
How X Works — continue through the wider subject library.
Concentration and the calibration chain
Serial dilution foundations — review why stepwise concentration factors multiply.
Spectrophotometry and calibration curves — connect concentration assignment with detector response.
Standard addition with changing spike volume — inspect another calculation whose volume model matters.
Other sources of analytical uncertainty
Recovery corrections and uncertainty — carry a justified recovery factor into a reported result.
Spike timing and matrix response — separate extraction loss from response changes.
Commutability of calibration materials — check whether a material behaves like the samples it represents.
Authoritative sources and scope
Sources checked on 21 October 2026. All numerical cases in this article are original, simplified teaching examples. They are not reported experimental results or official economic statistics.
Eurachem/CITAC: Quantifying Uncertainty in Analytical Measurement, third edition — covariance, calibration uncertainty and shared stock effects; see sections 8.2 and Appendix E.
NIST TN 1297: Law of Propagation of Uncertainty — sensitivity coefficients and covariance in a first-order model.
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