Why Two Laboratories Disagree by More Than Their Methods

A certificate reports 99.2%. The specification says 98.0 to 102.0%. The batch is compliant by a wide margin and nobody asks a further question. Now report 98.3% on an impurity at 0.48% against a 0.50% limit, and the margin is gone. The number that decides whether the batch is acceptable at that moment is not on the certificate.

That number is the measurement uncertainty: an interval, at a stated confidence, within which the true value is expected to lie. Every result has one. Most results do not print it.

Uncertainty is a budget, not a default

The figure is assembled from contributions, each estimated separately and combined in quadrature. The exercise is unglamorous and it usually produces a surprise: the term that dominates is rarely the instrument.

Contribution	Typical relative size	How it is estimated
Reference standard purity	0.5–1.0%	From the certificate's assigned value and its interval
Sample homogeneity	0.2–1.5%	Replicate determinations on separate portions
Weighing	0.05–0.3%	Balance calibration and minimum weight
Volumetric dilution	0.1–0.4%	Glassware tolerance, temperature, operator
Calibration curve	0.2–0.8%	Residuals across the working range
Injection repeatability	0.1–0.3%	Replicate injections of one solution

The table is illustrative and the ranges are wide on purpose: they vary with concentration, with the balance, and with whether the standard is a pharmacopoeial reference material or an in-house qualified one. What does not vary is the ordering. The anchor is usually the largest single term, and the instrument is usually not it.

Why the reference standard dominates

An assay is a comparison. The sample response is divided by the standard response, and both are multiplied by the standard's assigned purity. If that assigned purity is 99.5% with an interval of plus or minus one percent, then one percent of uncertainty enters the result directly, regardless of how good the chromatography is.

This is why two competent laboratories can disagree by more than either one's repeatability suggests. They are not disagreeing about the sample. They are disagreeing about the anchor, or about the interval attached to it, and the interval is often inherited from a certificate rather than calculated from data.

Expanded, standard, and the factor of two

Contributions are combined as standard uncertainties, then multiplied by a coverage factor to give an expanded uncertainty at a confidence level. A factor of two corresponds roughly to ninety-five percent confidence for a normal distribution. So a combined standard uncertainty of 1.0% becomes an expanded uncertainty of 2.0%, and that is where the familiar figure comes from.

The factor is not a fudge. It is a statement about how often the interval is expected to contain the true value. Quoting an expanded uncertainty without the factor and the confidence level is quoting half a sentence.

Decision rules, and the guard band

Once the uncertainty is known, the question becomes how to use it at the specification limit. Two rules are common.

Shared risk. Accept any result inside the limit, and accept that roughly half of the results sitting exactly at the limit belong to material that is marginally outside. Simple, and it puts the risk on the customer.

Guard band. Accept only results that are inside the limit by at least the uncertainty. Material must be comfortably conforming to be accepted. It protects the customer and rejects some conforming material, which is a cost the supplier carries.

Neither is wrong. What is wrong is not having chosen. A quality agreement that states a limit and no decision rule leaves the borderline case to be negotiated after the result exists, which is the worst possible time to negotiate anything.

Where this connects to the rest of the file

Weighing uncertainty is a separate budget, and it is the one that catches small-scale work: a twenty milligram sample on a balance with a hundredth-of-a-milligram readout carries a relative uncertainty that a two hundred milligram sample does not. That arithmetic is set out in the minimum weight note.

Sampling uncertainty sits outside the laboratory entirely and is often larger than everything inside it. A result from a single increment of a two hundred kilogram lot describes that increment. The sampling plan is what converts it into a statement about the lot, and if the plan is a scoop from the top of the drum, no laboratory budget can rescue the conclusion.

There is also the arithmetic trap at the other end. An assay of 100.4% is not automatically an error: it is a result whose uncertainty comfortably contains 100, combined with a reference standard whose assigned purity is itself an estimate. The note on results above 100% works through the terms.

What a certificate should carry

At minimum, a statement of the decision rule applied at the limit. Better, the expanded uncertainty for the method, with the coverage factor and the confidence level. Best, the budget's dominant terms named, so that a customer comparing two suppliers can see whether a difference is an anchor difference or a sample difference.

We describe our results as estimates with an interval, and we are explicit about which terms dominate, because a customer running a borderline impurity through their own laboratory will meet the same arithmetic. The framework for building and using the budget is set out in ILAC guidance and the ISO standards catalogue, with the accreditation expectations in EA documents.

The question to ask is not whether the laboratory is accredited. It is: for this method, what is your expanded uncertainty, what dominates it, and at what point inside the specification do you stop calling a result conforming?

An uncertainty statement a buyer can actually use

Three numbers make a result usable: the result itself, the expanded uncertainty at a stated confidence, and the decision rule applied at the limit. A certificate with the first and neither of the others invites the customer to invent the missing two, and customers invent them conservatively, which is how conforming material gets rejected.

The rule can be written in one line: a result is accepted when it lies inside the specification, and where it lies within the expanded uncertainty of a limit, the batch is referred for a confirmatory determination before disposition. That sentence removes the negotiation, because it says what happens next rather than leaving it to whoever answers the email.

Where the arithmetic bites

Consider a related substance with a limit of 0.50% and an expanded uncertainty of 0.04% at the relevant level. A result of 0.48% is inside the limit and inside the uncertainty band that touches it. A result of 0.44% is inside and clear of the band. The two results are two hundredths apart and belong to different decisions.

Without the uncertainty, both are simply conforming, and the batch ships. With the uncertainty and no decision rule, both are argued about. With both, the first is re-tested and the second is released, and the re-test either confirms the result or reveals a preparation problem, which is information worth having either way.

The numbers above are illustrative. The structure is not: a limit, an uncertainty, and a rule that says what happens in the band between them.

How uncertainty propagates into the customer's file

The customer's specification usually carries the same limit, and their incoming test carries its own uncertainty. Two independent determinations, each with an interval, produce a difference whose expected magnitude is the combination of both. That is why a supplier and a customer can both be right and disagree by more than either one's repeatability.

The practical implication is that a specification agreed without a decision rule will produce a disagreement at some point, and the disagreement will be settled commercially. The clause that prevents it is short, and it belongs in the quality agreement rather than in a certificate.

Two mistakes worth naming

Quoting uncertainty as if it were tolerance. An uncertainty of plus or minus two percent does not mean the specification can be widened by two percent. It means the true value may be two percent away from the reported one. Widening the specification on that basis doubles the error rather than accounting for it.

Assuming a better instrument reduces the budget. A newer chromatograph improves the injection term, which is usually the smallest. The dominant terms are the standard's assigned value and the sample's homogeneity, and neither improves when the detector does.

We report results as estimates, state which term dominates the budget for the method in question, and apply a written decision rule at the limit. The conventions for building the budget come from ILAC guidance and the ISO standards catalogue, with accreditation expectations in EA documents, and the arithmetic of a small sample is set out in the minimum weight note.

Why the budget is worth the effort

A budget turns three arguments into one calculation. Whether a batch at the limit is conforming, whether two laboratories can both be right, and whether a specification is achievable by the method that measures it. Each of those questions is currently settled by seniority or by patience, and each of them has a number attached that would settle it in a paragraph.

The effort is a few days of work for a method, repeated when the method changes, and it produces a statement that survives staff turnover because it is written down. That is the real return: not precision for its own sake, but a decision that does not have to be re-made by whoever is answering the email this year.

Frequently asked questions

What is measurement uncertainty?

A quantified interval around a result within which the true value is expected to lie, at a stated confidence level. It is not error, which is a specific deviation from a specific true value, and it is not tolerance, which is what the specification permits.

Where do the contributions come from?

Sample homogeneity and sampling, weighing, volumetric dilution, reference standard purity assignment, calibration curve fit, response repeatability, and occasionally operator and environmental terms. The relative size of each varies with the method, and the point of a budget is to find which one dominates.

Why is the figure often around two percent for an assay?

Because the reference standard's assigned purity typically carries an uncertainty near one percent at high confidence, and that term propagates directly into the sample result. A method can be highly repeatable and still carry a two percent expanded uncertainty if its calibration anchor is uncertain to one percent.

Does an accredited laboratory have to report it?

An accredited laboratory has to have a documented budget and to apply a stated decision rule. Reporting the expanded uncertainty on the certificate is a customer-facing choice, and asking for it is a reasonable request when a result sits near a limit.

What is a guard band?

A narrowing of the acceptance interval by the uncertainty, so that a result is only accepted when it is comfortably inside. It protects the customer at the cost of rejecting some conforming material. The alternative, sharing the risk, accepts borderline results and accepts the corresponding probability of being wrong.

How does this affect a borderline impurity result?

Directly. A related substance at 0.48% against a 0.50% limit is inside the specification by an amount smaller than the method's uncertainty. Whether that counts as conforming depends on the decision rule, and the rule should be agreed before the result arrives, not after.uncertainty budget