Building an uncertainty budget for a standard solution: from certificate to mg/mL

The certificate covers the vial, not the solution

A certificate states an assigned value and an expanded uncertainty for the material as supplied: that solid, in that container, under the storage it names, at the time it was characterised. Weigh a portion and dissolve it and you have created a different quantity, the concentration of your solution, which carries its own uncertainty. It is never smaller than the certificate's, and it is usually larger by an amount that depends on your balance and your glassware rather than on the producer.

The failure this prevents is specific. A laboratory copies the plus or minus 0.50 per cent onto the label of a 1 mg/mL working solution. On the preparation below the correct value is 0.58 per cent at the same coverage factor; from a 5 mg portion it is 1.9 per cent. The certificate cannot say which applies, because it does not know how the portion was taken.

The model, in relative terms

For a standard weighed and made to volume, C = m x P / V, with m the mass in milligrams, P the assigned mass fraction as a decimal, V the volume in millilitres and C the result in mg/mL. Multiplication and division mean the relative uncertainties add in quadrature:

(u(C)/C)^2 = (u(m)/m)^2 + (u(P)/P)^2 + (u(V)/V)^2

This is the root-sum-of-squares combination set out in the GUM, and it holds while the three inputs are independent. That condition is where most budgets quietly break.

Component one: the mass taken

Four inputs, only one of which is on the certificate.

Buoyancy is the term usually left out. A balance reports conventional mass, defined against weights of density 8.0 g/cm3 in air of 1.2 mg/cm3. An organic solid near 1.3 g/cm3 occupies more volume than the weights balancing it and has a true mass about 0.09 per cent above the reading. Left uncorrected it is a bias; entered as a rectangular distribution it is a small term; treating the reading as exact is neither. The definition is in NISTIR 6969, and the weighing step carries its own minimum-weight arithmetic.

Component two: the assigned value

The largest term, and the one most often misread. Two distinctions decide it.

Percentage points, or per cent of the value. A certificate reading 99.30 per cent with an expanded uncertainty of 0.50 per cent may mean 0.50 points on the value or 0.50 per cent of it. At 99.30 per cent the difference is 0.0035 points and does not matter; at 50 per cent it is a factor of two.

As supplied, or dried. A value assigned on the material as received includes the water it holds. A value assigned after drying at a stated condition applies to dried material, so weighing as received requires you to apply the loss on drying yourself, which adds a component you must measure. The two bases are not interchangeable, and on a hygroscopic salt the gap reaches 0.5 per cent, larger than every other term here.

Here P = 0.9930 with U = 0.50 points at k = 2, so u(P) = 0.0025 and u(P)/P = 0.252 per cent.

One thing in this slot is not a component: a chromatographic area percentage. Peak-area normalisation is not a mass fraction, so putting it where P belongs introduces a bias of unknown size. Mass balance or qNMR is what produces a value that can occupy this position.

Component three: the volume

A one-mark flask contributes three things: the tolerance it was made to, the temperature you filled it at against the temperature its capacity is defined at, and how reproducibly you stop at the mark.

The tolerance deserves a caution. ISO 1042 defines the capacity of a one-mark flask at 20 degrees C and sets two accuracy classes, so the figure stamped on the glass is an upper bound on the error of any flask that passed, not the error of your flask. ISO 4787 exists because that difference matters: it gives methods for testing an individual instrument and determining its error at one or more points. A tested flask carries a smaller and defensible Type B input; an untested one carries the class tolerance.

Temperature is often dismissed and should not be. Water expands by roughly 0.00021 per degree C, so a flask filled at 28 degrees C holds about 0.17 per cent more volume than one filled at 20 degrees C, some two and a half times the tolerance term.

Combining, and what the total says

Combined relative uncertainty 0.292 per cent, so U = 0.58 per cent at k = 2 and the solution is 0.9930 plus or minus 0.0058 mg/mL.

The shares matter more than the total. The assigned value carries 74 per cent of the variance, the weighing 16 per cent and the flask 10 per cent. A five-place balance and a tested flask have taken the two smaller terms close to their floor. What remains cannot be improved by any balance because it is the certificate: when a method needs a solution better than 0.6 per cent, the lever is a tighter certificate, not a better laboratory.

One pipette, two deliveries

Take a 1:100 dilution done as two 1:10 steps with the same 10 mL pipette. It is tempting to enter the pipette's 0.07 per cent twice and combine in quadrature for 0.10 per cent. That is wrong. The calibration error of one physical pipette is the same error in both steps: if it delivers 0.07 per cent high the first time it does so the second time, and the dilution factor is biased by 0.14 per cent rather than by a random quantity near 0.10. Correlated components add linearly, and the quadrature sum applies only to independent ones.

The test is whether both deliveries come from the same physical instrument. Two pipettes of the same class have independent errors and combine in quadrature; one pipette used twice does not.

Bias is not uncertainty

The GUM requires a correction for every recognised systematic effect, and the uncertainty entered in the budget is the uncertainty of that correction rather than the size of the effect. Four things here are corrections, and entering them as random terms hides them:

• Using a peak-area percentage as the assigned value. Correct it by using an assay value, or do not claim a mass fraction.
• Ignoring the salt or hydrate factor when a concentration is stated on the free base. The conversion factors belong in the equation, not in the uncertainty.
• Drying a standard whose certificate says use as is. That alters the material, not the uncertainty.
• Filling at 28 degrees C and calling the volume 25.00 mL. Control the temperature, or carry the volume at the temperature actually used.

When k = 2 is not enough

Expanded uncertainty is the combined standard uncertainty multiplied by a coverage factor, and k = 2 corresponds to about 95 per cent only when the combined uncertainty is approximately normal and its own uncertainty is negligible. That second condition is a statement about degrees of freedom. Where a large Type B term dominates, the effective degrees of freedom are effectively infinite and k = 2 is sound. Where a Type A term from a handful of replicates dominates it is not: at eight effective degrees of freedom the factor for 95 per cent is 2.3 rather than 2.0, and a budget reporting k = 2 claims coverage it does not have. In a standard-solution budget the Type A terms are usually the small ones, so k = 2 is normally defensible.

What a gravimetric preparation changes

Weighing the diluent instead of filling to a mark removes the volumetric component: 25 g of diluent on a five-place balance carries a relative uncertainty near 0.001 per cent, against 0.093 per cent for the flask. It also changes the unit to mg/g, so the density of the solution never enters the budget, which is why gravimetric reference materials are commonly certified in mg/kg.

What it does not remove is the dominant term. The total falls only from 0.58 to 0.55 per cent, because the same 74 per cent of the variance is still the certificate. A gravimetric preparation buys robustness, not a smaller number.

The budget has a date

Every figure above describes the solution on the day it was made. A stored solution acquires a stability component from adsorption to the container wall, evaporation through a septum or hydrolysis of the analyte, and that component is on no certificate, because the producer characterised a solid and not your liquid. Where a hold time has been established it enters as a Type B term from the study; where it has not, the honest statement is that the uncertainty after storage is unknown. Establishing a hold time is the measurement that converts the second case into the first.

Five questions for a supplier

• Is the stated uncertainty expanded, and at what coverage factor?
• Is the assigned value on an as-supplied or a dried basis, and what is the drying condition?
• Is the purity a mass fraction from an assay, or a chromatographic area percentage?
• Does the certificate state metrological traceability, and to what?
• What is the smallest unit, and does the assigned value apply to the whole unit or to a portion?

ISO 17034:2016 is the standard a producer is assessed against for competence and consistent operation, and it covers the production of all reference materials including certified ones. A certificate that cannot answer them is not evidence that the material is wrong; it is evidence that the uncertainty of your solution cannot be calculated from it.