How to calculate measurement uncertainty

Nordtest TR 537 calculates uncertainty from the quality control and validation data a laboratory already produces; this is the top-down approach. The steps:

• Define the measurand and the measuring range. Use absolute uncertainty (mg/L) at the low end of the range and relative uncertainty (%) above it.
• For u(Rw), take long-term results of a stable control sample that goes through the same steps as real samples: u(Rw) = s / x̄. Nordtest recommends data covering at least a year, ideally more than 60 results.
• If the control sample skips steps (a ready-made standard solution, for example), add the repeatability from duplicate analyses of routine samples: u(Rw) = √(s_Rw² + s_r²).
• For u(bias), use a certified reference material (CRM), proficiency tests (PT) or spike recovery. In PT, each round's bias is b_i = (result − assigned value) / assigned value × 100; Nordtest recommends at least 6 rounds.
• So that positive and negative biases do not cancel out, take the root mean square: RMS_bias = √(Σb_i² / n). Combine it with the uncertainty of the reference values: u(bias) = √(RMS_bias² + u(Cref)²).
• Combine and expand: u_c = √(u(Rw)² + u(bias)²), U = 2 · u_c.

In PT, u(Cref_i) is the standard uncertainty of the assigned value; when the report gives s_R and the number of participants instead, u(Cref_i) = s_R / √n_lab. With a single CRM measured in at least 5 separate runs, u(bias) = √(bias² + (s_bias / √n)² + u(Cref)²).

The example is nitrate in drinking water by ion chromatography. A control sample at about 25 mg/L was measured 64 times over 14 months: x̄ = 24.86 mg/L, s = 0.46 mg/L. The last six proficiency test rounds are in the table.

The calculation takes six steps:

• u(Rw) = 0.46 / 24.86 × 100 = 1.85%
• RMS_bias = √((2.61² + 0.90² + 1.42² + 2.12² + 1.49² + 1.89²) / 6) = 1.82%
• u(Cref) = (1.10 + 0.95 + 1.40 + 1.05 + 1.20 + 0.90) / 6 = 1.10%
• u(bias) = √(1.82² + 1.10²) = 2.13%
• u_c = √(1.85² + 2.13²) = 2.82%
• U = 2 × 2.82 = 5.64%, rounded up to two significant digits: 5.7%

For a sample measured at 38.4 mg/L, U = 38.4 × 0.057 = 2.19 mg/L; rounded up, the result is reported as 38.4 ± 2.2 mg/L (k = 2). This U holds for the range the data cover, about 10–48 mg/L. You can run the same steps on your own data with the measurement uncertainty Excel template.

In the GUM approach (JCGM 100:2008), a model equation links the result to its inputs; the standard uncertainty of each input is found and combined through sensitivity coefficients. This is the bottom-up approach. The Eurachem/CITAC QUAM guide applies the same method to chemical analysis.

• Write the model equation, for example c = 1000 · m · P / V for a standard solution.
• Calculate Type A components from repeated measurements: if the mean of n readings is used, u = s / √n; if a single reading is used, u = s; the degrees of freedom are ν = n − 1.
• Take Type B components from a certificate, a tolerance or a calibration report and divide according to the distribution (see the table).
• Find each sensitivity coefficient c_i = ∂f / ∂x_i, which shows how much the result changes per unit change of the input. The component's contribution is c_i · u(x_i).
• For independent inputs, combine: u_c = √Σ(c_i · u(x_i))².
• Find the effective degrees of freedom with the Welch–Satterthwaite formula: ν_eff = u_c⁴ / Σ((c_i · u(x_i))⁴ / ν_i). k comes from the t-distribution: k = 2 for a large ν_eff, and k = 2.28 at 95.45% coverage for ν_eff = 10.

An example budget: 250.4 mg of metal with a purity of 0.9995 is dissolved in a 250.0 mL volumetric flask, so c = 1000 × 250.4 × 0.9995 / 250.0 = 1001.1 mg/L.

The sensitivity coefficients come from the model: c_m = c / m = 4.00 (mg/L)/mg, c_P = c / P = 1001.6 mg/L and c_V = −c / V = −4.00 (mg/L)/mL. The combined uncertainty is u_c = √(0.400² + 0.289² + 0.245² + 0.200²) = 0.586 mg/L. Only the filling repeatability has finite degrees of freedom, so ν_eff = 0.586⁴ / (0.200⁴ / 9) ≈ 660 and k = 2.00. U = 2 × 0.586 = 1.17 mg/L; rounded up, the result is 1001.1 ± 1.2 mg/L (k = 2). The share column shows where to start improving: almost half of the uncertainty comes from the weighing.

Nordtest suits routine test methods, and the GUM suits work with a clear model equation. When the control sample goes through the same steps as real samples, Nordtest data already contain day-to-day variation and every step of the analysis. The GUM shows which component dominates.

U is given with at most two significant digits, and the result is rounded to the same decimal place (JCGM 100, 7.2.6). The GUM also gives rounding up as an option where appropriate; a cautious laboratory writes 5.64% as 5.7%.

A report line reads: "Nitrate: 38.4 ± 2.2 mg/L. The expanded uncertainty is stated with a coverage factor k = 2, corresponding to a coverage probability of about 95%." On a test report, uncertainty is included when it matters for the validity or use of the result, when the customer asks for it, or when it affects conformity with a specification limit (ISO/IEC 17025 §7.8.3.1). On a calibration certificate it accompanies every result (§7.8.4.1).

The decision rule says how uncertainty is taken into account when a result is compared with a specification limit. ISO/IEC 17025 requires the rule to be agreed with the customer beforehand (§7.1.3) and stated on the report (§7.8.6). ILAC-G8 describes three common rules; the table compares a result of 47.9 mg/L, with U = 2.8 mg/L, against a specification whose upper limit is 50 mg/L:

With simple acceptance, the probability of a false accept can reach 50% when the result equals the limit. A guard band of w = U keeps it below 2.5% for a one-sided limit and a normal distribution.

Most mistakes come from data that miss part of the routine variation and from mixed units. Run the calculation through these checks:

• Does u(Rw) come from long-term control data? Replicates from a single day make the uncertainty look smaller than it is.
• If the control sample skips sample preparation, has the component from duplicate analyses of routine samples been added?
• Are the biases combined as an RMS, with u(Cref) included? A bias that is not statistically significant still counts as a component.
• Has the certificate's U been divided by its own k to give a standard uncertainty?
• Are u(Rw) and u(bias) in the same form, both in % or both in mg/L?
• Is there a separate absolute U for the low end of the measuring range?
• Does the report state k and the coverage probability?

PureField LIMS is laboratory software that works in line with ISO/IEC 17025 and ISO 15189. In its measurement uncertainty module, uncertainty is handled as a study and the calculation runs inside the application.

• Study types: repeatability, intermediate precision, trueness (CRM replicates, spike recovery or PT results), Nordtest TR 537 and a Eurachem/CITAC GUM budget.
• The study owner sets up the design and invites the people who will measure. Each participant enters their own values, and the statistics are recalculated on the server after every save.
• In a Nordtest study, u(Rw) can come from at least 20 results of a control sample over a chosen period or from an approved intermediate precision study, and u(bias) from an approved trueness study.
• In a GUM budget, the model equation is written in the method's formula language. Contributions are calculated with the Kragten method, ν_eff with the Welch–Satterthwaite formula and k for 95.45% coverage.
• The owner locks the data. An authorised person who did not design the study, lock it or enter values approves it with an e-signature; before approval, the results are recalculated from the record.
• Applying an approved Nordtest or GUM study to its method rounds U up to two significant digits (6.43% becomes 6.5%) and creates a new version of the method. Results entered after that, and their reports, carry this uncertainty with k = 2.
• The study's A4 report shows the design, the participants, every value with who entered it, the measurement date and instrument, the results with their formulas, the evaluation and the approval.