Measurement uncertainty calculator: free uncertainty budget

What the calculator does

It builds the uncertainty budget that NABL 141 clause 3.7.1 describes: for each source of uncertainty, the standard uncertainty, the sensitivity coefficient, the contribution to the result and the degrees of freedom. From those it works out the combined standard uncertainty u_c, the effective degrees of freedom, the coverage factor k and the expanded uncertainty U. It rounds U and the result the way NABL 141 and the GUM ask, and writes the explanatory note in the wording of NABL 141 clauses 3.6.1 and 3.6.2. The CSV download opens in Excel, so you can keep the budget with the method's validation records.

Measurement uncertainty calculation formula, step by step

Step 1: Type A, from repeat results

For n results x₁ to x_n, the tool works out the mean, the standard deviation s and the standard uncertainty:

s = √[ Σ(x_i − mean)² / (n − 1) ]    u = s / √n    degrees of freedom = n − 1

Which one to use depends on what the report gives (NABL 141 3.3.2; GUM 4.2):

• The report gives the mean of these n results, as with the mean of three cubes: use u = s/√n.
• The report gives a single determination, and your repeats are a precision study, such as ten replicates of a check sample on different days: use u = s. Eurachem/CITAC gives the same example: a result with an observed standard deviation of 4.1 mg/L has a standard uncertainty from precision of 4.1 mg/L.

The n in s/√n is the number of results averaged on the report. If your s comes from a precision study and the report gives the mean of duplicates, the standard uncertainty is s/√2 (GUM 4.2.4). Work that out, and enter it as a calibration-certificate contribution with k = 1 and the study's degrees of freedom.

Step 2: Type B, from certificates, tolerances and resolution

Why half the resolution: a display that reads to 0.1 shows the same figure for anything within 0.05 either side of it. So the half-width a is 0.05, not 0.1. NABL 141's own example is a thermometer reading to 0.1 °C: a = 0.05 °C and u = 0.05/√3 = 0.029 °C. In the calculator you type the step (0.1) and it halves it.

Step 3: Put every contribution in the same terms

Before contributions can be combined, each must be expressed as an uncertainty in the result. The sensitivity coefficient c_i does that.

• A value in its own unit is multiplied by c_i, the change in the result per unit change in that input. For a load resolution in kN on a 150 mm cube, c_i = 1000 N per kN ÷ 22 500 mm² = 0.0444 N/mm² per kN. If the value is already in the unit of the result, c_i = 1.
• A value in % is a relative uncertainty. When the result is calculated only by multiplying and dividing, as in strength = load ÷ area or concentration = titre × normality × factor ÷ volume, relative standard uncertainties combine directly (Eurachem/CITAC QUAM 8.2.6, rule 2; NABL 141 4.7.1.2). Then c_i = 1, or 2 for a quantity that is squared in the formula. The calculator turns each % into the unit of the result by multiplying it by the result, so % and absolute contributions can sit in the same budget.

Do not use a % for an input that is added or subtracted in the formula, such as a blank titre. A % of the blank is not a % of the result. Convert it to the unit of the result first.

Step 4: Combined standard uncertainty

u_c = √[ Σ (c_i · u(x_i))² ]

This is the law of propagation of uncertainty for independent inputs (NABL 141 3.4; GUM 5.1). The budget also shows each contribution's share of u_c², which tells you where improvement is worth the effort.

Step 5: Effective degrees of freedom and coverage factor

ν_eff = u_c⁴ / Σ [ (c_i · u(x_i))⁴ / ν_i ]

This is the Welch–Satterthwaite formula (NABL 141 Appendix B; GUM G.4.1). Type A contributions have ν = n − 1. Type B contributions are taken as infinite, which drops them from the sum, unless you set a value (NABL 141 B.3; GUM G.4.3). If you judge a Type B figure reliable to only about 25 %, the GUM gives ν = 8; for about 50 %, ν = 2 (GUM G.4.2).

NABL 141 uses k = 2 for about 95 % when the effective degrees of freedom are “of significant size”. When they are not, Appendix B takes k from the t-distribution at 95.45 %, with ν_eff truncated to the next lower integer. The calculator gives the same values as NABL 141 Table B.1, for example k = 2.87 for 4 degrees of freedom and 2.28 for 10. Eurachem/CITAC notes that k = 2 may be too small below about six degrees of freedom (QUAM 8.3.3).

Step 6: Expanded uncertainty and rounding

U = k · u_c    report y ± U

U is given to at most two significant figures, and the result is rounded to the same last digit as U (NABL 141 3.6.3; NABL 143 6.3; GUM 7.2.6). Round only after all calculations are finished. NABL 141 adds that if rounding lowers U by more than 5 %, you should round up. With two significant figures, ordinary rounding never lowers U by more than about 4.8 %, so that rule only matters if you round to one figure.

Worked example: compressive strength of concrete cubes

A set of three 150 mm cubes is tested at 28 days. The strengths are 33.0, 34.0 and 33.4 N/mm², and the report gives their mean. The cube area is 150 × 150 = 22 500 mm². Three contributions are considered:

• Repeatability. Mean 33.467 N/mm², s = 0.5033 N/mm², u = s/√3 = 0.2906 N/mm², with 2 degrees of freedom.
• CTM calibration. The certificate gives U = 0.5 % of the load at k = 2, so u = 0.25 %. Strength is load ÷ area, so the relative uncertainty passes straight to the result: 0.25 % of 33.467 = 0.08367 N/mm².
• CTM display resolution. The load reads to 0.1 kN, so u = 0.1/(2√3) = 0.02887 kN. With c_i = 0.0444 N/mm² per kN, that is 0.00128 N/mm².

The effective degrees of freedom are ν_eff = 0.3024⁴ / (0.2906⁴ / 2) = 2.35.

Repeatability from only three cubes makes up 92 % of u_c², and it has only 2 degrees of freedom. Eurachem/CITAC warns that k = 2 may be too small in exactly this case, which is why NABL 141 Appendix B reads k from the t-distribution instead. The calculator flags it. A better fix than a large k is a better estimate of the scatter: a pooled standard deviation from many past sets has many degrees of freedom and brings k back close to 2 (GUM 4.2.4).

A full budget for cube strength would also cover the measurement of the cube dimensions, which give the area, and any other effect your procedure identifies. They are left out here to keep the example short. Press Reset to example in the calculator to load these figures.

When must a test report show measurement uncertainty?

A testing lab must evaluate measurement uncertainty (ISO/IEC 17025 clause 7.6.3), but it does not have to print it on every report. Clause 7.8.3.1 c, as reproduced in NABL 220, asks for it on test reports

“where applicable, the measurement uncertainty presented in the same unit as that of the measurand or in a term relative to the measurand (e.g. percent) when it is relevant to the validity or application of the test results; a customer's instruction so requires, or the measurement uncertainty affects conformity to a specification limit”.

Two notes to clause 7.6.3 matter in practice. If a well-recognised test method specifies limits to the major sources of uncertainty and how to present results, following the method satisfies 7.6.3. And once the uncertainty of a method has been established and verified, it need not be evaluated for each result, provided the lab can show that the critical influencing factors are under control. For what else a report must contain, and how uncertainty enters a statement of conformity, see our NABL test report format guide.

Common mistakes in uncertainty budgets

• Dividing by √n when the report gives a single result, or not dividing when it gives a mean. The n is the number of results averaged on the report.
• Using a certificate's U as if it were u. Divide by the k printed on the certificate.
• Taking the whole resolution step as the half-width. A display reading to 0.1 gives a = 0.05.
• Adding a % to a figure in mg/L or N/mm². Convert one to the other first, as the calculator does.
• Counting an effect twice. If your repeats were done by different analysts on different days, analyst and day-to-day variation is already in s. Do not add it again as a Type B term (NABL 141 3.3.3.11).
• Using k = 2 when a few repeats dominate the budget, as in the cube example.
• Too many digits, such as U = 0.6048 N/mm², or a result not rounded to the last digit of U.
• One absolute uncertainty for a wide range of results, when uncertainty grows with the level. State it as a % or by range (QUAM 8.2.9).
• Leaving out major sources, such as sampling, recovery or the cube dimensions, because they are hard to estimate.

Assessors look for these. Our guide to common NABL non-conformities covers uncertainty findings with the rest.

What this calculator does not do

• It does not replace a validated method or your lab's own documented procedure for measurement uncertainty. The answer is only as good as the contributions you enter, and choosing them is the hard part.
• It assumes the contributions are independent. It does not handle correlated inputs (GUM 5.2).
• It does not derive sensitivity coefficients from your formula. You supply c_i, or use % values for a result that only multiplies and divides.
• It does not do top-down estimates from validation and proficiency testing data (ISO 21748), asymmetric intervals or Monte Carlo simulation.
• It does not apply a decision rule or say whether a result complies with a limit.
• It is not reviewed or approved by NABL. NABL accredits laboratories, not tools or software.

Where Tanush LIMS fits

Tanush LIMS does not calculate measurement uncertainty today. It keeps the duplicate results, RPD and run history the estimate needs: the worksheet records each replicate and shows the RPD of duplicates, and every earlier run is kept when a test is re-analysed. An MU module can be built for your lab. See Tanush LIMS for NABL labs or book a demo.

Sources

• JCGM, JCGM 100:2008, Evaluation of measurement data: Guide to the expression of uncertainty in measurement (GUM), clauses 4.2, 4.3, 5.1, 7.2.6, F.2.2.1 and G.4
• NABL, NABL 141, Guidelines for Estimation and Expression of Uncertainty in Measurement, Issue 04, 10 February 2020
• NABL, NABL 143, Policy on CMC and Measurement Uncertainty in Calibration, Issue 04, 26 March 2021, clause 6.3
• NABL, NABL 220, Document Review Checklist (as per ISO/IEC 17025:2017), Issue 02, Amendment 02, 22 September 2026, clauses 7.6 and 7.8.3.1
• Eurachem/CITAC, Quantifying Uncertainty in Analytical Measurement, third edition (QUAM:2012.P1), clauses 8.2, 8.3 and 9.5