Understanding Type A and Type B Measurement Uncertainty

Type A and Type B describe the evaluation method

Type A and Type B are classifications for *how an uncertainty contribution is evaluated*. They do not classify the source as random or systematic, and they do not indicate whether one contribution is more credible than the other.

This distinction matters because uncertainty budgets often become misleading when Type A is treated as “random uncertainty” and Type B as “systematic uncertainty.” A repeatability study may reveal random variation and be evaluated by Type A. A correction obtained from a reference certificate may address a systematic effect, while the certificate’s stated uncertainty is normally evaluated by Type B. But either category can contain elements of both kinds of measurement behaviour.

The framework comes from the *Guide to the Expression of Uncertainty in Measurement* (GUM). It provides a common way to express different forms of technical evidence as standard uncertainties, so that relevant contributions can be combined consistently.

Type A evaluation: evidence from statistical analysis

A Type A evaluation is based on statistical analysis of a series of observations. The familiar example is repeatability.

Suppose a laboratory calibrates a pressure indicator at 500 kPa using a stable pressure controller and obtains five corrected indication errors, in kPa:

0.8, 0.7, 0.9, 0.8, 0.8

The arithmetic mean is the estimate of the error at that test point. The observed standard deviation, \(s\), describes the dispersion of individual results. If the reported result is the mean of the five observations, the relevant repeatability contribution is usually the standard deviation of the mean:

\[ u(\bar{x}) = \frac{s}{\sqrt{n}} \]

where \(n\) is the number of observations.

This calculation relies on an important operational judgement: the observations must represent the measurement variation relevant to the reported result. Five rapid readings under highly stable conditions may estimate short-term indication repeatability, but they may not represent variation due to operator technique, fixture repositioning, thermal stabilisation, range changes, or separate measurement runs.

Type A evidence can also come from regression analysis, pooled repeatability studies, control-chart data, or method validation data. The statistical model must match the measurand and procedure. A laboratory should not automatically insert a standard deviation merely because a spreadsheet has a field labelled “repeatability.”

The number of observations also matters. A small dataset can provide a valid Type A estimate, but it has limited degrees of freedom. This affects the confidence associated with the estimate and can affect the calculation of an expanded uncertainty where the effective degrees of freedom are low.

Type B evaluation: evidence other than the current statistical series

A Type B evaluation uses technical information other than statistical analysis of the particular set of repeated observations being considered. It is not an estimate made by intuition. A sound Type B evaluation is evidence-based and documented.

Typical sources include:

• stated uncertainty from a calibration certificate for a reference standard;
• manufacturer specifications, where their applicability has been assessed;
• resolution of a digital indicator or scale division of an analogue instrument;
• environmental limits and known sensitivity coefficients;
• published material properties or physical constants;
• prior measurement data, such as a validated stability history;
• limits defined by a controlled procedure.

For example, a digital multimeter reading to 0.01 V may be subject to quantisation uncertainty. If the displayed value is assumed equally likely to lie anywhere within plus or minus half a digit, the limits are \(\pm 0.005\) V. Under a rectangular probability distribution, the standard uncertainty is:

\[ u = \frac{a}{\sqrt{3}} \]

where \(a\) is the half-width of the interval. The standard uncertainty from resolution is therefore \(0.005/\sqrt{3} = 0.0029\) V.

The distribution is not selected because rectangular distributions are convenient. It reflects the available knowledge. A rectangular distribution is commonly appropriate when only upper and lower bounds are justified and values within those bounds are considered equally plausible. Other evidence may support a normal, triangular, U-shaped, or other distribution. The choice should be technically defensible and recorded.

A calibration certificate also requires careful interpretation. If a reference thermometer certificate states an expanded uncertainty of 0.06 °C with coverage factor \(k = 2\), the associated standard uncertainty is normally 0.03 °C, provided the certificate’s stated conditions and traceability are applicable to the laboratory’s use. If no coverage factor or confidence statement is given, the laboratory needs further information before simply dividing by two.

Building a usable uncertainty budget

An uncertainty budget translates the measurement model into a controlled technical record. Each line should identify the input quantity or influence, the estimate where relevant, the uncertainty evaluation method, probability distribution, divisor, sensitivity coefficient, resulting standard uncertainty, and any correlation with other inputs.

For a basic direct comparison calibration, a simplified model might be:

\[ E = I{\mathrm{UUT}} - I{\mathrm{ref}} + C{\mathrm{res}} + C{\mathrm{env}} \]

where \(E\) is the error of the unit under test, \(I{\mathrm{UUT}}\) is its indication, \(I{\mathrm{ref}}\) is the reference value, and the correction terms account for resolution and environmental effects. The budget should reflect the actual procedure, not a generic template copied from another discipline or range.

Once each contribution is expressed as a standard uncertainty and adjusted by its sensitivity coefficient, independent contributions are commonly combined by the root-sum-of-squares method:

\[ uc = \sqrt{\sum (ci u(x_i))^2} \]

This assumes the inputs are uncorrelated. That assumption should be examined, particularly where the same reference certificate, common environmental sensor, correction model, or data set influences more than one term. Ignoring material correlation can understate or overstate the combined uncertainty.

The combined standard uncertainty may then be multiplied by an appropriate coverage factor to report expanded uncertainty. In many routine applications, \(k = 2\) is used to give an interval with an approximately 95% level of coverage, subject to the model, distributions and effective degrees of freedom. It should not be presented as an exact 95% statement without the supporting conditions.

What makes the evaluation defensible

A defensible budget is not defined by the number of entries or by whether every contribution has been labelled Type A or Type B. It is defined by whether it represents the measurement process, uses applicable evidence, and remains valid as conditions change.

Laboratories should review the budget when reference standards are recalibrated, certificates report materially different uncertainties, procedures change, new software or fixtures are introduced, environmental controls are revised, or intermediate-check data identifies instability. A Type B stability allowance based on two years of history may no longer be suitable after repair, transport damage, adjustment, or a change in usage pattern.

Type A and Type B evaluations are complementary tools. The first extracts uncertainty information from relevant observations; the second converts other reliable technical knowledge into a standard uncertainty. Used together, they allow the uncertainty statement on a certificate to be understood as evidence about the reported measurement result rather than as a final administrative calculation.

References

• JCGM 100:2008, *Evaluation of measurement data - Guide to the expression of uncertainty in measurement (GUM 1995 with minor corrections)*, Joint Committee for Guides in Metrology.
• JCGM 200:2012, *International vocabulary of metrology - Basic and general concepts and associated terms (VIM)*, 3rd edition.
• ISO/IEC 17025:2017, *General requirements for the competence of testing and calibration laboratories*, particularly clause 7.6 on evaluation of measurement uncertainty.