Methods and instruments: how to combine contributions to measurement uncertainty

From individual components to combined uncertainty

A measurement does not usually determine the exact value of the quantity being examined. Its result is an estimate and, for the description to be complete, it must be accompanied by a quantitative expression of its uncertainty. This is how the National Institute of Standards and Technology of the United States (NIST) frames the subject in its technical note on measurement uncertainty. The document describes a method for combining contributions; it does not report a recent finding. NIST: classification of uncertainty components

The quantity being combined is not just any figure associated with each source: NIST represents each contribution by a standard uncertainty, equivalent to an estimated standard deviation. The result of combining them is called the combined standard uncertainty, with the suggested symbol uc</sub>. NIST describes it as an estimate of the standard deviation of the measurement result. NIST: combined standard uncertainty

Contributions may come from Type A or Type B evaluations. The distinction depends on how their numerical values are estimated: a Type A evaluation uses statistical analysis of series of observations; a Type B evaluation is obtained by other means, based on available information and an assumed probability distribution. By definition, these are not synonyms for “random” and “systematic” uncertainty: NIST cautions that the correspondence is not necessarily simple and that the classification depends on how the quantity in question appears in the mathematical model of the measurement. NIST: classification of uncertainty components

The combination rule and the role of covariances

NIST states that individual standard uncertainties are combined using the usual method for combining standard deviations, known as the law of propagation of uncertainty. In the usual case of combining uncorrelated components, the root sum of squares method, abbreviated RSS, is used: the contributions are squared, added, and the square root is taken. But the general rule also takes covariances into account when appropriate. Therefore, applying RSS without considering relationships between components might not represent the prescribed procedure when those relationships are relevant. NIST: combined standard uncertainty NIST: classification of uncertainty components

The table summarizes the distinctions described by NIST; it is not an exhaustive list of sources of uncertainty or a recipe for evaluating a particular measurement. The excerpt refers to an equation in appendix A, but does not include its expression. Therefore, this summary explains the general principle without reconstructing that equation or providing a numerical calculation. NIST: combined standard uncertainty NIST: classification of uncertainty components

Corrections, systematic effects, and limits on what is combined

For each recognized systematic effect that significantly influences the result, NIST starts from the recommendation to apply a correction—or a correction factor—and associate the standard uncertainty of that correction with it. The correction may be positive, negative, or zero; its uncertainty may be evaluated by Type A or Type B. The important point is that the uncertainty of a correction is not the systematic error caused by the effect. It expresses the uncertainty in the result that remains because the required correction value is not fully known. Confusing “error” with “uncertainty” obscures an essential distinction in the method. NIST: combined standard uncertainty

NIST recommends applying corrections for recognized significant systematic effects, but acknowledges that limited resources may mean this is not always practical. It also notes that every effort should be made to identify these effects. The reported uncertainty should conform to the guidelines as far as possible, even when applying a correction is not practical. This does not turn an unidentified source into a quantified contribution: the passage concerns recognized effects and the uncertainty associated with the corrections considered. NIST: combined standard uncertainty

What the result allows us to interpret

In many practical situations, NIST says, the probability distribution described by the result y and its combined standard uncertainty uc</sub>(y) is approximately normal (Gaussian). To associate the interval y − uc</sub>(y) to y + uc</sub>(y) with an approximate confidence level of 68 %, NIST adds that the combined uncertainty itself must have negligible uncertainty. The distribution is approximately normal when the conditions of the central limit theorem are met: for example, when y is obtained from a significant number of other quantities with appropriate probability distributions, the standard uncertainties of those quantities contribute magnitudes comparable to uc</sub>(y), and the linear approximation implicit in the propagation is adequate. These conditions support the normal approximation; the negligible uncertainty of uc</sub>(y) is an additional requirement for the approximate interpretation of the interval. Therefore, NIST does not present 68 % as an automatic property of every combined uncertainty. NIST: combined standard uncertainty

A statement based on uc</sub> must also be distinguished from a confidence interval in the specific statistical sense. According to NIST, this term is appropriate only when certain conditions are met, including that all components contributing to uc</sub> come from Type A evaluations. The note therefore proposes describing, more generally, the fraction p of the distribution covered by an interval as its coverage probability or confidence level, and making explicit the basis for that interpretation when reporting it. NIST: combined standard uncertainty NIST: reporting uncertainty

When reporting results, the NIST material recommends reporting the expanded uncertainty U together with the coverage factor k used to obtain it, or reporting uc</sub>. It also says the report should include the standard uncertainty components, the resulting value of uc</sub>, how each component was evaluated and, where appropriate, its degrees of freedom; if k is not 2, how it was chosen should be described. If a probabilistic interpretation is offered, its basis should be explained. Combined uncertainty therefore summarizes contributions, but its meaning to someone reading a result also depends on how it is justified and reported. NIST: reporting uncertainty