Measurement Uncertainty

Measurement uncertainty is a non-negative parameter describing the dispersion of quantity values attributed to a measurand—the quantity intended to be measured—on the basis of available information. In metrology, it forms part of a measurement result rather than merely describing an instrument. It may be expressed as a standard deviation or through an interval with a stated coverage probability. Its evaluation accounts for both observed variability and incomplete knowledge of relevant measurement conditions. (jcgm.bipm.org)

Meaning and conceptual distinctions

Uncertainty is distinct from measurement error, the difference between a measured value and a reference value. An error can have a sign; uncertainty characterizes dispersion and cannot be negative. Correcting a recognized systematic effect does not eliminate uncertainty, because the correction itself may be imperfectly known. Closely agreeing repeated readings can therefore coexist with substantial uncertainty arising from a shared calibration correction. (nist.gov)

A widely used framework is the Guide to the Expression of Uncertainty in Measurement (GUM), maintained by the Joint Committee for Guides in Metrology. It provides methods for translating information about input quantities into uncertainty associated with an output quantity. These methods depend on a specified measurement model and the information used to characterize its inputs. (bipm.org)

Uncertainty also supports metrological traceability: relating a measurement result to a reference through a documented, unbroken chain of calibrations, each contributing uncertainty. Traceability alone does not guarantee that uncertainty is sufficiently small for a particular purpose. (jcgm.bipm.org)

Type A and Type B evaluation

The GUM distinguishes two methods of evaluating uncertainty components, not two kinds of physical error. Type A evaluation uses statistical analysis of measurement data. Examples include repeated observations, curve fitting, and analysis of variance. For $n$ independent observations of a stable quantity, with sample standard deviation $s$, the standard uncertainty of the sample mean is

This is the standard error of the mean, not the dispersion of individual readings. The formula requires the independence assumptions underlying it. (nist.gov)

Type B evaluation uses other relevant information, such as calibration certificates, manufacturer specifications, previous measurements, reference data, and experience with instruments or materials. This information is represented by an appropriate probability distribution. If a quantity is modeled by a uniform distribution over an interval with half-width $a$, its standard uncertainty is

A certificate reporting expanded uncertainty $U$ with coverage factor $k$ commonly supplies the standard uncertainty $U/k$. (nist.gov)

Type A is not synonymous with “random,” nor Type B with “systematic.” Both describe evaluation methods, and both yield standard uncertainties that can be combined. Their reliability depends on the quality of the information; a Type A evaluation based on limited data need not be superior to a well-supported Type B evaluation. (nist.gov)

Measurement models and propagation

A measurement model expresses the output quantity as a function of inputs:

The estimated result is $y=f(x_1,\ldots,x_N)$. In a first-order approximation, the combined standard uncertainty is obtained from

The coefficients $c_i$, defined by partial derivatives, describe sensitivity to each input; $u(x_i,x_j)$ denotes covariance. The equation propagates input variances and covariances into the output variance. (nist.gov)

For uncorrelated inputs, the covariance terms vanish, giving a root-sum-of-squares combination. Shared standards or corrections can introduce correlation, so treating all contributions as independent may misstate the uncertainty. An uncertainty budget documents input estimates, uncertainty components, evaluation methods, sensitivity coefficients, and relevant dependencies. (nist.gov)

When first-order propagation is inadequate, propagation of distributions provides an alternative. A Monte Carlo method samples input distributions and evaluates the model repeatedly to approximate the output distribution. Correlated inputs require an appropriate joint distribution. Output estimates, standard uncertainties, and potentially asymmetric coverage intervals can then be calculated. (bipm.org)

Expanded uncertainty and coverage

Expanded uncertainty is commonly defined as

where $k$ is a coverage factor. The interval $y\pm U$ is associated with a stated coverage probability under the adopted assumptions. For an approximately normal distribution and a sufficiently well-known combined standard uncertainty, $k=2$ gives approximately 95% coverage. It does not provide that coverage universally. (nist.gov)

When uncertainty estimates have limited degrees of freedom, coverage factors may be obtained using Student’s t-distribution. A coverage interval is not automatically a confidence interval in the strict statistical sense; its interpretation depends on how uncertainty components and distributions were established. Relative standard uncertainty, $u_c(y)/|y|$, is defined only for nonzero $y$. (itl.nist.gov)

Reporting and illustrative calculation

A report identifies the measured quantity, estimated value, units, and whether the accompanying uncertainty is standard or expanded. For expanded uncertainty, it states the coverage factor and explains any claimed coverage probability. Supporting documentation describes the components and their evaluation sufficiently to make the calculation interpretable. (nist.gov)

As an illustrative calculation, suppose two uncorrelated standard-uncertainty contributions to a mass result are $0.003$ g and $0.004$ g. Their combination is $0.005$ g; with $k=2$, the expanded uncertainty is $0.010$ g. A result could therefore be reported as $100.000$ g with $U=0.010$ g, $k=2$. Any associated coverage statement would still require justified distributional assumptions. (nist.gov)