Essentials of expressing measurement uncertainty
This is a
brief summary of the method of evaluating and expressing uncertainty in
measurement adopted widely by U.S. industry, companies in other countries, NIST,
its sister national metrology institutes throughout the world, and many
organizations worldwide. These "essentials" are adapted from NIST
Technical Note 1297 (TN 1297), prepared by B.N. Taylor and
C.E. Kuyatt and entitled Guidelines for Evaluating and Expressing the
Uncertainty of NIST Measurement Results, which in turn is based on the
comprehensive International Organization for Standardization (ISO) Guide to
the Expression of Uncertainty in Measurement. Users requiring more detailed
information may access TN 1297 online, or if
a comprehensive discussion is desired, they may
purchase the ISO Guide.
Background information on the development of the ISO Guide, its
worldwide adoption, NIST TN 1297, and the NIST policy on expressing
measurement uncertainty is given in the section
International and U.S. perspectives on
To assist you in reading these guidelines, you may wish to consult a short
Background information on the development of the ISO Guide, its worldwide adoption, NIST TN 1297, and the NIST policy on expressing measurement uncertainty is given in the section International and U.S. perspectives on measurement uncertainty.
To assist you in reading these guidelines, you may wish to consult a short glossary. Additionally,a companion publication to the ISO Guide, entitled the International Vocabulary of Basic and General Terms in Metrology, or VIM, gives definitions of many other important terms relevant to the field of measurement. Users may also purchase the VIM.
The case of interest is where the quantity Y being measured, called the measurand, is not measured directly, but is determined from N other quantities X1, X2, . . . , XN through a functional relation f, often called the measurement equation:
Included among the quantities Xi are corrections (or correction factors), as well as quantities that take into account other sources of variability, such as different observers, instruments, samples, laboratories, and times at which observations are made (e.g., different days). Thus, the function f of equation (1) should express not simply a physical law but a measurement process, and in particular, it should contain all quantities that can contribute a significant uncertainty to the measurement result.
An estimate of the measurand or output quantity Y, denoted by y, is obtained from equation (1) using input estimates x1, x2, . . . , xN for the values of the N input quantities X1, X2, . . . , XN. Thus, the output estimate y, which is the result of the measurement, is given by
For example, as pointed out in the ISO Guide, if a potential difference V is applied to the terminals of a temperature-dependent resistor that has a resistance R0 at the defined temperature t0 and a linear temperature coefficient of resistance b, the power P (the measurand) dissipated by the resistor at the temperature t depends on V, R0, b, and t according to
Classification of uncertainty components
The uncertainty of the measurement result y arises from the uncertainties u (xi) (or ui for brevity) of the input estimates xi that enter equation (2). Thus, in the example of equation (3), the uncertainty of the estimated value of the power P arises from the uncertainties of the estimated values of the potential difference V, resistance R0, temperature coefficient of resistance b, and temperature t. In general, components of uncertainty may be categorized according to the method used to evaluate them.
method of evaluation of uncertainty by the statistical analysis of series of observations,
Type B evaluation
Representation of uncertainty components
Standard uncertainty: Type A
Standard uncertainty: Type B
Continue to Evaluating uncertainty components: Type A