Section: 2 | Expression of Uncertainty of Measurements |
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In general, the result of a measurement is only an approximation or estimate of the true value of the quantity subject to measurement, and thus the result is of limited value unless accompanied by a statement of its uncertainty. Much (but not all) of the scientific data appearing in the literature does include some indication of the uncertainty, but this may be stated in many different ways and is often explained poorly. In an effort to encourage consistency in uncertainty statements, the International Committee for Weights and Measures (CIPM) of the Bureau International des Poids et Mesures initiated a project, in collaboration with several other international organizations, to prepare a set of guidelines expressing international consensus on the recommended method of stating uncertainties. This project resulted in the publication of the *Guide to the Expression of Uncertainty in Measurement* (Refs. 1 and 2), which is often referred to as *GUM*. The recommendations of *GUM* have been summarized by the National Institute of Standards and Technology in NIST Technical Note 1297*, Guidelines for Evaluating the Uncertainty of NIST Measurement Results* (Ref. 3).

In the notation of *GUM*, we are concerned with the **measurand**, i.e., the quantity that is being measured. In physics and chemistry this is usually called a **physical quantity **and represents some inherent characteristic of a material, system, or process that can be expressed in numerical terms — specifically as the product of a number and a reference, commonly called a **unit. **Thus, the density of water at room temperature is (approximately) 0.998 g/mL (grams per milliliter) or, alternatively 998 kg m^{–3} (kilograms per meter cubed). This statement gives the most likely value of the measurand, to this level of precision, but gives no information on how much the stated value might differ from the true value. A more detailed discussion of measurement terminology is given in the* International Vocabulary of Metrology* (VIM) (Ref. 4).

It is important to differentiate between the terms **error** and **uncertainty**. The error in a measurement is the difference between the measured value and the true value; the error can be stated if the true value is known (to some level of accuracy). The uncertainty is an estimate of the maximum reasonable extent to which the measured value is believed to deviate from the true value, in a situation where the true value is not known (most often the case). The result of a measurement can unknowably be very close to the true value, and thus have negligible error, even though its uncertainty is large.

The uncertainty of the result of a measurement generally consists of several components, which may be grouped in two types according to the method used to estimate their numerical values:

**Type A. Those which are evaluated by statistical methods**

**Type B. Those which are evaluated by other means**

The terms “random uncertainty” and “systematic uncertainty” are often used, but these terms do not always correspond in a simple way to the A and B categories. This is because the nature of an uncertainty component is conditioned by how the quantity appears in the mathematical model that describes the current measurement process. An uncertainty component arising from a systematic effect may in some cases be evaluated by methods of Type A while in other cases by methods of Type B.

In the *GUM* formulation, each component of uncertainty, whether in the A or B category, is represented by an estimated standard deviation, termed **standard uncertainty**, symbol *u _{i} *, and equal to the positive square root of the estimated variance

For an uncertainty component of Type A, *u _{i} *=

In a similar manner, each uncertainty component of Type B is represented by a quantity *u _{j} *, which is obtained from an assumed probability distribution based on all the available information about the measurement process. Because

- Previous measurement data
- Experience with, or general knowledge of, the behavior and properties of relevant materials and instruments
- Manufacturer’s specifications
- Data provided in calibrations and other reports
- Uncertainties assigned to reference data taken from handbooks

The specific approach to evaluating the standard uncertainty *u _{j} *of a Type B uncertainty will depend on the detailed model of the measurement process. The following are examples of steps that may be used:

- Convert a quoted uncertainty (for example, in a calibration factor) that is a stated multiple of an estimated standard deviation to a standard uncertainty by dividing the quoted uncertainty by the multiplier.
- Convert a quoted uncertainty that defines a “confidence interval” having a stated level of confidence, such as 95% or 99%, to a standard uncertainty by treating the quoted uncertainty as if a normal distribution had been used to calculate it (unless otherwise indicated) and dividing it by the appropriate factor for such a distribution. These factors are 1.960 and 2.576 for the two levels of confidence given.
- Model knowledge of the quantity in question by a normal distribution and estimate lower and upper limits
*a*_{–}*a*_{+}*a*_{+}+*a*_{–})/2 (i.e., the midpoint of the limits) and there is 1 chance out of 2 (i.e., a 50% probability) that the value of the quantity lies in the interval*a*_{–}to*a*_{+}.*u*_{j}_{ }≈ 1.48*a*,*a*= (*a*_{+}–*a*_{–})/2 is the half-width of the interval. - Model knowledge of the quantity in question by a normal distribution and estimate lower and upper limits
*a*_{–}*a*_{+}*a*_{+}+*a*_{–})/2*a*_{–}to*a*_{+}. Then*u*_{j}_{ }≈*a*, where*a*= (*a*_{+}–*a*_{–}*.* - Estimate lower and upper limits
*a*_{–}and*a*_{+}for the value of the quantity in question such that the probability that the value lies in the interval*a*_{–}to*a*_{+}is, for all practical purposes, 100%. Provided that there is no contradictory information, treat the quantity as if it is equally probable for its value to lie anywhere within the interval*a*to_{–}*a*_{+}; that is, model it by a uniform or rectangular probability distribution. The best estimate of the value of the quantity is then (*a*+_{+}*a*_{–})/2 with*u*=_{j }*a*/√3 where*a*= (*a*_{+}–*a*)/2. If the distribution used to model the quantity is triangular rather than rectangular, then_{–}*u*=_{j}*a*/√6. The rectangular distribution is a reasonable default model in the absence of any other information. But if it is known that values of the quantity in question near the center of the limits are more likely than values close to the limits, a triangular or a normal distribution may be a better model.

When all the standard uncertainties of Type A and Type B have been determined in this way, they should be combined to produce the **combined standard uncertainty **(suggested symbol *u*_{c}), which may be regarded as the estimated standard deviation of the measurement result. This process, often called the *law of propagation of uncertainty* or “root-sum-of-squares,” involves taking the square root of the sum of the squares of all the *u _{i} *. In many practical measurement situations, the probability distribution characterized by the measurement result

In fundamental metrological research (involving physical constants, calibration standards, and the like) the combined standard uncertainty *u _{c} *is normally used as the statement of uncertainty in a measurement. In most cases, however, it is desirable to use a measure of uncertainty that defines an interval about the measurement result

It should be noted that the International Union of Pure and Applied Chemistry (IUPAC) is reviewing recommendations on metrological and quality concepts in analytical chemistry (Ref. 10).

- Group the uncertainty components into Type A (can be evaluated by statistical methods) and Type B (must be evaluated by other means).
- Determine the standard uncertainty for each component of Type A by statistical methods and for each component of Type B by other suitable methods, based on modeling the measurement process.
- Take the square root of the sum of the squares of all the standard uncertainties to get the combined standard uncertainty
*u*_{c}. - Specify a coverage factor
*k*which, when multiplied by*u*_{c}, gives the expanded uncertainty*U*. In fundamental metrological research*k*= 1 is usually chosen; in other cases,*k*= 2 (corresponding to a confidence level of about 95%) is the most common choice.

*Evaluation of Measurement Data - Guide to the Uncertainty in Measurement*,*JCGM 100:2008*, BIPM, Sevres, 2008, <bipm.org/utils/common/documents/jcgm/JCGM_1002008_E.pdf>- ISO,
*Guide to the Expression of Uncertainty in Measurement*, International Organization for Standardization, Geneva, Switzerland, 1993. Several supplements have been published; see Bich, W., Cox, M. C., and Harris, P. M., “Evolution of the*Guide to the Expression of Uncertainty in Measurement,*”*Metrologia*43, S161, 2006. [https://doi.org/10.1088/0026-1394/43/4/S01] - Taylor, B. N., and Kuyatt, C. E.,
*Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results*, NIST Technical Note 1297, National Institute of Standards and Technology, Gaithersburg, MD, 1994; available for free download at <physics.nist.gov/cuu/Uncertainty/bibliography.html>. *International Vocabulary of Metrology - Basic and General Concepts and Associated Terms, JCGM 200:2012, Third Edition*, BIPM, Sevres, 2012.- Bell, S.,
*A Beginner’s Guide to Uncertainty of Measurement*, National Physical Laboratory, Teddington, Middlesex, UK, 2001; available on the Internet at <www.npl.co.uk/server.php?show=ConWebDoc.1785>. - Eisenhart, C., “Realistic Evaluation of the Precision and Accuracy of Instrument Calibration Systems,”
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