7   Reporting uncertainty

7.1   General guidance

7.1.1   In general, as one moves up the measurement hierarchy, more details are required on how a measurement result and its uncertainty were obtained. Nevertheless, at any level of this hierarchy, including commercial and regulatory activities in the marketplace, engineering work in industry, lower‑echelon calibration facilities, industrial research and development, academic research, industrial primary standards and calibration laboratories, and the national standards laboratories and the BIPM, all of the information necessary for the re‑evaluation of the measurement should be available to others who may have need of it. The primary difference is that at the lower levels of the hierarchical chain, more of the necessary information may be made available in the form of published calibration and test system reports, test specifications, calibration and test certificates, instruction manuals, international standards, national standards, and local regulations.

7.1.2   When the details of a measurement, including how the uncertainty of the result was evaluated, are provided by referring to published documents, as is often the case when calibration results are reported on a certificate, it is imperative that these publications be kept up‑to‑date so that they are consistent with the measurement procedure actually in use.

7.1.3   Numerous measurements are made every day in industry and commerce without any explicit report of uncertainty. However, many are performed with instruments subject to periodic calibration or legal inspection. If the instruments are known to be in conformance with their specifications or with the existing normative documents that apply, the uncertainties of their indications may be inferred from these specifications or from these normative documents.

7.1.4   Although in practice the amount of information necessary to document a measurement result depends on its intended use, the basic principle of what is required remains unchanged: when reporting the result of a measurement and its uncertainty, it is preferable to err on the side of providing too much information rather than too little. For example, one should

1. describe clearly the methods used to calculate the measurement result and its uncertainty from the experimental observations and input data;
2. list all uncertainty components and document fully how they were evaluated;
3. present the data analysis in such a way that each of its important steps can be readily followed and the calculation of the reported result can be independently repeated if necessary;
4. give all corrections and constants used in the analysis and their sources.

A test of the foregoing list is to ask oneself “Have I provided enough information in a sufficiently clear manner that my result can be updated in the future if new information or data become available?”

7.2   Specific guidance

7.2.1   When reporting the result of a measurement, and when the measure of uncertainty is the combined standard uncertainty u_c(y), one should

1. give a full description of how the measurand Y is defined;
2. give the estimate y of the measurand Y and its combined standard uncertainty u_c(y); the units of y and u_c(y) should always be given;
3. include the relative combined standard uncertainty u_c(y)⁄│y│, │y│ ≠ 0, when appropriate;
4. give the information outlined in 7.2.7 or refer to a published document that contains it.

If it is deemed useful for the intended users of the measurement result, for example, to aid in future calculations of coverage factors or to assist in understanding the measurement, one may indicate

• the estimated effective degrees of freedom v_eff (see G.4);
• the Type A and Type B combined standard uncertainties u_cA(y) and u_cB(y) and their estimated effective degrees of freedom v_effA and v_effB (see G.4.1, Note 3).

7.2.2   When the measure of uncertainty is u_c(y), it is preferable to state the numerical result of the measurement in one of the following four ways in order to prevent misunderstanding. (The quantity whose value is being reported is assumed to be a nominally 100 g standard of mass m_S; the words in parentheses may be omitted for brevity if u_c is defined elsewhere in the document reporting the result.)

1. “m_S = 100,021 47 g with (a combined standard uncertainty) u_c = 0,35 mg.”
2. “m_S = 100,021 47(35) g, where the number in parentheses is the numerical value of (the combined standard uncertainty) u_c referred to the corresponding last digits of the quoted result.”
3. “m_S = 100,021 47(0,000 35) g, where the number in parentheses is the numerical value of (the combined standard uncertainty) u_c expressed in the unit of the quoted result.”
4. “m_S = (100,021 47 ± 0,000 35) g, where the number following the symbol ± is the numerical value of (the combined standard uncertainty) u_c and not a confidence interval.”

   NOTE   The ± format should be avoided whenever possible because it has traditionally been used to indicate an interval corresponding to a high level of confidence and thus may be confused with expanded uncertainty (see 7.2.4). Further, although the purpose of the caveat in 4. is to prevent such confusion, writing Y = y ± u_c(y) might still be misunderstood to imply, especially if the caveat is accidentally omitted, that an expanded uncertainty with k = 1 is intended and that the interval y − u_c(y) ≤ Y ≤ y + u_c(y) has a specified level of confidence p, namely, that associated with the normal distribution (see G.1.3). As indicated in 6.3.2 and Annex G, interpreting u_c(y) in this way is usually difficult to justify.

7.2.3   When reporting the result of a measurement, and when the measure of uncertainty is the expanded uncertainty U = ku_c(y), one should

1. give a full description of how the measurand Y is defined;
2. state the result of the measurement as Y = y ± U and give the units of y and U;
3. include the relative expanded uncertainty U⁄│y│, │y│ ≠ 0, when appropriate;
4. give the value of k used to obtain U [or, for the convenience of the user of the result, give both k and u_c(y)];
5. give the approximate level of confidence associated with the interval y ± U and state how it was determined;
6. give the information outlined in 7.2.7 or refer to a published document that contains it.

7.2.4   When the measure of uncertainty is U, it is preferable, for maximum clarity, to state the numerical result of the measurement as in the following example. (The words in parentheses may be omitted for brevity if U, u_c, and k are defined elsewhere in the document reporting the result.)

“m_S = (100,021 47 ± 0,000 79) g, where the number following the symbol ± is the numerical value of (an expanded uncertainty) U = ku_c, with U determined from (a combined standard uncertainty) u_c = 0,35 mg and (a coverage factor) k = 2,26 based on the t‑distribution for v = 9 degrees of freedom, and defines an interval estimated to have a level of confidence of 95 percent.”

7.2.5   If a measurement determines simultaneously more than one measurand, that is, if it provides two or more output estimates y_i (see H.2, H.3, and H.4), then, in addition to giving y_i and u_c(y_i), give the covariance matrix elements u(y_i,  y_j) or the elements r(y_i,  y_j) of the correlation coefficient matrix (C.3.6, Note 2) (and preferably both).

7.2.6   The numerical values of the estimate y and its standard uncertainty u_c(y) or expanded uncertainty U should not be given with an excessive number of digits. It usually suffices to quote u_c(y) and U [as well as the standard uncertainties u(x_i) of the input estimates x_i] to at most two significant digits, although in some cases it may be necessary to retain additional digits to avoid round‑off errors in subsequent calculations.

In reporting final results, it may sometimes be appropriate to round uncertainties up rather than to the nearest digit. For example, u_c(y) = 10,47 mΩ might be rounded up to 11 mΩ. However, common sense should prevail and a value such as u(x_i) = 28,05 kHz should be rounded down to 28 kHz. Output and input estimates should be rounded to be consistent with their uncertainties; for example, if y = 10,057 62 Ω with u_c(y) = 27 mΩ, y should be rounded to 10,058 Ω. Correlation coefficients should be given with three‑digit accuracy if their absolute values are near unity.

7.2.7   In the detailed report that describes how the result of a measurement and its uncertainty were obtained, one should follow the recommendations of 7.1.4 and thus

1. give the value of each input estimate x_i and its standard uncertainty u(x_i) together with a description of how they were obtained;
2. give the estimated covariances or estimated correlation coefficients (preferably both) associated with all input estimates that are correlated, and the methods used to obtain them;
3. give the degrees of freedom for the standard uncertainty of each input estimate and how it was obtained;
4. give the functional relationship Y = f(X₁, X₂, ..., X_N) and, when they are deemed useful, the partial derivatives or sensitivity coefficients ∂f⁄∂x_i. However, any such coefficients determined experimentally should be given.

   NOTE   Since the functional relationship f may be extremely complex or may not exist explicitly but only as a computer program, it may not always be possible to give f and its derivatives. The function f may then be described in general terms or the program used may be cited by an appropriate reference. In such cases, it is important that it be clear how the estimate y of the measurand Y and its combined standard uncertainty u_c(y) were obtained.