## 7. Reporting Uncertainty

7.1 The stated NIST policy regarding reporting uncertainty is (see Appendix C):

Report U together with the coverage factor k used to obtain it, or report uc.

When reporting a measurement result and its uncertainty, include the following information in the report itself or by referring to a published document:

• A list of all components of standard uncertainty, together with their degrees of freedom where appropriate, and the resulting value of uc. The components should be identified according to the method used to estimate their numerical values:
1. those which are evaluated by statistical methods,
2. those which are evaluated by other means.

• A detailed description of how each component of standard uncertainty was evaluated.

• A description of how k was chosen when k is not taken equal to 2.

It is often desirable to provide a probability interpretation, such as a level of confidence, for the interval defined by U or uc. When this is done, the basis for such a statement must be given.

7.2 The NIST requirement that a full description of what was done be given is in keeping with the generally accepted view that when reporting a measurement result and its uncertainty, it is preferable to err on the side of providing too much information rather than too little. However, when such details are provided to the users of NIST measurement results by referring to published documents, which is often the case when such results are given in calibration and test reports and certificates, it is imperative that the referenced documents be kept up-to-date so that they are consistent with the measurement process in current use.

7.3 The last paragraph of the NIST policy on reporting uncertainty (see subsection 7.1 above) refers to the desirability of providing a probability interpretation, such as a level of confidence, for the interval defined by U or uc. The following examples show how this might be done when the numerical result of a measurement and its assigned uncertainty is reported, assuming that the published detailed description of the measurement provides a sound basis for the statements made. (In each of the three cases, the quantity whose value is being reported is assumed to be a nominal 100 g standard of mass ms.)

ms = (100.021 47 ± 0.000 70) g, where the number following the symbol ± is the numerical value of an expanded uncertainty U = kuc, with U determined from a combined standard uncertainty (i.e., estimated standard deviation) uc = 0.35 mg and a coverage factor k = 2. Since it can be assumed that the possible estimated values of the standard are approximately normally distributed with approximate standard deviation uc, the unknown value of the standard is believed to lie in the interval defined by U with a level of confidence of approximately 95 percent.

ms = (100.021 47 ± 0.000 79) g, where the number following the symbol ± is the numerical value of an expanded uncertainty U = kuc, with U determined from a combined standard uncertainty (i.e., estimated standard deviation) uc = 0.35 mg and a coverage factor k = 2.26 based on the t-distribution for ν = 9 degrees of freedom, and defines an interval within which the unknown value of the standard is believed to lie with a level of confidence of approximately 95 percent.

ms = 100.021 47 g with a combined standard uncertainty (i.e., estimated standard deviation) of uc = 0.35 mg. Since it can be assumed that the possible estimated values of the standard are approximately normally distributed with approximate standard deviation uc, the unknown value of the standard is believed to lie in the interval ms ± uc with a level of confidence of approximately 68 percent.

When providing such probability interpretations of the intervals defined by U and uc, subsection 5.5 should be recalled. In this regard, the interval defined by U in the second example might be a conventional confidence interval (at least approximately) if all the components of uncertainty areobtained from Type A evaluations.

7.4 Some users of NIST measurement results may automatically interpret U = 2uc and uc as quantities that define intervals having levels of confidence corresponding to those of a normal distribution, namely, 95 percent and 68 percent, respectively. Thus, when reporting either U = 2uc or uc, if it is known that the interval which U = 2uc or uc defines has a level of confidence that differs significantly from 95 percent or 68 percent, it should be so stated as an aid to the users of the measurement result. In keeping with the NIST policy quoted in subsection 6.5, when the measure of uncertainty is expanded uncertainty U, one may use a value of k that does lead to a value of U that defines an interval having a level of confidence of 95 percent if such a value of U is necessary for a specific application dictated by an established and documented requirement.

7.5 In general, it is not possible to know in detail all of the uses to which a particular NIST measurement result will be put. Thus, it is usually inappropriate to include in the uncertainty reported for a NIST result any component that arises from a NIST assessment of how the result might be employed; the quoted uncertainty should normally be the actual uncertainty obtained at NIST.

7.6 It follows from subsection 7.5 that for standards sent by customers to NIST for calibration, the quoted uncertainty should not normally include estimates of the uncertainties that may be introduced by the return of the standard to the customer's laboratory or by its use there as a reference standard for other measurements. Such uncertainties are due, for example, to effects arising from transportation of the standard to the customer's laboratory, including mechanical damage; the passage of time; and differences between the environmental conditions at the customer's laboratory and at NIST. A caution may be added to the reported uncertainty if any such effects are likely to be significant and an additional uncertainty for them may be estimated and quoted. If, for the convenience of the customer, this additional uncertainty is combined with the uncertainty obtained at NIST, a clear statement should be included explaining that this has been done.

Such considerations are also relevant to the uncertainties assigned to certified devices and materials sold by NIST. However, well-justified, normal NIST practices, such as including a component of uncertainty to account for the instability of the device or material when it is known to be significant, are clearly necessary if the assigned uncertainties are to be meaningful.