Reporting Uncertainty

9. Reporting Uncertainty

• state the estimated number of degrees of

9.1. General

freedom for the standard uncertainty of each

9.1.1. The information necessary to report the input value (methods for estimating degrees result of a measurement depends on its intended

of freedom are given in the ISO Guide use. The guiding principles are:

[H.2]).

• present sufficient information to allow the

NOTE: Where the functional relationship is

result to be re-evaluated if new information

extremely complex or does not exist

or data become available explicitly (for example, it may only exist as a

computer program), the relationship may be

• it is preferable to err on the side of providing

described in general terms or by citation of

too much information rather than too little.

appropriate references. In such cases, it must be clear how the result and its uncertainty

9.1.2. When the details of a measurement,

were obtained.

including how the uncertainty was determined, depend on references to published

9.2.4. When reporting the results of routine documentation, it is imperative that the

analysis, it may be sufficient to state only the documentation to hand is kept up to date and

value of the expanded uncertainty and the value consistent with the methods in use.

of k.

9.2. Information required

9.3. Reporting standard uncertainty

9.2.1. A complete report of a measurement result

should include or refer to documentation

9.3.1. When uncertainty is expressed as the containing,

combined standard uncertainty uc (that is, as a • a description of the methods used to

single standard deviation), the following form is calculate the measurement result and its

recommended:

uncertainty from the experimental "(Result): x (units) [with a] standard uncertainty observations and input data

of uc (units) [where standard uncertainty is as • defined in the International Vocabulary of Basic

the values and sources of all corrections and and General terms in Metrology, 2nd ed., ISO constants used in both the calculation and 1993 and corresponds to one standard the uncertainty analysis

deviation.]"

• a list of all the components of uncertainty

NOTE The use of the symbol ± is not recommended

with full documentation on how each was

when using standard uncertainty as the

evaluated

symbol is commonly associated with

9.2.2. The data and analysis should be presented

intervals corresponding to high levels of

in such a way that its important steps can be confidence. readily followed and the calculation of the result

Terms in parentheses [] may be omitted or repeated if necessary.

abbreviated as appropriate.

9.2.3. Where a detailed report including

EXAMPLE:

intermediate input values is required, the report

Total nitrogen: 3.52 %w/w

should • Standard uncertainty: 0.07 %w/w *

give the value of each input value, its standard uncertainty and a description of

*Standard uncertainty corresponds to one

how each was obtained

standard deviation.

• give the relationship between the result and

9.4. Reporting expanded uncertainty

the input values and any partial derivatives, covariances or correlation coefficients used

9.4.1. Unless otherwise required, the result x to account for correlation effects

should be stated together with the expanded uncertainty U calculated using a coverage factor

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Quantifying Uncertainty Reporting Uncertainty

k =2 (or as described in section 8.3.3.). The • The limits may have been set with some following form is recommended:

allowance for measurement uncertainties. "(Result): (x ± U) (units)

Consideration should be given to both factors in any assessment. The following paragraphs give

[where] the reported uncertainty is [an expanded examples of common practice. uncertainty as defined in the International

Vocabulary of Basic and General terms in

9.6.2. Assuming that limits were set with no metrology, 2nd ed., ISO 1993,] calculated using

allowance for uncertainty, four situations are

a coverage factor of 2, [which gives a level of apparent for the case of compliance with an confidence of approximately 95%]"

upper limit (see Figure 2):

Terms in parentheses [] may be omitted or

i) The result exceeds the limit value plus the abbreviated as appropriate. The coverage factor

expanded uncertainty.

should, of course, be adjusted to show the value

ii) The result exceeds the limiting value by less actually used. than the expanded uncertainty.

EXAMPLE:

iii) The result is below the limiting value by

Total nitrogen: (3.52 ±

0.14) %w/w *

less than the expanded uncertainty

*The reported uncertainty is an expanded

iv) The result is less than the limiting value

uncertainty calculated using a coverage factor

minus the expanded uncertainty.

of 2 which gives a level of confidence of approximately 95%.

Case i) is normally interpreted as demonstrating clear non-compliance. Case iv) is normally interpreted as demonstrating compliance. Cases

ii) and iii) will normally require individual

9.5. Numerical expression of results

consideration in the light of any agreements with

9.5.1. The numerical values of the result and its the user of the data. Analogous arguments apply uncertainty should not be given with an

in the case of compliance with a lower limit. excessive number of digits. Whether expanded

9.6.3. Where it is known or believed that limits uncertainty U or a standard uncertainty u is

have been set with some allowance for given, it is seldom necessary to give more than

uncertainty, a judgement of compliance can two significant digits for the uncertainty. Results

reasonably be made only with knowledge of that should be rounded to be consistent with the

allowance. An exception arises where uncertainty given.

compliance is set against a stated method operating in defined circumstances. Implicit in

9.6. Compliance against limits

such a requirement is the assumption that the

9.6.1. Regulatory compliance often requires that uncertainty, or at least reproducibility, of the

a measurand, such as the concentration of a toxic stated method is small enough to ignore for substance, be shown to be within particular

practical purposes. In such a case, provided that limits. Measurement uncertainty clearly has

appropriate quality control is in place, implications for interpretation of analytical

compliance is normally reported only on the results in this context. In particular:

value of the particular result. This will normally

be stated in any standard taking this approach. • The uncertainty in the analytical result may

need to be taken into account when assessing compliance.

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Quantifying Uncertainty Reporting Uncertainty

Upper Control Limit

Result plus

Result

Result below

Result minus

uncertainty

above limit

limit but limit

uncertainty

above limit

but limit

within

below limit

Figure 2: Uncertainty and compliance limits

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Quantifying Uncertainty Appendix A. Examples