# 8   Summary of procedure for evaluating and expressing uncertainty

The steps to be followed for evaluating and expressing the uncertainty of the result of a measurement as presented in this Guide may be summarized as follows:

1. Express mathematically the relationship between the measurand Y and the input quantities Xi on which Y depends: Y = f(X1X2, ..., XN). The function f should contain every quantity, including all corrections and correction factors, that can contribute a significant component of uncertainty to the result of the measurement (see 4.1.1 and 4.1.2).
2. Determine xi, the estimated value of input quantity Xi, either on the basis of the statistical analysis of series of observations or by other means (see 4.1.3).
3. Evaluate the standard uncertainty u(xi) of each input estimate xi. For an input estimate obtained from the statistical analysis of series of observations, the standard uncertainty is evaluated as described in 4.2 (Type A evaluation of standard uncertainty). For an input estimate obtained by other means, the standard uncertainty u(xi) is evaluated as described in 4.3 (Type B evaluation of standard uncertainty).
4. Evaluate the covariances associated with any input estimates that are correlated (see 5.2).
5. Calculate the result of the measurement, that is, the estimate y of the measurand Y, from the functional relationship f using for the input quantities Xi the estimates xi obtained in step 2 (see 4.1.4).
6. Determine the combined standard uncertainty uc(y) of the measurement result y from the standard uncertainties and covariances associated with the input estimates, as described in Clause 5. If the measurement determines simultaneously more than one output quantity, calculate their covariances (see 7.2.5, H.2, H.3, and H.4).
7. If it is necessary to give an expanded uncertainty U, whose purpose is to provide an interval y − U to y + U that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand Y, multiply the combined standard uncertainty uc(y) by a coverage factor k, typically in the range 2 to 3, to obtain U = kuc(y). Select k on the basis of the level of confidence required of the interval (see 6.2, 6.3, and especially Annex G, which discusses the selection of a value of k, that produces an interval having a level of confidence close to a specified value).
8. Report the result of the measurement y together with its combined standard uncertainty uc(y) or expanded uncertainty U as discussed in 7.2.1 and 7.2.3; use one of the formats recommended in 7.2.2 and 7.2.4. Describe, as outlined also in Clause 7, how y and uc(y) or U were obtained.