An internally consistent procedure of uncertainty analysis will give the same results no matter how the calculations are legitimately rearranged. This paper identifies an apparent internal inconsistency arising with the full procedure described in the Guide to the Expression of Uncertainty in Measurement. The phenomenon occurs when the measurand is expressible as a function of quantities with common sources, and is related to the calculation of an effective number of degrees of freedom. One common situation in which this inconsistency arises is where a point on a line is measured using estimates of the intercept and slope derived from the same dataset; the numbers of degrees of freedom obtained differ according to whether this point is measured directly from the dataset or from the estimates of the intercept and slope. An example is given where the widths of the corresponding expanded uncertainty intervals differ by a factor of approximately 2. Statistical principles explaining this issue are discussed, and a principle for the calculation of an effective number of degrees of freedom is proposed.
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