On higher-order corrections for propagating uncertainties

Abstract

The Guide to the Expression of Uncertainty in Measurement (GUM) recommends the use of a first-order Taylor series expansion for propagating errors and uncertainties. The GUM also requires the use of a second-order Taylor series approximation for calculating uncertainties when the first-order approximation alone is not adequate. When only first-order Taylor expansions are used as a basis for propagating uncertainties, the first-order approximation to the standard deviation coincides with the first-order approximation to the root mean square error of an estimate. However, this is generally not true when using higher-order approximations. In this paper, we derive the second-order approximation of the standard deviation and the root mean square error of an estimate. We compare them with the correction terms given in the GUM and demonstrate their use via examples. We also provide a computer program that uses symbolic derivatives to calculate the second-order approximations of the uncertainty in measurement results.

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