Asymptotic expansion technique can evaluate the measurement uncertainty by classifying an output quantity into a measured value and its correction values. The asymptotic expansion technique combines simultaneous observations of input quantities into the output measured value. The asymptotic expansion technique is useful in evaluating a multi-variate output quantity such as the moist-air density formula (CIPM-2007), in which covariances among input quantities could complicate the evaluation of measurement uncertainty.

This study demonstrates that both the Taylor’s series expansion and the chain rule of differentiation are enough to calculate the sensitivity coefficients for the CIPM-2007 air density formula. The measurement uncertainty is found to be greater than the original CIPM-2007 formula by two orders of magnitude. It is because the uncertainty of correction values come from a commercial instrument for monitoring laboratory environments. Nevertheless, the asymptotic expansion technique is useful for measurement uncertainty evaluation to avoid subtle problems of ignoring covariance of input quantities in the literature.

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