Accuracy of error propagation exemplified with ratios of random variables

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Abstract
The method of error propagation provides a convenient tool for calculating mean and variance of a measurand from means and variances of primarily measured quantities. However, being based on a (usually first-order) Taylor approximation of the measurement function, it only yields approximate results with unknown accuracy. We develop a method for estimating the accuracy of (Nth-order) error propagation for an arbitrary number of correlated random quantities, and apply our findings to the ratio of two random variables (RVs). A comparison with some analytically solved expressions for certain probability density functions (PDFs) as well as with some computer simulations reveals the excellent quality of our estimates as long as the involved PDFs are not significantly skew. For the ratio of two RVs it turns out that conventional, first-order error propagation is safely applicable (with about 1% accuracy) as long as the denominator’s mean is larger than about 12 times its standard deviation. Using second-order error propagation, the approximation for the ratio’s mean can be refined, yielding 1% accuracy if the denominator’s mean is larger than about four times its standard deviation.