The measurement uncertainty of ratios which share uncertainty components in numerator and denominator

Abstract

When calculations are made of ratios of values of the same kind of quantity, correlations often arise from the use of the same instrument to make the measurement. The combined uncertainty in this case is usually much smaller than that from a treatment in which the numerator and denominator are assumed to be independent. For a simple ratio $R{a \over b}$, with a and b having the identical uncertainty u, the relative uncertainty in R is given by $\left| {{u \over a}\left( {1 - R} \right)} \right|$. This type of positive correlation usually reduces uncertainty and may cancel when the ratio is unity. An example from the EURACHEM guide, Quantification of Uncertainty in Analytical Measurement, overestimates the combined uncertainty of a titration by assuming complete independence of terms in a ratio.