Should we report a ratio of averages (RA) or an average of ratios (AR)? The answer depends on what we want to average and how.
By definition, things that are not additive cannot be averaged in the arithmetic sense. This includes things like standard deviations, correlation coefficients, slopes, tangents, sines, cosines,…
In the case of properties of matter, these can be classified as extensive or intensive. Extensive properties can be added together and averaged. By contrast, intensive properties are not additive so they cannot be averaged.
The ratio of two intensive or two extensive properties is classified as an intensive ratio. These ratios are not additive either.
For instance mass, volume, and moles are extensive properties so density (mass/volume) and molar concentrations (moles/volume) are intensive ratios. Do not average them in the arithmetic sense.
Some times we find online chemistry articles and lab instructions where these ratios are arbitrarily averaged on the grounds that, anyway, their RAs and ARs agree up to certain number of digits. This sends the wrong message that it does not matter if properties are extensive or intensive.
Ratio of averages or average of ratios? That is the question.
Let R be the ratio of two extensive properties, A and B. Assume we have a set of A’s and a set of B’s. Because R is intensive we cannot average R values.
What we can do is this: average all A’s and all B’s separately and take their ratio. If the number of instances of A and B are the same, simply divide the sum of all A’s by the sum of all B’s.
Thus for density determinations we can report the sum of masses divided by the sum of volumes; i.e.
density = sum of masses/sum of volumes
In the case of determinations of concentrations by titrations, we may report the sum of volumes of the titrant divided by the sum of volumes of the titrated analyte and then multiply this ratio by the concentration of the titrant.
For instance if the titration of Va volume of a strong acid consumes Vb volume of a strong base of known concentration, both monoprotics, we get from several titrations that
acid concentration = base concentration*(Sum of Vb’s/Sum of Va’s).
If A and B are large data sets of same size, R can be computed from the slope of the curve obtained by plotting A vs. B.
IR Thoughts 