User:Dlawyer/sandbox

Taking averages of rates and ratios
Suppose there exists a set of rates $$r_i=\frac{a_i}{b_i}$$ and one wants to find an average. If the rates are benefit-cost ratios each $$a_i$$ is a benefit and each $$b_i$$ is a cost. Just taking an average of the $$r_i$$'s fails to find what is the total benefit-cost ratio: $$\frac{\sum a_i}{\sum b_i}$$, the sum of the benefits divided by the sum of the costs. If one knows all the $$a_i$$'s and $$b_i$$'s. then one may directly find the total benefit-cost ratio. But if only knows only the rates $$r_i$$'s and either the $$a_i$$'s or the $$b_i$$'s then one needs to take a weighted averages of the $$r_i$$'s so as to obtain the ratio of the sums, the overall benefit-cost ratio. What weights should be used? And should one use the arithmetic mean or the harmonic mean? If one knows in addition to the $$r_i$$'s, only the $$b_i$$'s but not the $$a_i$$'s, then one uses the arithmetic mean weighted by the $$b_i$$'s such that the sum of the weights add to unity (1). Conversely, if one knows in addition to the $$r_i$$'s, only the $$a_i$$'s but not the $$b_i$$'s, then one uses the harmonic mean weighted by the $$a_i$$'s such that the sum of the weights add to unity (1). Summarizing: If weighted by the denominators, use the arithmetic mean. If weighted by the numerators, use the harmonic mean.