Log-ratios and the logarithmic mean

Logarithmically transformed ratios have an important role to play in some branches of statistics and economics. Focus of this paper is on an unfamiliar interpretation of these logratios, which are in fact absolute differences being divided by the logarithmic mean. It is demonstrated that familiarity with the logarithmic mean and its relation to more common mean values leads to view a problem from a new angle whenever logratios are involved. Examples investigated are Fisher’s z-transformation, the likelihood-ratio test of goodness-of-fit, Theil’s measure of inequality, and the log-ratio itself as one important device for measuring relative change within timeseries.