Mean relative error and standard relative deviation

Statistics on the natural log scale, such as differences, regression coefficients, and standard deviations, frequently arise when analyzing log‐transformed data or modeling binary, count, or time‐to‐event data. The statistical properties of log‐transformed estimators (for example, log odds ratio estimators) frequently appear in statistical articles. Understanding the magnitude of these quantities can be useful. Remarkably, many can be readily interpreted, without exponentiation. In this note, we introduce four new interpretations. While a log‐scale standard deviation can be interpreted as an approximate coefficient of variation describing variation about an arithmetic mean, we argue it can be more useful to interpret it exactly as a “standard relative deviation” describing variation about a geometric mean. For positive‐valued estimators, we show how the standard error of a log‐transformed estimator can be interpreted as the “standard relative error” describing the precision of the untransformed estimator. We also show how the bias and root mean squared error of a the log‐transformed estimator can be interpreted as the “mean relative error” and “root mean squared relative error” of the untransformed estimator. We illustrate the usefulness of our interpretations for (i) the usual scale parameter of a lognormal distribution, (ii) statistical properties of a log odds ratio estimator, and (iii) a single measure of the precision of an odds ratio estimator which agrees with its usual, asymmetric 95% confidence interval.