A New Way to View the Magnitude of the Difference between the Arithmetic Mean and the Geometric Mean and the Difference between the Slopes When a Continuous Dependent Variable Is Expressed in Raw Form Versus Logged Form

Abstract

In Trond Petersen’s “Multiplicative Models for Continuous Dependent Variables: Estimation on Unlogged versus Logged Form” (this volume, pp. 113–164), the following phenomenon is noted: With the arithmetic and geometric means of a nonnegative quantitative variable measured for two groups (say, groups 1 and 2), it is possible for the arithmetic mean $({\mathrm{A}}_1)$ for group 1 to be larger than the arithmetic mean $({\mathrm{A}}_2)$ for group 2 and the geometric mean $({\mathrm{G}}_1)$ for group 1 to be smaller than the geometric mean $({\mathrm{G}}_2)$ for group 2. In the present note, some new formulas are introduced that will help to make clear when this phenomenon will occur and when it will not occur. The phenomenon considered here is of interest in situations where, for example, as noted in Petersen’s paper, results obtained using the raw form for a continuous dependent variable are compared with the corresponding results obtained using the logged form for the dependent variable. In addition, the formulas that are introduced in the present note also provide a new way to view the magnitude of the difference between the arithmetic mean and the geometric mean.