# Geometric to Arithmetic Mean

Statistical analysts often model positive reponses (such as in our case fish densities) using a log-normal distribution. By default the expected values for such models represent the geometric mean ($$\mu_G$$) but readers are typically most interested in the arithmetic mean ($$\mu_A$$). The difference can be important particularly because $$\mu_G \leq \mu_A$$.

The geometric mean of a log-normal distribution can be converted to its arithmetic mean using the equation

$\mu_A = exp(log(\mu_G) +\sigma_L^2 / 2)$

where $$\sigma_L$$ is the standard deviation on a log-scale.

As a simple demonstration consider the following code.

set.seed(102)

g2amean <- function (gm, logsd) exp(log(gm) + logsd^2 / 2)

logMu <- 5
logSD <- 0.5

x <- rlnorm(10000, logMu, logSD)

exp(logMu)
## [1] 148.4132
mean(x)
## [1] 168.2861
g2amean(exp(logMu), logSD)
## [1] 168.1741