Statistical analysts often model positive responses (such as 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)
##  148.4132
##  168.2861
##  168.1741