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)

## [1] 148.4132

mean(x)

## [1] 168.2861

g2amean(exp(logMu), logSD)

## [1] 168.1741