# Geometric to Arithmetic Mean

By Joe Thorley

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`

`## [1] 168.2861`

`## [1] 168.1741`