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.
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