# Abstract

We often assume that an abundance estimate is log-normally distributed and use that as a basis for constructing confidence intervals. This is an aide memoire for me on that topic…

# Setup

We’ve estimated the abundance of some population using some method1 and now have an estimated abundance, which we’ll call $\hat{N}$ and an associated estimate of the variance $\widehat{\text{Var}}(\hat{N})$.

So we assume: $N_\text{truth} \sim \log\mathcal{N}(\hat{N}, \text{Var}(\hat{N}))$ for R’s sake (for things like qlnorm and plnorm etc) we need the following parameters: $\mu = \mathbb{E}(\log N) \quad \text{and} \quad \sigma^2 = \text{Var}(\log N)$

# Some maths

So if we know2 that: $\mathbb{E}(N) = \exp(\mu + \sigma^2/2)$ and $\text{Var}(N) = \exp(2\mu +\sigma^2)(\exp\sigma^2 -1)$ we can get back to expressions about $\mu$ and $\sigma$. For $\sigma$ first… % and for $\mu$: %

These values can then be fed to qlnorm to obtain confidence intervals, hurrah!

1. Maybe it was a density surface model? Maybe it was a Horvitz-Thompson estimate? Voodoo?

2. taken from the *lnorm help in R