... mean,¹

Just in case you haven't heard the word ``mean'' in this context before, it's just another word for ``average.'' Statisticians tend to like to talk about the ``mean'' of a distribution or the ``mean'' of a sample, rather than the average. Since I have an adjunct appointment in our Department of Statistics, that's the terminology I'll use.

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... mean.²

Similarly, the long-term growth rate of your retirement portfolio is determined by the geometric mean of your annual returns, not the arithmetic mean.

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... mean.³

Notice that I said that this example ``illustrates'' that the geometric mean is always less than the arithmentic mean. The proof, for anyone who cares, follows from Jensen's inequality.

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... mean.⁴

The math isn't too hard. It's in appendix A of [3], if you'd like to see it. By the way, the same calculations apply to your retirement portfolio. The value of your retirement investment will decline if the variance in annual rate of return is more than about twice the mean annual rate of return. That's one reason why it's important to pay attention not only to what the annual rate of return on an investment is, but on how variable that return is, i.e., how risky it is.

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... tell.⁵

Actuallly, that's not quite true. By fitting a hierarchical Bayesian model to a time series of population sizes, it is possible to distinguish between intrinsice process variability and variability that is the result of measurement error. But doing this isn't easy, and the approach hasn't yet been documented in the literature. I'm working on it right now, and I expect to have an example of it by the end of the year. If you're interested, ask me for details.

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... set.⁶

Those of you familiar with population genetics will recognize this as a trick analogous to the one population geneticists use for defining the effective size of populations with respect to genetic drift.

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... stochasticity.⁷

Notice that we are implcitly assuming density-independent population dynamics here (except for the ``hard'' cap on population size, $N_m$), or at least that the magnitude of variation in population growth rate is overwhelmingly determined by factors unrelated to the current size of populations.

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... individuals:⁸

age/stage structure can complicate this a lot, if reproduction is heavily concentrated in one age or stage

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... stochasticity:⁹

Holds true with age or stage structure

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... time.¹⁰

$0.7797 = \sqrt{6}/\pi$, $-0.5772 = \Gamma'(1)$

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....¹¹

Remember that we can't calculate persistence time for density-independent models in this case.

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