playing a mean harmonica

(yes -- the punny blog titles will just keep getting worse until someone starts posting comments!!)

as promised: part II of my engrossing saga of the two lesser-known cousins of the arithmetic mean. previously, we considered the geometric mean, and a fairly obscure use it might be put to in (slightly) more accurately summarizing population change over time. i'd be interested to hear (comments, anyone?) about any other uses of a biological nature to which it can be put.

so, on to the harmonic mean: S&R show you how to calculate it (p. 44) and if you google "harmonic mean + use" the interwebs will tell you that it might be useful for figuring out how fast you went on average under certain very unnatural driving conditions. evidently it also has some uses in calculating electrical resistance and maybe in petroleum geology as well. but -- we're all biologists... why should we care?

as it turns out, this is a fairly important measure in conservation biology as well, used in calculating effective population size over time. a number of papers and books (including Gotelli and Ellison, 2004, referenced in my previous blog post) outline or advocate for its use in 'averaging' population sizes over time.

as a hypothetical example (modified from Gotelli and Ellison, 2004): over a decade, a population has the following sizes: 986, 1067, 95, 221, 489, 821, 961, 1017, 1039, 1126. obviously something pretty bad happened there in year #3, from which it took several years to recover.

the arithmetic mean population size for the decade is still a pretty high 782.2.


> x = c(986, 1067, 95, 221, 489, 821, 961, 1017, 1039, 1126)
> mean(x)
[1] 782.2


however, the scenario i've laid out above, most of you will immediately recognize is a "bottleneck" of the type you learned about in reference to genetic drift. in terms of genetic diversity, the presence of such an event has a pronounced negative effect. the harmonic mean, not coincidentally, emphasizes the smaller values in a series, and gives them greater weight:


> 1/mean(1/x)
[1] 414.2493


most of the references that i consulted don't actually provide a citation to the original use of the harmonic mean for this purpose, however, using my amazing sleuthing skills, i managed to trace it back to at least the 1930s (Wright, 1938). i'd be curious if there were any references that pre-date this.

Reference

Wright, S. 1938. Size of population and breeding structure in relation to evolution. Science 87:430-431.