# What is the proper method for calculating the Coefficient of Variation of a rate?

I have been asked to evaluate the variation between movement rates of fish within different systems (rates are in km/day). I have between 25-100 samples in each system. I understand that when reporting the average rate for a given system, it is more appropriate to use the harmonic mean than the arithmetic mean.

If that's the case, how would one calculate the Coefficient of Variation ($CV = \sigma/\mu$) for the movement rates in each system? Does the standard deviation become the (arithmetic) mean difference of each sample from the harmonic mean?

• The harmonic mean is equivalent to applying the arithmetic mean to the reciprocals of the rates: that is, rather than expressing the data in km/day, express them in day/km. Then proceed as usual to compute the CV. You should verify that the univariate distributions of the data, when re-expressed in this way, are approximately symmetric: that is what would justify the re-expression and give meaning to the CV in the first place. – whuber Jun 3 '17 at 16:06
• Thank you. Is it appropriate to report 1/sd(rate in day/km) as the standard deviation of the rates in each system? – Von Jun 4 '17 at 16:18
• Hm, I'm sorry, I'm realizing that's probably not right. For reporting purposes, is it possible to re-express the standard deviation of the movement rates in each system in km/day? – Von Jun 4 '17 at 16:27