# Which average to use with range

Say I wanted to calculate how much rain in terms of volume falls on the Earth each day or hour, etc. In order to do so I'd first need to know how much water is in a single raindrop. I know that and it ranges from $10^{-6}$ to $10^{-3}$ cubic inches of water per drop. This range is a large range and finding the mean I get $10^{-4} * 5.01$. This doesn't seem quite right because this is closer to $10^{-3}$ than to $10^{-6}$. By calculating the geometric mean, I get $3 * 10^{-5}$, which now is closer to $10^{-6}$ than to $10^{-3}$. The logarithmic mean gives $1 * 10^{-4}$. Which one is the best to use in this situation? Arithmetic, Geometric, or Logarithmic average? Or is there some other mean I'm not aware of?

Edit: I'd also like to know the best average to use for any range.

## 1 Answer

"Ranges from ... to ..." hardly allows one to make an educated guess about the average size, i.e. about the sum of all sizes divided by their count. Where did you get the numbers from? What prevents the existence of raindrops with $0.9\cdot 10^{-3}$ or $1.1\cdot 10^{-6}$ cubic inches? Or: What range would yo assume if you measured in SI units?

If I wer4e t make a bold guess, I would notice that smaller radii correspond to more surface energy and assume that the distribution of raindrops among energy levels follows some Boltzman statistic. But real physicits are much better at such guesses :)

• Thanks, but I'd like to know the best average to use for any range. Also I got the numbers from here – Milo Apr 24 '14 at 6:09