Playing around with the Harmonic Mean of two numbers I stumbled on an interesting ratio, and looked at the others as well.
Here are the definitions, for numbers a and b
If we use m for the mean, then
for the arithmetic mean we have the ratio (b-m)/(m-a) = 1
for the geometric mean we have b/m = m/a
for the harmonic mean we have (b-m)/(m-a) = b/a
and for the RMS mean we have (b2 – m2)/( m2 – a2) = 1
I am quite sure that there is a way of seeing these which ties them all together, perhaps Mr. Joseph Nebus can find it !
The harmonic mean can be used to explain the harmonic tuning of a keyboard instrument (as opposed to equal temper tuning). I am planning a post on this for later.
The formula I gave for the harmonic mean is not the usual one – use a bit of algebra ! – but it is easier to calculate with.
The RMS mean is used extensively in Statistics, Rigid Body Dynamics and Electrical Engineering. The well known 110 volts in your house electric system is the RMS mean of the alternating voltage actually supplied. The Standard Deviation is the RMS average of the distances of the data values from the arithmetic mean value.
A non formal view of these means (the first three) is that the arithmetic mean is about the positions of the two numbers, the geometric mean is about the absolute sizes of the numbers and the harmonic mean is about the relative sizes of the numbers.
if we take the zero, the two numbers, and the harmonic mean the four values have a cross ratio of -1 (see part 3 of the Christmas Tale)