What exactly does the Geometric Mean actually mean ? When should I use it?

Question

I am working signal demodulation and the RC value requires to be the geometric mean of the periods of the message signal and the carrier wave. I wonder why I need the geometric mean for this or why in general or under what circumstances do people use Geometric means ? Thanks for your inputs and time.

Answer 1

Geometric mean is based on the products of numbers being compared; analogous to how the arithmetic mean is based on their sum.

The geometric mean of $n$ variables is the $n$th root of their product: $\sqrt[\Large n]{\prod_{i=1}^n x_i}$

It's usually used to give a meaningful measure of "average" between figures with different dimensions or range of values.

For instance, a carrier wave usually has a much higher frequency than the message signal, so if you used an arithmetic mean of the periods the influence of changes in the message signal's period would dominate the measurement. Where as when using the geometric mean then a certain percentage change in the period of either the message or carrier signals would have the same impact on the measurement of this 'average'.

If the signal has period $T_s$ and the carrier $T_c$, then $$\begin{align}\text{Arithmetic mean: }&\bar T =\frac{T_S+T_C}{2}\\[2ex]\text{Geometric mean: }&GM(T)=\sqrt{T_s\cdot T_c}\end{align}$$

To visualise it: the geometric mean of the length of two lines is the side of a square with the same area as a rectangle with length and width equal to the two lines.

Answer 2

Geometric means tend to be used in cases where you're looking at quantities that relate to each other as ratios rather than as differences. Part of this comes from the fact that the logarithm of the geometric mean is the arithmetic mean of the logarithm: $$\log\left(\prod_{i=1}^n A_i\right)^{1/n}=\frac{1}{n}\log\left(\prod_{i=1}^n A_i\right)=\frac{1}{n}\left(\sum_{i=1}^n \log{A_i}\right)$$

On a related note, if, for example, you have a bank account with a variable rate of interest, then the geometric mean of the annual interest rates (expressed as $(1+r)$ terms) gives you the fixed annual rate that would give you the same final amount, which is a more meaningful measure of "average" than the arithmetic one. So similarly anything that relates to rates of growth, or anything else where the terms act more multiplicatively than additively.