A Candidate Definition for Geometric Variance and Its Limits

Summary

The document asks how to define a geometric counterpart to arithmetic variance. One answer applies a geometric mean to one plus each squared deviation from a geometric mean of one-plus observations. This gives a candidate formula, but the exchange does not establish that it is a standard or uniquely appropriate definition.

An addendum questions whether a logarithmic approach to geometric variance and standard deviation yields a consistent relationship between the two. A further answer points to geometric standard deviation as a related concept, without developing the derivation in the document. The material is useful as a prompt for distinguishing possible definitions, but it supplies no worked numerical validation or consensus resolution. Readers should treat the proposed formula as tentative and check what properties they need from a dispersion measure before applying it.

Key ideas

• A proposed geometric variance uses a geometric mean of one plus squared deviations.
• The candidate formula measures deviations from the geometric mean of one-plus observations.
• The exchange questions whether a logarithmic variance and standard deviation definition is internally consistent.
• A geometric standard deviation is mentioned as related, but its derivation is not included.
• The discussion does not establish a standard definition or validate the candidate with examples.

Tags

# Geometric Variance


# Geometric Variance












If the arithmetic mean is:

$ \frac { \Sigma (x_i) }{n}$

and the geometric mean is

$ (\prod (1+x_i) ) ^{1/n}$

The arithmetic variance is

$ \frac { \Sigma(x_i-\mu)^2 } {n} $

then what is the geometric variance?

[I actually have an answer, while it gets a decent result I have to think about a way to check it, and it looks funny]

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/17283

For a random variable $\xi$, the variance is defined by $$mean\Big(\big(\xi -mean (\xi)\big)^2\Big).$$ Then the geometric variance should be defined by $$\prod_{i=1}^n\Bigg(1+ \bigg[x_i-\prod_{j=1}^n(1+x_j)^{1/n}\,\bigg]^2\, \Bigg)^{1/n}.$$

Addendum ----

The definition given in the link below is only a way of thinking. However, it does not provide a consistent definition. For example, for the variance var, it would be defined by something like $$\ln var = \frac{\sum_{i=1}^n (\ln A_i - \ln u_g)^2}{n}.$$ If the standard deviation is defined by $$\ln \sigma_g = \sqrt{\frac{\sum_{i=1}^n (\ln A_i - \ln u_g)^2}{n}},$$ Then what is the relationship between $var$ and $\sigma_g$?

## Answer by mconsidine (score 3)

https://quant.stackexchange.com/a/17311

Perhaps this works : http://en.wikipedia.org/wiki/Geometric_standard_deviation

In particular, see under "Derivation"

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.