# geometric standard deviation

The geometric mean is like the arithmetic mean on a log scale. with the arithmetic mean it is often useful to find the standard deviation. Can the same sort of thing be done to create a geometric version.

With the arithmetic mean you can find the standard deviation by:

1. Taking the "arithmetic distance" (difference) from the mean for each data point,
2. Make them all positive by some function ($n^2$),
3. taking the arithmetic mean of the results to get the variance and
4. do the inverse of the step 2 operation ($\sqrt n$) to get a standard deviation.

$$\frac 1n{\sum_{i=0}^n (\overline x-x_i)^2}$$

Can you do a similar processes to get a "geometric standard deviation"? I think it would look something like this:

1. Taking the "geometric distance" (ratio) from the mean for each data point,
2. Make them all positive by some function ($|\ln(n)|$?),
3. taking the geometric mean of the results to get the "geometric variance" and
4. do the inverse of the step 2 operation ($e^n$?) to get a standard deviation.

$$e^\sqrt[n] {\prod_{i=0}^n\ln(\overline x/x_i)}$$

Is their a reasonable mathematical way to define a geometric standard deviation? And has it already bean done?