Calculating a Geometric Standard Deviation

Jim has a set of data on which he needs to calculate some statistical information. He uses built-in Excel functions to calculate many of these, such as the geometric mean. He cannot seem to figure out how to calculate the geometric standard deviation, however.

The place that a geometric mean is most often used (and, therefore, a geometric standard deviation) is when calculating investment returns over time, especially when the returns involve compound interest. How you calculate the geometric mean is rather easy-you use the GEOMEAN function built into Excel. How you calculate a geometric standard deviation, however, depends on which resource you are referencing.

One reference that explains the math behind a geometric standard deviation is found on Wikipedia:

http://en.wikipedia.org/wiki/Geometric_standard_deviation

Let’s assume that you have calculated the compound annual growth rate for an investment for four years. Over those four years the rate is expressed as 1.15 (+15%), 0.9 (-10%), 1.22 (+22%), and 1.3 (+30%). If you place these values in cells A1:A4, then apply the simplest form of calculating geometric standard deviation found on the Wikipedia page, you would enter the following as an array formula:

=EXP(STDEV(LN(A1:A4)))

This provides a result of 1.1745, rounded to four decimal places. However, there is some muddiness, as evidenced in this mathematical treatise at the Motley Fool:

http://www.fool.com/workshop/2000/workshop000309.htm

Note that it references the results of the above formula as the “standard deviation of the log values,” insisting that you need to add the average of the log values to the standard deviation and then use the EXP function, in this manner:

=EXP(STDEV(LN(A1:A4))+AVERAGE(LN(A1:A4)))

Again, this must be entered as an array formula. It provides a result of 1.3294, which is significantly different from what is returned using the simpler formula from Wikipedia. Which is the actual geometric standard deviation is apparently a matter of debate and, perhaps, dependent on a definition of terms.