Calculating a Geometric Standard Deviation

The calculation of geometric mean (GM)is not difficult and easy to calculate.

What I need to understand is the following;

If instead of finding the GM of true value, you have values with measurement uncertainty std dev, how do you calculate the std deviation of the error of the GM

Thus if we have
X1,S1
X2,S2
X3,S3
X4,S4

Is it still the case that the GM is the forth root of X1*X2*X3*X4?

What is the Std Dev of the GM based on the uncertainty of each measurement?

I figured out why Tom's formula gives different values. He has an error in the formula. You cannot take the LN of a range of numbers. Column B must contain the LN of each row in column A.

Example: B2 =LN(A2); B3 =LN(A3); ... B10 =LN(A10)

You would then replace "LN(A2:A10)" with "B2:B10".
Then the correct formula would be
=EXP(AVERAGE(B2:B10))

Sorry, my previous comment contained a different range because I copied and pasted from my own Excel spreadsheet and forgot to change the range to A2:A10 before submitting.

Comment for Tom. I get different results for the following two formulas:

=GEOMEAN(D3:D47)
=EXP(AVERAGE(LN(D3:D47)))

You state that they should give the same value, unless "equivalent" is not the same as "equal".

Just a point of clarification:

The Motley Fool page you link to explicitly calculates "a return one GSD above average", which is why it adds the (geometric) average to the geometric standard deviation.

It is simply calculating the +1 SD geometric mean. It is not claiming that the result (32.15%) is the geometric standard deviation. The example it provides specifically calculates the geometric mean that is one standard deviation above the mean.

this is entered as an array formula. I do not know how to do that. WShy not explain it.

Angus

I have a question. Suppose I am trying to compute the probability that a manager outperforms its benchmark by a particular amount over different periods of time. For example, I know the standard error over 87 months between the manager and the benchmark is 1.65%, I know the manager underperformed the benchmark by 22.3%. This was calculated by taking the difference between the cumulative returns (geometrically linked monthly returns)of the manager and the benchmark. The monthly mean difference between the manager's returns and the benchmark returns is -.19%. I think we could assume the difference between the manager and the benchmark should be zero (population mean of difference between the benchmark and the manager. This is a tricky problem because it could involve log normal calculations and potentially annualized performance and standard deviation. I want to keep the solution as simple and understandable as possible.

Try a helper column. if the data is in A1:A4 is 1.15, .9,1.22,1.3 and the stddev is 1.174489.
With the items in in the range B1:B7 i.e. thrree blanks in the range

In c1 type =IF(B1>0,LN(B1),"") and copy down

In C8 use the formula =EXP(STDEV(C1:C7)). this is not an array formula.

Further thoughts on the original article, the fools method surely refers to the upper boundry (the error level

I have a table with chemical data for several rock samples. The rock sample numbers are ordered as rows and the chemical elements analyzed as columns.

Data for the analyzed elements is not available for all the samples. In other words, for each of the analyzed elements there are some blank cells. Replacement of those cells by a numerical value such as half the limit of detection to avoid the blank cells, is meaningless for me.

Is there a function (or script) in excel that can calculate the geometric mean and geometric standard deviation for each of the variables but considering that not all the cell are filled with numerical data (in other words some cell are blank).

Besides, I need to calculate this for each of the elements analyzed.

Adding to Tom C's comments, the geometric mean will always be lower than the average. This is because the high value affect the average, but the mean can be used to calculate the total. In your example 1.1425 * 4 = 4.57, the total of your percentages.

Conversely, the standard deviation of the geometric mean will be higher than a normal standard deviation.

With regard to adding the mean to the std Dev, I think that should refer to confidence levels. Adding and subtracting 1.96std Dev to the mean gives a 95% confidence limit. The amount that we can expect a value to be.

Excels GEOMEAN function is equivalent to =10^AVERAGE(LOG(A2:A10))assuming your data is in the range A2:A10. It's entered as an array formula (Ctrl-Shift-Enter).

Similarly, the geometric standard deviation is calculated by the following formula:
=10^STDEV(LOG(A2:A10)). Again this is entered as an array formula.

Since data is being multiplied instead of added, the logarithms of the data are used since adding logarithms is equivalent to multiplying the raw numbers. Substituting EXP for 10^ and LN for LOG will give the same results.