Using GEOMEAN With a Large List

Ken wrote about a problem he was having with the GEOMEAN function. When he attempts to use the function on a large number of values (3,500 rows of data), he gets a #NUM error value returned.

The GEOMEAN function is used to return the geometric mean of a series of values. The GEOMEAN of n numbers is the n-th root of the product of the numbers. For example, if the series of values contains four values (A through D), then the product of those numbers is A * B * C * D, and the GEOMEAN is the fourth root of that product.

The #NUM error would be return if any of three conditions were met: any of the values was equal to zero, any of the values was negative, or the limits of Excel were exceeded. It is likely that it is this last condition that Ken is running into, particularly if any of his 3,500 values are large.

Since GEOMEAN finds the product of the 3,500 numbers (multiplies them all by each other) and then takes the nth root, the product may easily be too large for Excel. The largest positive number in Excel is 9.99999999999999 * 10^307 (in scientific notation this is written as 9.99999999999999E+307) If the product gets larger than this number you will get a #NUM error for the function.

The solution is to use logs to do the calculation. This is easiest to understand when you look at a transformation of the GEOMEAN function:

GEOMEAN = (X1*X2*X3*...*Xn)^ (1/n)
ln(GEOMEAN) = ln((X1*X2*X3*...*Xn)^ (1/n))
ln(GEOMEAN) = (1/n) * ln(X1*X2*X3*...*Xn)
ln(GEOMEAN) = (1/n) * (ln(X1)+ln(X2)+ln(X3)+...+ln(Xn))
ln(GEOMEAN) = average(ln(X1)+ln(X2)+ln(X3)+...+ln(Xn))
GEOMEAN = exp(average(ln(X1)+ln(X2)+ln(X3)+...+ln(Xn)))

If you follow through the above, you see that GEOMEAN is equivalent to the exponent of the average of the logs of the values. You can calculate the desired result by using the following array formula instead of the GEOMEAN function:

=EXP(AVERAGE(LN(A1:A3500)))

This assumes that the desired values are in the range A1:A3500. Since it is an array formula, you must enter it into a cell by using Ctrl+Shift+Enter.