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How to Use Geometric Mean

The geometric mean is a type of average. It is the nth root of the product of n numbers. For example, the geometric mean of 3, 6 and 9 is (3x6x9)^(1/3), or about 5.45. More intuitively, it is the arithmetic mean of the logarithmic values of a data set, which is then converted back to base -10. This makes it useful to financial analysts and scientists, especially biologists. Specifically, the geometric mean is useful in analyzing exponential trends and population growth. It also dampens the effects of extreme data points. Because this mean is the average of logs, you should not include negative number in the data set you are averaging. This doesn̵7;t mean you can̵7;t analyze negative growth, since you̵7;ll average not the rates but the rates added to 1.

Instructions

- 1
  Use the geometric mean for financial growth as follows: Suppose an investment fund returns 12 percent, -3 percent and then 8 percent for three successive years. You can determine the effective rate over the three years by taking the geometric mean of the rates plus 1. (1.12x0.97x1.08)^(1/3) = 1.0547, or 5.47 percent. Note that the arithmetic mean would instead return 5.67 percent, exaggerating the return. On the other hand, 1.0547^3 = 1.12x0.97x1.08; so the geometric mean correctly identifies what constant rate of return would produce the same returns that the fund actually returned.
- 2
  Use the geometric mean for population growth as follows. Suppose a growing tree produces 100 oranges one year, then 180 the next year, then 210 and finally 300. The total growth is of course 200 percent. Convert the numbers to percent growth. You̵7;ll get 80 percent, 16.7 percent and 42. percent. Add 1 to each. The geometric mean is therefore (1.80x1.167x1.429)^(1/3) = 1.4425. So the average annual rate of growth is 44.25 percent. And as you can see, 100x1.4425^3 = 300, so 44.25 percent gives the right result.
- 3
  Use the geometric mean in geometry to find an equivalent volume. For example, a plank of wood that is a quarter foot by a third of a foot by 10 feet is equivalent to a cube of wood that is [(0.25)(0.333)10]^(1/3) = 0.941 feet on each side. This is intuitively obvious though because width x depth x height = volume and (equivalent cube̵7;s side)^3 = volume.

Numerology