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08/13/2015

Help with Statistics: The Geometric Mean

Say you want to compare two companies on two different metrics- their corporate social responsibility rating (from 0 to 5) and their financial stability rating (from 0 to 100). If you compared the average, or arithmetic mean, of the two company's ratings, over time changes in their financial stability would skew the comparison greater than changes in their social responsibility rating. To normalize this weight, you will want to find the geometric mean. This post will teach you how to do it and how it works.

The Formula

The geometric mean=     The capital PI symbolizes taking the product of a set of numbers. The geometric mean formula says to multiply k numbers and to take the k-th root of the product. The result is the geometric mean. It probably looks and sounds more complicated than it actually is, so let's look at some examples.

Example 1:

What is the geometric mean of 1, 5, and 25? First, multiply our numbers: 1*5*25=125. Next, since there are three numbers, we take the cube root. = 5. So the geometric mean of 1, 5, and 25 is 5. Now we will try another one.

Example 2:

What is the geometric mean of 4, 9, 6, and 15? Multiply our numbers: 4*9*6*15=3240. Take the fourth root of 3240. = 7.5446, which is our geometric mean.

No Negative Numbers

It is important to note that the geometric mean only works with positive numbers. Using negative numbers will result in imaginary results. It is usually not a problem given the typical uses of the geometric mean.

Applications

The geometrical mean better represents growth rates, as arithmetic means tend to overstate growth. Business calls the geometric mean of growth rates the Compound Annual Growth Rate (CAGR). It is also the only correct mean to use with normalized results, presented as ratios to reference values. As such, it is a more accurate way of comparing achievements.

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