Difference between Arithmetic and Geometric Mean

I was wondering what the difference is between the arithmetic mean and the geometric mean. Wikipedia has some pretty good definitions that really helped clarify the subject.

Wikipedia states for the arithmetic mean:

"In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. The arithmetic mean is what students are taught very early to call the "average". If the list is a statistical population, then the mean of that population is called a population mean. If the list is a statistical sample, we call the resulting statistic a sample mean."

Wikipedia states for the geometric mean:

"The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average," except that instead of adding the set of numbers and then dividing the sum by the count of numbers in the set, n, the numbers are multiplied and then the nth root of the resulting product is taken.

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root (i.e., the second root) of their product, 16, which is 4. As another example, the geometric mean of 1, ½, and ¼ is simply the the cube root of their product, 0.125, which is ½.

The geometric mean can be understood in terms of geometry. The geometric mean of two numbers, a and b, is simply the length of the square whose area is equal to that of a rectangle with dimensions a and b. That is, what is n such that n² = a × b? Similarly, the geometric mean of three numbers, a, b, and c, is the length of a cube whose volume is the same as a box with sides equal to the three given numbers. This geometric interpretation of the mean is very likely what gave it its name."

So why would you use the arithmetic mean instead of the geometric mean, or vice versa? Well, first of all, it is only possible to use the geometric mean on positive numbers. As Wikipedia continues to state, "It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment." As Wikipedia infers, the user of the means must understand the data that they are evaluating and utilize the most appropriate mean.

page revision: 0, last edited: 10 Nov 2007 17:43