What's the difference between the geometric and the arithmetic mean, besides the calculation itself?

Answer 1

There's more to this than meets the eye. Before answering, I looked in an advanced algebra book.

For the easy cases first, the arithmetic mean is the average. Add up n numbers and divide by n. So the arithmetic mean of 6, 9, and 18 is (6+9+18)/3 = 33/3 = 11.

This number 11 "balances" all the numbers: (6-11) + (9-11) + (18-11) = (-5) + (-2) + (+7) = 0.

For the geometric mean, you the n numbers together, then take the nth root. The geometric mean of 6, 9, and 18 is (6x9x18)^(1/3) = 972^(1/3) = 9.9 (approximately).

This number 9.9 "balances" all the numbers like this: (6/9.9)(9/9.9)(18/9.9) = 0.606 x 0.908 x 1.817 = 1.

Often, the geometric mean is only used for two numbers. Suppose we pick 9 and 16 (not by accident). The arithmetic mean is (9+16)/2 =25/2 = 12.5. The geometric mean is sqrt(9x16) = sqrt(144) = 12. And note that (9/12)(16/12) = 1.

In statistics, we use the arithmetic mean often; we use the geometric mean less often. One common statistical use is sqrt[n(n-1)] where n is the number of data points, and the (n-1) denotes loss of a "degree of freedom".

Okay. There's another meaning of these terms also. Suppose we have two endpoints, say, 60 and 100, and we want to have that range broken into five equally spaced intervals. To do that, we must insert "arithmetic means". Take (100-60) and divide by <5> (n+1): (100-60)/5 = 8. The four arithmetic means to be inserted are 68, 76, 84, and 92. With those numbers inserted, we've broken the original interval into five equal segments.

You can do the same thing with geometric intervals. Let's break that same interval, 60 to 100, into five segments. To do this, I like to use logarithms. Calculate (log 100 - log 60)/5 = (2.00000 - 1.77815)/5 = 0.22185/5 = 0.04437 and take the antilog, which is 1.10757 (the common ratio).

That's the factor we'll apply to get these segments (to the nearest tenth): 60.0, 66.5, 73.6, 81.5, 90.3, 100.0

There are five segments here, and we've inserted five geometric means. You can go from one to the next by multiplying by 1.10757.

This technique has many applications in areas as diverse as economics, ecology, population studies, and so on.

To give you just one example, several years ago I was doing a study of pricing policy at a prominent auto manufacturer. Let's say their entry-level vehicle was priced at $15,000, and their luxury sedan was $80,000. They want three intermediate car lines. How should they be priced?

Using the arithmetic model, (80,000 - 15,000)/4 = $16,250. The new vehicles would be priced at $31,250, $47,500, and $63,750.

But the geometric model is better: (log 80000 - log 15000)/4 gives a multiplier of 1.52 -- each model will be 52% higher than the one before it. The new vehicle prices will be $22,800, $34,600, and $52,600.

Notice the spacing between the arithmetic and geometric models. The auto manufacturer will have greater sales using the geometric model, because of the way people look at new car prices.

Well, that's enough. I just told you everything I know about arithmetic and geometric means.

Answer 2

The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members.

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. If one particular number occurs more times than others in the list, it is called a mode. 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.

visit the web for more:
http://en.wikipedia.org/wiki/Geometric_mean
http://www.thinkingapplied.com/means_folder/deceptive_means.htm

Answer 3

Hey go to this website cos I can't explain.

Answer 4

http://en.wikipedia.org/wiki/Geometric_mean