Geometric Mean Article Index for
Geometric 		Website Links For
Geometric 		 

Information About

Geometric Mean
  CATEGORIES ABOUT GEOMETRIC MEAN
   
means
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



The geometric '''mean''' of a Set of Positive Data is defined as the ''n''th Root of the Product of all the members of the set, where n is the number of members.


CALCULATION


The geometric mean of a set {''a1'', ''a2'', ..., ''an''} is:
:\bigg(\prod_{i=1}^n a_i \bigg)^{1/n} = (a_1 \cdot a_2 \dotsb a_n)^{1/n} = \sqrt {Link without Title} {a_1 \cdot a_2 \dotsb a_n}.

The geometric mean of a data set Is Always Smaller Than Or Equal To the set's Arithmetic Mean (the two means are equal if and only if all members of the data set are equal). This allows the definition of the Arithmetic-geometric Mean , a mixture of the two which always lies in between.

The geometric mean is also the arithmetic-harmonic mean in the sense that if two Sequences (''a''''n'') and (''h''''n'') are defined:
:a_{n+1} = rac{a_n + h_n}{2}, \quad a_1= rac{x + y}{2}
and
:h_{n+1} = rac{2}{ rac{1}{a_n} + rac{1}{h_n}}, \quad h_1= rac{2}{ rac{1}{x} + rac{1}{y}}
then ''a''''n'' and ''h''''n'' will converge to the geometric mean of ''x'' and ''y''.


RELATIONSHIP WITH ARITHMETIC MEAN OF LOGARITHMS


By using Logarithmic Identities to transform the formula, we can express the multiplications as a sum and the power as a multiplication.

:\bigg(\prod_{i=1}^nx_i \bigg)^{1/n} = \exp\left[ rac1n\sum_{i=1}^n\ln x_i ight].

This is simply computing the Arithmetic Mean of the logarithm transformed values of x_i (i.e. the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale. I.e., it is the Generalised F-mean with f(x) = ln x.

Therefore the geometric mean is related to the Log-normal Distribution .
The log-normal distribution is a distribution which is normal for the logarithm
transformed values. We see that the
geometric mean is the exponentiated value of the mean of the log transformed
values, i.e. emean(ln(X)).


WHEN TO USE THE GEOMETRIC MEAN


The geometric mean is useful to determine "average factors". For example, if a stock rose 10% in the first year, 20% in the second year and fell 15% in the third year, then we compute the geometric mean of the factors 1.10, 1.20 and 0.85 as (1.10 × 1.20 × 0.85)1/3 = 1.0391... and we conclude that the stock rose 3.91 percent per year, on average.

Put another way...

The arithmetic mean is relevant any time several quantities add together to produce a total. The arithmetic mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same total?"

In the same way, the geometric mean is relevant any time several quantities multiply together to produce a product. The geometric mean answers the question, "if all the quantities had the same value, what would that value have to be in order to achieve the same product?"

For example, suppose you have an investment which earns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.50, and the third year it was multiplied by 1.30. The relevant quantity is the geometric mean of these three numbers. Source .


SEE ALSO




EXTERNAL LINKS