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Encyclopedia > Geometric mean

The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. A negative number is a number that is less than zero, such as −3. ... In mathematics, an nth root of a number a is a number b, such that bn=a. ... A data set (or dataset) is a collection of data, usually presented in tabular form. ...

Contents

Calculation

The geometric mean of a data set [a1, a2, ..., an] is given by

bigg(prod_{i=1}^n a_i bigg)^{1/n} = sqrt[n]{a_1 cdot a_2 cdot dots cdot a_n}.

The geometric mean of a data set is smaller than or equal to the data 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. In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM-GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal... 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. ... In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i. ...


The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (an) and (hn) are defined: For other senses of this word, see sequence (disambiguation). ...

a_{n+1} = frac{a_n + h_n}{2}, quad a_1=frac{x + y}{2}

and

h_{n+1} = frac{2}{frac{1}{a_n} + frac{1}{h_n}}, quad h_1=frac{2}{frac{1}{x} + frac{1}{y}}

then an and hn 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. In mathematics, there are several logarithmic identities. ...

bigg(prod_{i=1}^nx_i bigg)^{1/n} = expleft[frac1nsum_{i=1}^nln x_iright]

This is simply computing the arithmetic mean of the logarithm transformed values of xi (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. 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. ... In mathematics and statistics, the generalised f-mean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(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 arithmetic mean of the log transformed values, i.e. emean(ln(X)). In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed. ...


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 signify 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, which is about 1.28966 or about a 29% annual rate of return.


In the scientific community, when reporting experimental results, it is also important to know whether arithmetic mean or geometric mean should be used. If, for example, you are averaging ratios (i.e. ratio = new method/old method) over many experiments, geometric mean should be used. This becomes evident when considering the two extremes. If one experiment yields a ratio of 10,000 and the next yields a ratio of 0.0001, an arithmetic mean would misleadingly report that the average ratio is 5000 when rounded off (5000.0005 to be exact). The geometric mean is 1, which would be a more accurate representation.


See also

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. ... In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i. ... In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ... A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. ... The geometric standard deviation describes how spread out are a set of numbers whose preferred average is the geometric mean. ... In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ... The Heronian mean of two non-negative real numbers and is given by . ... In mathematics, hyperbolic coordinates are a useful method of locating points in Quadrant I of the Cartesian plane {(x,y) : x > 0, y > 0} = Q. Hyperbolic coordinates take values in HP = {(u,v) : u ∈ R, v > 0 }. For (x,y) in Q take u = −1/2 log(y... In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM-GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed. ... In mathematics, Muirheads inequality, also known as the bunching method, generalizes the inequality of arithmetic and geometric means. ... This article or section is in need of attention from an expert on the subject. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted geometric mean is calculated as Note that if all the weights are equal, the weighted geometric mean is the same as the geometric mean. ...

External links


  Results from FactBites:
 
How to Calculate Geometric Mean (3272 words)
A geometric mean, unlike an arithmetic mean, tends to dampen the effect of very high or low values, which might bias the mean if a straight average (arithmetic mean) were calculated.
Geometric mean is often used to evaluate data covering several orders of magnitude, and sometimes for evaluating ratios, percentages, or other data sets bounded by zero.
For example, to calculate the geometric mean of the values +12%, -8%, and +2%, instead calculate the geometric mean of their decimal multiplier equivalents of 1.12, 0.92, and 1.02, to compute a geometric mean of 1.0167.
Geometric mean - Psychology Wiki (612 words)
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.
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).
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.
  More results at FactBites »

 
 

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