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From Wikipedia

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 the numbers are multiplied and then the nth root (where n is the count of numbers in the set) of the resulting product is taken.

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is '. As another example, the geometric mean of three numbers 1, ½, ¼ is thecube root of their product (1/8), which is 1/2; that is '.

The geometric mean can also be understood in terms of geometry. The geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. Similarly, the geometric mean of three numbers, a, b, and c, is the length of one side of a cube whose volume is the same as that of a right cuboid with sides whose lengths are equal to the three given numbers.

The geometric mean only applies to positive numbers. 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. The geometric mean is also one of the three classic Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.

Calculation

The geometric mean of a data set \{a_1,a_2 , \ldots,a_n\} is given by:

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

The geometric mean of a data set is less 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.

The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (an) and (hn) are defined:

a_{n+1} = \frac{a_n + h_n}{2}, \quad a_0=x

and

h_{n+1} = \frac{2}{\frac{1}{a_n} + \frac{1}{h_n}}, \quad h_0=y

then an and hn will converge to the geometric mean of x and y.

This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by Bolzano-Weierstrass theorem) and the fact that geometric mean is preserved:

\sqrt{a_ih_i}=\sqrt{\frac{a_i+h_i}{\frac{a_i+h_i}{h_ia_i}}}=\sqrt{\frac{a_i+h_i}{\frac{1}{a_i}+\frac{1}{h_i}}}=\sqrt{a_{i+1}h_{i+1}}

Replacing arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.

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}^na_i \bigg)^{1/n} = \exp\left[\frac1n\sum_{i=1}^n\ln a_i\right]

This is sometimes called the log-average. It is simply computing the arithmetic mean of the logarithm transformed values of a_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) = log x.

For normal frequency:

GM = Antilog\left[\frac1n\sum\ logx\right]

For frequency distribution:

GM = Antilog\left[\frac1n\sum\ f logx\right]

Applications

Proportional growth

The geometric mean is more appropriate than the arithmetic mean for describing proportional growth, both exponential growth (constant proportional growth) and varying growth; in business this is known as the compound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount.

Suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, so the growth is 80%, 16.7% and 42.9% for each year respectively. Using the arithmetic mean calculates a (linear) average growth of 46.5% (80% + 16.7% + 42.9% divided by 3). However, if we start with 100 oranges and let it grow 46.5% each year, the result is 314 oranges, not 300, so the linear average over-states the year-on-year growth.

Instead, we can use the geometric mean. Growing with 80% corresponds to multiplying with 1.80, so we take the geometric mean of 1.80, 1.167 and 1.429, i.e. \sqrt[3]{1.80 \times 1.167 \times 1.429} = 1.443, thus the "average" growth per year is 44.3%. If we start with 100 oranges and let the number grow with 44.3% each year, the result is 300 oranges.

Aspect ratios

The geometric mean has been used in choosing a compromise aspect ratio in film and video: given two aspect ratios, the geometric mean of them provides a compromise between them, distorting or cropping both in some sense equally. Concretely, two equal area rectangles (with the same center and parallel sides) of different aspect ratios intersect in a rectangle whose aspect ratio is the geometric mean, and their hull (smallest rectangle which contains both of them) likewise has aspect ratio their geometric mean.

In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.

The harmonic mean H of the positive real numbers x1, x2, ..., xn > 0 is defined to be

H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} = \frac{n \cdot \prod_{i=1}^n x_i }{ \sum_{j=1}^n \frac{\prod_{i=1}^n x_i}{x_j}}.

Equivalently, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. From the third formula in the above equation it is more apparent that the harmonic mean is related to the arithmetic and geometric means.

Relationship with other means

The harmonic mean is one of the three Pythagorean means. For all positive data sets containing at least one pair of nonequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between. (If all values in a nonempty dataset are equal, the three means are always equal to one another; e.g. the harmonic, geometric, and arithmetic means of {2, 2, 2} are all 2.)

It is the special case M−1 of the power mean.

Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones.

The arithmetic mean is often mistakenly used in places calling for the harmonic mean. In the speed example below for instance the arithmetic mean 50 is incorrect, and too big.

The harmonic mean is related to the other Pythagorean means, as seen in the third formula in the above equation. This is noticed if we interpret the denominator to be the arithmetic mean of the product of numbers n times but each time we omit the jth term. That is, for the first term we multiply all n numbers but omit the first, for the second we multiply all n numbers but omit the second and so on. The numerator, excluding the n, which goes with the arithmetic mean, is the geometric mean to the power n. Thus the nth harmonic mean is related to the nth geometric and arithmetic means.

Weighted harmonic mean

If a set of weights w_1, ..., w_n is associated to the dataset x_1, ..., x_n, the weighted harmonic mean is defined by

\frac{\sum_{i=1}^n w_i }{ \sum_{i=1}^n \frac{w_i}{x_i}}.

The harmonic mean as defined is the special case where all of the weights are equal to 1, and is equivalent to any weighted harmonic mean where all weights are equal.

Examples

In physics

In certain situations, especially many situations involving rates and ratios, the harmonic mean provides the truest average. For instance, if a vehicle travels a certain distance at a speed x (e.g. 60 kilometres per hour) and then the same distance again at a speed y (e.g. 40 kilometres per hour), then its average speed is the harmonic mean of x and y (48 kilometres per hour), and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the vehicle travels for a certain amount of time at a speed x and then the same amount of time at a speed y, then its average speed is the arithmetic mean of x and y, which in the above example is 50 kilometres per hour. The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same distance, then the average speed is the harmonic mean of all the sub-trip speeds, and if each sub-trip takes the same amount of time, then the average speed is the arithmetic mean of all the sub-trip speeds. (If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed.)

Similarly, if one connects two electrical resistors in parallel, one having resistance x (e.g. 60Ω) and one having resistance y (e.g. 40Ω), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of x and y (48Ω): the equivalent resistance in either case is 24Ω (one-half of the harmonic mean). However, if one connects the resistors in series, then the average resistance is the arithmetic mean of x and y (with total resistance equal to the sum of x and y). And, as with previous example, the same principle applies when more than two resistors are connected, provided that all are in parallel or all are in series.

In other sciences

In Information retrieval and some other fields, the harmonic mean of the precision and the recall is often used as an aggregated performance score: the F-score (or F-measure).

An interesting consequence arises from basic algebra in problems of working together. As an example, if a gas-powered pump can drain a pool in 4 hours and a battery-powered pump can drain the same pool in 6 hours, then it will take both pumps (6 Â· 4)/(6 + 4), which is equal to 2.4 hours, to drain the pool together. Interestingly, this is one-half of the harmonic mean of 6 and 4.

In hydrology the harmonic mean is used to average hydraulic conductivity values for flow that is perpendicular to layers (e.g. geologic or soil). On the other hand, for flow parallel to layers the arithmetic mean is used.

In sabermetrics, the Power-speed number of a player is the harmonic mean

Mean difference

The mean difference is a measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. A related statistic is the relative mean difference, which is the mean difference divided by the arithmetic mean. An important relationship is that the relative mean difference is equal to twice the Gini coefficient, which is defined in terms of the Lorenz curve.

The mean difference is also known as the absolute mean difference and the Gini mean difference. The mean difference is sometimes denoted by Δ or as MD. Themean deviation is a different measure of dispersion.

Calculation

For a population of size n, with a sequence of values yi, i = 1 to n:

MD = \frac{1}{n(n-1)} \Sigma_{i=1}^n \Sigma_{j=1}^n | y_i - y_j | .

For a discrete probability functionf(y), where yi, i = 1 to n, are the values with nonzero probabilities:

MD = \Sigma_{i=1}^n \Sigma_{j=1}^n f(y_i) f(y_j) | y_i - y_j | .

For a probability density functionf(x):

MD = \int_{-\infty}^\infty \int_{-\infty}^\infty f(x)\,f(y)\,|x-y|\,dx\,dy .

For a cumulative distribution function F(x) with quantile function x(F):

MD = \int_0^1 \int_0^1 |x(F_1)-x(F_2)|\,dF_1\,dF_2 .

Relative mean difference

When the probability distribution has a finite and nonzero arithmetic mean, the relative mean difference, sometimes denoted by ∇ or RMD, is defined by

RMD = \frac{MD}{\mbox{arithmetic mean}}.

The relative mean difference quantifies the mean difference in comparison to the size of the mean and is a dimensionless quantity. The relative mean difference is equal to twice the Gini coefficient which is defined in terms of the Lorenz curve. This relationship gives complementary perspectives to both the relative mean difference and the Gini coefficient, including alternative ways of calculating their values.

Properties

The mean difference is invariant to translations and negation, and varies proportionally to positive scaling. That is to say, if X is a random variable and c is a constant:

  • MD(X + c) = MD(X),
  • MD(-X) = MD(X), and
  • MD(cX) = |c| MD(X).

The relative mean difference is invariant to positive scaling, commutes with negation, and varies under translation in proportion to the ratio of the original and translated arithmetic means. That is to say, if X is a random variable and c is a constant:

  • RMD(X + c) = RMD(X) · mean(X)/(mean(X) + c) = RMD(X) / (1+c / mean(X)) for c≠ -mean(X),
  • RMD(-X) = −RMD(X), and
  • RMD(cX) = RMD(X) for c> 0.

If a random variable has a positive mean, then its relative mean difference will always be greater than or equal to zero. If, additionally, the random variable can only take on values that are greater than or equal to zero, then its relative mean difference will be less than 2.

Compared to standard deviation

Both the standard deviation and the mean difference measure dispersion—how spread out are the values of a population or the probabilities of a distribution. The mean difference is not defined in terms of a specific measure of central tendency, whereas the standard deviation is defined in terms of the deviation from the arithmetic mean. Because the standard deviation squares its differences, it tends to give more weight to larger differences and less weight to smaller differences compared to the mean difference. When the arithmetic mean is finite, the mean difference will also be finite, even when the standard deviation is infinite. See the examples for some specific comparisons. The recently introduced distance standard deviation plays similar role than the mean difference but the distance standard deviation works with centered distances. See also E-statistics.

Sample estimators

For a random sample S from a random variable X, consisting of n values yi, the statistic

MD(S) = \frac{\sum_{i=1}^n \sum_{j=1}^n | y_i - y_j |}{n(n-1)}

is a consistent and unbiasedestimator of MD(X). The statistic:

RMD(S) = \frac{\sum_{i=1}^n \sum_{j=1}^n | y_i - y_j |}{(n-1)\sum_{i=1}^n y_i}

is a consistentestimator of RMD(X), but is not, in general, unbiased.

Confidence intervals for RMD(X) can be calculated using bootstrap sampling techniques.

There does not exist, in general, an unbiased estimator for RMD(X), in part because of the difficulty of finding an unbiased estimation for multiplying by the inverse of the mean. For example, even where the sample is known to be taken from a random variable X(p) for an unknown p, and X(p) - 1 has the Bernoulli distribution, so that Pr(X(p) = 1) = 1 âˆ’ p and , then

RMD(X(p)) = 2p(1 âˆ’ p)/(1 + p).

But the expected value of any estimator R(S) of RMD(X(p)) will be of the form:


From Yahoo Answers

Question:Also; what is the harmonic mean in algebra & music?

Answers:Harmonic Mean Formula: Harmonic Mean = N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN) For eg harmonic mean of following no's We have following no's = 1,1 ,1 ,1 So harmonic mean will be = Sum of No's / Total no of No's 1+1+1+1/ 4 = 1 , So harmonic mean is 1 Geometric mean is a kind of average of a set of numbers that is different from the arithmetic average. The geometric mean is well defined only for sets of positive real numbers. This is calculated by multiplying all the numbers (call the number of numbers n ), and taking the nth root of the total. A common example of where the geometric mean is the correct choice is when averaging growth rates. Formula: Geometric Mean : Geometric Mean = ((X1)(X2)(X3)........(XN))1/N where X = Individual score N = Sample size (Number of scores) Geometric Mean Example: To find the Geometric Mean of 1,2,3,4,5. Step 1: N = 5, the total number of values. Find 1/N. 1/N = 0.2 Step 2: Now find Geometric Mean using the formula. ((1)(2)(3)(4)(5))0.2 = (120)0.2 So, Geometric Mean = 2.60517

Question:What are the advantages and disadvantages of Arithmetic Mean,Geometric Mean,Harmonic Mean,Median and Mode in Statistics.Please give me some help !

Answers:I use statistics quite a bit in my work, but you asked a big question with a big answer! There's much to be said about the different pythagorean means (arithmetic, geometric, harmonic) and why and how they are used. I think the web page below gives a very good description of each.

Question:for a takehome test there was this problem about geomertic mean. she didn't even teach us how to do it! i need help please! Use the given information to find all possible values of x. (assume the given quantities must be positive) 1) The geometric mean of x - 3 and x + 4 is x 2) The geometric mean of x and x^2 is 8 could you please explain how to solve it so i can do it by myself next time? mucho appreciation :D

Answers:The geometric mean of 2 and 18 is 6 because 6/2 = 18/6. So 1) x/(x-3) = (x+4)/x x^2 = (x-3)(x+4) x^2 = x^2 +x - 12 12 = x 2) 8/x = x^2/8 64 = x^3 x = 4 I hope this helps!

Question:I missed a week of school and I need help..... I need to know how to figure out the Geometric mean of several problems... Also why do sometimes the answers come out in a regular number (i.e. 8) and sometimes like 4sqrt(2) 4 and 8 2 and 7 9 and 12 8 and 9 3 and 9 2 and 32 4 and 16 15 and 20 5 and 45 12 and 4 THANKS

Answers:The geometric mean of two numbers is the square root of their product. If the product is a perfect square, then the GM will be an integer, otherwise it will be a surd GM of 4 and 8 = sqrt(4*8) = sqrt(32) = 4sqrt(2) GM of 2 and 32 = sqrt(2*32) = sqrt(64) = 8

From Youtube

How To Find the Geometric Mean :Expand the description and view the text of the steps for this how-to video. Check out Howcast for other do-it-yourself videos from Saysa and more videos in the Mathematics Tests category. You can contribute too! Create your own DIY guide at www.howcast.com or produce your own Howcast spots with the Howcast Filmmakers Program at www.howcast.com The geometric mean of two numbers is the square root of their product. More generally, it is the "nth" root of the product of n numbers. To complete this How-To you will need: A set of numbers A calculator with advanced math functions Step 1: Take the square root for two numbers Take the square root of the product if you are only dealing with two numbers. This is the geometric mean when there are only two numbers. Tip: The geometric mean of 2 and 72 is the square root of their product, or 12. Step 2: Determine logarithms for more than two numbers If you are dealing with more than two numbers, determine the logarithm of each number that will be multiplied. Use the logarithmic function key on your calculator to calculate these values. Step 3: Add the values Add each of these logarithmic values together. Step 4: Divide by the number of values Divide the sum of the logarithmic values by the total number of values. Step 5: Determine the antilog value Determine the antilog value of the average using the antilogarithm function key on your calculator. This is the geometric mean for the general case. Thanks for watching How To Find the ...

Geometric Mean.flv :