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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of 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. 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 In Mathematics, an n th root of a Number a is a number b such that bn = a. In Mathematics, a product is the Result of multiplying, or an expression that identifies factors to be multiplied

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 the cube root (i. e. , the third root) of their product, 0. 125, which is ½.

The geometric mean can be understood in terms of geometry. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The geometric mean of two numbers, a and b, is simply the side length of the square whose area is equal to that of a rectangle with side lengths a and b. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as That is, what is n such that n² = a × b? Similarly, the geometric mean of three numbers, a, b, and c, is the side length of a cube whose volume is the same as that of a rectangular prism with side lengths equal to the three given numbers. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong This geometric interpretation of the mean is very likely what gave it its name.

The geometric mean only applies to positive numbers. [1] 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 world population is the total number of living Humans on Earth at a given time The geometric mean is also one of the three classic Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. The three classical Pythagorean means are the Arithmetic mean ( A) the Geometric mean ( G) and the Harmonic mean ( H) In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average.

Calculation

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

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

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). 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 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 This allows the definition of the arithmetic-geometric mean, a mixture of the two which always lies in between. In Mathematics, the arithmetic-geometric mean (AGM of two positive Real numbers x and y is defined as follows First compute the Arithmetic

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. In Mathematics, a sequence is an ordered list of objects (or events

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. Algebraic identities Using simpler operations Logarithms can be used

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

This is sometimes called the log-average. It is simply computing the arithmetic mean of the logarithm transformed values of ai (i. 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 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 quasi-arithmetic mean or generalised f -mean is one generalisation of the more familiar Means such

Therefore the geometric mean is related to the log-normal distribution. In Probability and Statistics, the log-normal distribution is the single-tailed Probability distribution of any Random variable whose 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)).

Notes and references

1. ^ The geometric mean only applies to positive numbers in order to avoid taking the root of a negative product, which would result in imaginary numbers, and also to satisfy certain properties about means, which is explained later in the article. Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane