In statistics, given a set of data,

and corresponding weights,

the **weighted geometric mean** is calculated as

If all the weights are equal, the weighted geometric mean is the same as the geometric mean.

Weighted versions of other means can also be calculated. Probably the best known weighted mean is the weighted arithmetic mean, usually simply called the weighted mean. Another example of a weighted mean is the weighted harmonic mean.

The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values.

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In mathematics and statistics, the **arithmetic mean**, or simply the mean or the **average**, is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

In mathematics, the **geometric mean** is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the *n*th root of the product of n numbers, i.e., for a set of numbers *x*_{1}, *x*_{2}, ..., *x _{n}*, the geometric mean is defined as

In mathematics, **generalized means** are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means.

In mathematics, the **harmonic mean** is one of several kinds of average, and in particular, one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired.

In mathematics, the **logarithm** is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the *base* b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 10^{3}, the "logarithm base 10" of 1000 is 3, or log_{10}(1000) = 3. The logarithm of x to *base*b is denoted as log_{b}(*x*), or without parentheses, log_{b} *x*, or even without the explicit base, log *x*, when no confusion is possible, or when the base does not matter such as in big O notation.

There are several kinds of **mean** in mathematics, especially in statistics:

The **natural logarithm** of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln *x*, log_{e}*x*, or sometimes, if the base e is implicit, simply log *x*. Parentheses are sometimes added for clarity, giving ln(*x*), log_{e}(*x*), or log(*x*). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The **weighted arithmetic mean** is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

In probability theory and statistics, the **exponential distribution** is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

In probability theory, a **log-normal distribution** is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then *Y* = ln(*X*) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, *X* = exp(*Y*), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics.

In probability theory and statistics, the **geometric standard deviation** (**GSD**) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual *arithmetic* standard deviation, the *geometric* standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values. Thus, the geometric standard deviation may be more appropriately called **geometric SD factor**. When using geometric SD factor in conjunction with geometric mean, it should be described as "the range from to, and one cannot add/subtract "geometric SD factor" to/from geometric mean.

In mathematics and statistics, the **quasi-arithmetic mean** or **generalised f-mean** is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function . It is also called

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 if and only if every number in the list is the same.

In statistics, a **moving average** is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a **moving mean** (**MM**) or **rolling mean** and is a type of finite impulse response filter. Variations include: simple, and cumulative, or weighted forms.

**Estimation theory** is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.

In statistics and information theory, a **maximum entropy probability distribution** has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class, then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time.

A **diversity index** is a quantitative measure that reflects how many different types there are in a dataset and that can simultaneously take into account the phylogenetic relations among the individuals distributed among those types, such as *richness*, *divergence* or *evenness*. These indices are statistical representations of biodiversity in different aspects.

In mathematics, the **logarithmic mean** is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer.

In mathematics, the **Lehmer mean** of a tuple of positive real numbers, named after Derrick Henry Lehmer, is defined as:

In mathematics and statistics, the **Fréchet mean** is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. **Karcher mean** is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.

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