# average

## Information about average

In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. The most common method is the arithmetic mean, but there are more than one type of average (median being another common example).

Colloquially, people often use the term average to refer to an intuitive central tendency without having a specific measurement of central tendency in mind, or use terms such as "the average person". However, the phrase "there's no such thing as an average citizen" emphasizes that the average is a number, not a person or some other object. The average is calculated by combining the measurements related to a group of people or objects, to compute a number as being the average of the group.

Please see the table of mathematical symbols for explanations of the symbols used. In statistics, the term central tendency is used in some fields of empirical research to refer to what statisticians sometimes call "location". A "measure of central tendency" is either a location parameter or a statistic used to estimate a location parameter.

## Measures of central tendency

There are many kinds of averages. The fundamental concept they have in common is that they are all ways of preserving a property of a list, which is symetric with permutations of the list, when each element of the list is replaced by the constant value of the average. For instance, that arithmetic average preserves the sum of a list when each element of the list is replaced by the arithmetic average. Thus, the arithmetic average, A, of 2 and 8 is obtained by solving: 2 + 8 = A + A. The geometric average perserves the product of a list when each element of the list is replaced by the geometric average. Thus, the geometric average, G, of 2 and 8 is obtained by solving: 2*8 = G*G. Different kinds of averages are described below.

Name Equation or description
Arithmetic mean
MedianThe middle value that separates the higher half from the lower half of the data set
Geometric medianA rotation invariant extension of the median for points in Rn
ModeThe most frequent value in the data set
Geometric mean
Harmonic mean
(or RMS)
Generalized mean
Heronian mean where j≠i
Weighted mean
Truncated meanThe arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded
Interquartile meanA special case of the truncated mean, using the interquartile range
Midrange
Winsorized meanSimilar to the truncated mean, but, rather than deleting the extreme values, they are set equal to the largest and smallest values that remain
AnnualizationAnnualization of a set of returns is a variation on the geometric average that provides the intensive property of a return per year corresponding to a list of returns. Define the return factor as one plus the return. The annualized return factor of a list of returns is the Tth root of the product of their return factors, where T is the sum of the durations of the periods of the returns. For instance, if the return for one year is 10% and the return for a subsequent half a year is 5% then the annualized return of both of these returns together is obtained by taking the product of 1.1 and 1.05, and then taking the result to the power of one over 1.5 and then subtracting one, which gives approximately 10.08%.

## Other averages

Other more sophisticated averages are: trimean, trimedian, and normalized mean. These are usually more representative of the whole data set.

One can create one's own average metric using generalized f-mean:

where f is any invertible function. The harmonic mean is an example of this using f(x) = 1/x, and the geometric mean is another, using f(x) = log x. Another example, expmean (exponential mean) is a mean using the function f(x) = ex, and it is inherently biased towards the higher values. However, this method for generating means is not general enough to capture all averages. A more general method for defining an average, y, takes any function of a list g(x_1, x_2, ..., x_n), which is symmetric under permutation of the members of the list, and equates it to the same function with the value of the average replacing each member of the list: g(x_1, x_2, ..., x_n) = g(y, y, ..., y). This most general definition still captures the important property of all averages that the average of a list of identical elements is that element itself.

## Average applied to a data stream

The concept of an average can be applied to a stream of data as well as a bounded set, the goal being to find a value about which recent data is in some way clustered. The stream may be distributed in time, as in samples taken by some data acquisition system from which we want to remove noise, or in space, as in pixels in an image from which we want to extract some property. An easy-to-understand and widely used application of average to a stream is the simple moving average in which we compute the arithmetic mean of the most recent N data items in the stream. To advance one position in the stream, we add 1/N times the new data item and subtract 1/N times the data item N places back in the stream.

## Derivation of the name

The original meaning of the word average is "damage sustained at sea": the same word is found in Arabic as awar, in Italian as avaria and in French as avarie. Hence an average adjuster is a person who assesses an insurable loss.

Marine damage is either particular average, which is borne only by the owner of the damaged property, or general average, where the owner can claim a proportional contribution from all the parties to the marine venture. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean".

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
data set (or dataset) is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question.
expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample and the measures. Together with simple graphics analysis, they form the basis of virtually every quantitative analysis of data.
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. The arithmetic mean is what students are taught very early to call the "average".
median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
cross

·
multiplication 3 · 4 means the multiplication of 3 by 4.
Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.
Empirical research is any research that bases its findings on direct or indirect observation as its test of reality. Such research may also be conducted according to hypothetico-deductive procedures, such as those developed from the work of R. A. Fisher.
location parameter, since its value determines the "location" of the probability distribution.

In other words, when you graph the function, the location parameter determines where the origin will be located.
A statistic (singular) is the result of applying a function (statistical algorithm) to a set of data.
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. The arithmetic mean is what students are taught very early to call the "average".
median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
The geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central
rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming.

## Definition

In mathematics, an invariant is something that does not change under a set of transformations. The property of being an invariant is invariance.
median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.
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.
In mathematics, the harmonic mean (formerly 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.
In mathematics, root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g. waves.
A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means.

## Definition

If is a non-zero real number, we can define the generalized mean with exponent
The Heronian mean of two non-negative real numbers and is given by . It is named after Hero of Alexandria.

The volume of a frustum of a pyramid (or cone) is found by multiplying the height of the frustum by the Heronian mean of the areas of the opposing parallel faces.
See weight function for the continuous case.

The weighted mean, or weighted average, of a non-empty list of data

with corresponding non-negative weights

A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, and typically discarding an
The interquartile mean (IQM) is a statistical measure of central tendency, much like the mean (in more popular terms called the average), the median, and the mode.

The IQM is a truncated mean