Mean (mathematics)

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In mathematics and statistics, a mean or average is a measure of central tendency. It results from performing a certain series of operations with a set of numbers and that, under certain conditions, can represent the entire set by itself. There are different types of means, such as the geometric mean, the weighted mean and the harmonic mean, although in common language, both in statistics and in mathematics, the elementary of all of them is the term that generally refers to the arithmetic mean.

Examples of stockings

There are numerous examples of stockings. $\scriptstyle {\bar {x}}=m_{i}(x_{1},\dots ,x_{n})$ , one of the few properties shared by all means is that any mean is between the maximum value and the minimum value of the set of variables:

$\min\{x_{1},x_{2},\dots x_{n}\}\leq {\bar {x}}\leq \max\{x_{1},x_{2},\dots x_ {n}\}$

In addition, it must be fulfilled that:

${\bar {x}}=x_{1},\quad {\mbox{if}}\ x_{1}=x_{2}=\dots =x_{n}$ .

Arithmetic average

The arithmetic mean is a standard average that is often called the average .

${\bar {x}}={\frac {1}{n}}\sum{i=1}}^{n}{x_{i}}$

The mean is sometimes confused with the median or mode. The arithmetic mean is the average of a set of values, or their distribution; however, for skewed distributions, the mean is not necessarily the same value as the median or the exponential and Poisson mode.

For example, the arithmetic mean of 34, 27, 45, 55, 22, 34 (six values) is ${\tfrac {34+27+45+55+22+34}{6}}\ ={\tfrac {217}{6}}\approx 36.167$

Weighted arithmetic mean

Sometimes it can be useful to assign weights or values to the data depending on their relevance to a given study. In those cases a weighted average can be used. Yes $X_{1},X_{2},\ldots ,X_{n}$ is a data set or sample mean and $w_{1},w_{2},\ldots ,w_{n}$ are positive real numbers, called "weights" or weighting factors, the weighted average is defined, that is, it is relative to these weights as:

${\bar {X}}_{w}={\frac {X_{1}\cdot w_{1}+X_{2}\cdot w_{2}+\ldots +X_{n}\cdot w_{n}}{w_{1}+w_{2}+\ldots +w_{n}}}={\frac {\sum{i=1}^{n}X_{i}\cdot w_{ i}}{\sum i=1}^{n}w_{i}}}$

The mean is invariant against linear transformations, change of origin and scale, of the variables, that is, if X is a random variable and Y is another random variable that depends linearly on X , that is, Y = a XL + b ( where a represents the magnitude of the change of scale and b that of the change of origin) we have that:

${\bar {Y}}=a{\bar {X}}+b$

Geometric mean

The geometric mean is a very useful average on sets of numbers that are interpreted in order of their product, not their sum (as is the case with the arithmetic mean). For example, growth rates. ${\bar {x}}=\left(\prod{i=1}}^{n}{x_{i}}\right)^{{1/n}}$

For example, the geometric mean of the series of numbers 1,2,3,4,5,9 (six values) is $(1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 9)^{1/6}=1080^{1/6}\approx 3.203$

Harmonic mean

The harmonic mean is a very useful average on sets of numbers that are defined relative to some unit, for example speed (distance per unit of time). ${\bar {x}}=n\cdot \left(\sum{i=1}}^{n}{\frac {1}{x_{i}}}\right)^{{-1} }$

For example, the harmonic mean of the numbers: 34, 27, 45, 55, 22, and 34 is: ${\frac {6}{{\frac {1}{34}}+{\frac {1}{27}}+{\frac {1}{45}}+{\frac {1}{55}} +{\frac {1}{22}}+{\frac {1}{34}}}}\approx 33,018$

Generalizations of the mean

There are various generalizations of the above averages.

Generalized mean

Generalized means, also known as Hölder means, are an abstraction of square, arithmetic, geometric, and harmonic means. They are defined and grouped through the following expression:

${\bar {x}}(m)=\left({\frac {1}{n}}\cdot \sum{i=1}}^{n}{x_{i}^{m}} \right)^{{1/m}}$

Choosing an appropriate value of the parameter m , we have:

$m\rightarrow \infty$ - maximum,
$m=2\,$ - root mean square,
$m=1\,$ - arithmetic mean ,
$m\rightarrow 0$ - geometric mean,
$m=-1\,$ - harmonic mean,
$m\rightarrow -\infty$ - minimum.

Mean- generalized f

This mean can be generalized to a monotonic function as the generalized f-mean:

${\bar {x}}=f^{{-1}}\left({{\frac {1}{n}}\cdot \sum{i=1}}^{n}{f(x_ {i})}}\right)$

where $f:I\to I$ be an injective function and $I\subset \mathbb{R}$ an interval. Choosing particular forms for f yields some of the best known means:

$f(x)=x\,$ - arithmetic mean , $I=\mathbb{R}$
$f(x)={\frac {1}{x}}$ - harmonic mean, $I=(0,\infty )$
$f(x)=x^{m}\,$ - generalized mean, $I=(0,\infty )$
$f(x)=\ln x\,$ - geometric mean, $I=(0,\infty )$ .

Mean of a function

For a continuous function $F$ over an interval [a,b], the mean value of function can be calculated $F$ over [a,b] like:

${\bar {f}}={\frac {1}{ba}}\int a}^{b}f(t)dt$

In fact, the previous definition is valid even for a bounded function even if it is not continuous, provided that it is measurable.

Half statistic

The statistical mean is used in statistics for two different but numerically similar concepts:

The sample mean , which is a statistic that is calculated from the arithmetic mean of a set of values of a random variable.
The population mean , expected value, or mathematical expectation of a random variable.

In practice, given a sufficiently large statistical sample, the value of its sample mean is numerically very close to the mathematical expectation of the random variable measured in that sample. Said expected value can only be calculated if the probability distribution is known exactly, which rarely happens in reality. For this reason, for practical purposes, the so-called mean normally refers to the sample mean.

Sample mean

The sample mean is a random variable, since it depends on the sample, although it is a random variable in general with a smaller variance than the original variables used in its calculation. If the sample is large and well chosen, the sample mean can be treated as a numerical value that accurately approximates the population mean, characterizing an objective property of the population. It is defined as follows, if we have a statistical sample of values $(X_{1},X_{2},\ldots ,X_{n})$ for a random variable X with probability distribution F ( x ,θ) [where θ is a set of distribution parameters] the nth sample mean is defined as:

${\bar {X}}_{n}=T(X_{1},X_{2},\ldots ,X_{n})={\frac {1}{n}}\sum i=1}^{n}X_{i}={\frac {X_{1}+X_{2}+\ldots +X_{n}}{n}}$ .

Population mean

The population mean is technically not a mean but a fixed parameter that matches the mathematical expectation of a random variable. The name "population mean" is used to mean what numerical value of a sample mean is numerically close to the parameter population mean, for a suitable and sufficiently large sample.

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