In mathematics, mean has several different definitions depending on the context.
- In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution.^{[1]} In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving $\backslash mu\; =\; \backslash sum\; x\; P(x)$.^{[2]} An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite: for example, when the probability of the value $2^n$ is $\backslash tfrac\{1\}\{2^n\}$ for n = 1, 2, 3, ....
For a data set, the terms arithmetic mean, mathematical expectation, and sometimes average are used synonymously to refer to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x_{1}, x_{2}, ..., x_{n} is typically denoted by $\backslash bar\{x\}$, pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean (denoted $\backslash bar\{x\}$) to distinguish it from the population mean (denoted $\backslash mu$ or $\backslash mu\_x$).^{[3]}
For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.^{[4]}
Outside of probability and statistics, a wide range of other notions of "mean" are often used in geometry and analysis; examples are given below.
Types of mean
Pythagorean means
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Arithmetic mean (AM)
The arithmetic mean (or simply "mean") of a sample $x\_1,x\_2,\backslash ldots,x\_n$ is the sum the sampled values divided by the number of items in the sample:
- $\backslash bar\{x\}\; =\; \backslash frac\{x\_1+x\_2+\backslash cdots\; +x\_n\}\{n\}$
For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is
- $\backslash frac\{4\; +\; 36\; +\; 45\; +\; 50\; +\; 75\}\{5\}\; =\; \backslash frac\{210\}\{5\}\; =\; 42.$
The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data.
Nevertheless, many skewed distributions are best described by their mean – such as the exponential and Poisson distributions.
Geometric mean (GM)
The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.
- $\backslash bar\{x\}\; =\; \backslash left\; (\; \backslash prod\_\{i=1\}^n\{x\_i\}\; \backslash right\; )\; ^\backslash tfrac1n$
For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:
- $(4\; \backslash times\; 36\; \backslash times\; 45\; \backslash times\; 50\; \backslash times\; 75)^\{^1/\_5\}\; =\; \backslash sqrt[5]\{24\backslash ;300\backslash ;000\}\; =\; 30.$
Harmonic mean (HM)
The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).
- $\backslash bar\{x\}\; =\; n\; \backslash cdot\; \backslash left\; (\; \backslash sum\_\{i=1\}^n\; \backslash frac\{1\}\{x\_i\}\; \backslash right\; )\; ^\{-1\}$
For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is
- $\backslash frac\{5\}\{\backslash tfrac\{1\}\{4\}+\backslash tfrac\{1\}\{36\}+\backslash tfrac\{1\}\{45\}\; +\; \backslash tfrac\{1\}\{50\}\; +\; \backslash tfrac\{1\}\{75\}\}\; =\; \backslash frac\{5\}\{\backslash ;\backslash tfrac\{1\}\{3\}\backslash ;\}\; =\; 15.$
Relationship between AM, GM, and HM
AM, GM, and HM satisfy these inequalities:
- $AM\; \backslash ge\; GM\; \backslash ge\; HM\; \backslash ,$
Equality holds only when all the elements of the given sample are equal.
Generalized means
Power mean
The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric and harmonic means. It is defined for a set of n positive numbers x_{i} by
- $\backslash bar\{x\}(m)\; =\; \backslash left\; (\; \backslash frac\{1\}\{n\}\backslash cdot\backslash sum\_\{i=1\}^n\{x\_i^m\}\; \backslash right\; )\; ^\backslash tfrac1m$
By choosing different values for the parameter m, the following types of means are obtained:
ƒ-mean
This can be generalized further as the generalized f-mean
- $\backslash bar\{x\}\; =\; f^\{-1\}\backslash left(\{\backslash frac\{1\}\{n\}\backslash cdot\backslash sum\_\{i=1\}^n\{f(x\_i)\}\}\backslash right)$
and again a suitable choice of an invertible ƒ will give
Weighted arithmetic mean
The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from samples of the same population with different sample sizes:
- $\backslash bar\{x\}\; =\; \backslash frac\{\backslash sum\_\{i=1\}^n\{w\_i\; \backslash cdot\; x\_i\}\}\{\backslash sum\_\{i=1\}^n\; \{w\_i\}\}.$
The weights $w\_i$ represent the sizes of the different samples. In other applications they represent a measure for the reliability of the influence upon the mean by the respective values.
Truncated mean
Sometimes a set of numbers might contain outliers, i.e., data values which are much lower or much higher than the others.
Often, outliers are erroneous data caused by artifacts. In this case, one can use a truncated mean. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end, and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of total number of values.
Interquartile mean
The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.
- $\backslash bar\{x\}\; =\; \{2\; \backslash over\; n\}\; \backslash sum\_\{i=(n/4)+1\}^\{3n/4\}\{x\_i\}$
assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.
Mean of a function
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In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by
- $\backslash bar\{f\}=\backslash frac\{1\}\{b-a\}\backslash int\_a^bf(x)\backslash ,dx.$
Recall that a defining property of the average value $\backslash bar\{y\}$ of finitely many numbers $y\_1,\; y\_2,\; \backslash dots,\; y\_n$
is that $n\backslash bar\{y\}\; =\; y\_1\; +\; y\_2\; +\; \backslash cdots\; +\; y\_n$. In other words, $\backslash bar\{y\}$ is the constant value which when
added to itself $n$ times equals the result of adding the $n$ terms of $y\_i$. By analogy, a
defining property of the average value $\backslash bar\{f\}$ of a function over the interval $[a,b]$ is that
- $\backslash int\_a^b\backslash bar\{f\}\backslash ,dx\; =\; \backslash int\_a^bf(x)\backslash ,dx$
In other words, $\backslash bar\{f\}$ is the constant value which when integrated over $[a,b]$ equals the result of
integrating $f(x)$ over $[a,b]$. But by the second fundamental theorem of calculus, the integral of a constant
$\backslash bar\{f\}$ is just
- $\backslash int\_a^b\backslash bar\{f\}\backslash ,dx\; =\; \backslash bar\{f\}x\backslash bigr|\_a^b\; =\; \backslash bar\{f\}b\; -\; \backslash bar\{f\}a\; =\; (b\; -\; a)\backslash bar\{f\}$
See also the first mean value theorem for integration, which guarantees
that if $f$ is continuous then there exists a point $c\; \backslash in\; (a,\; b)$ such that
- $\backslash int\_a^bf(x)\backslash ,dx\; =\; f(c)(b\; -\; a)$
The point $f(c)$ is called the mean value of $f(x)$ on $[a,b]$. So we write
$\backslash bar\{f\}\; =\; f(c)$ and rearrange the preceding equation to get the above definition.
In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by
- $\backslash bar\{f\}=\backslash frac\{1\}\{\backslash hbox\{Vol\}(U)\}\backslash int\_U\; f.$
This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of f to be
- $\backslash exp\backslash left(\backslash frac\{1\}\{\backslash hbox\{Vol\}(U)\}\backslash int\_U\; \backslash log\; f\backslash right).$
More generally, in measure theory and probability theory, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.
There is also a harmonic average of functions and a quadratic average (or root mean square) of functions.
Mean of a probability distribution
See expected value.
Mean of angles
Sometimes the usual calculations of means fail on cyclical quantities such as angles, times of day, and other situations where modular arithmetic is used. For those quantities it might be appropriate to use a mean of circular quantities to take account of the modular values, or to adjust the values before calculating the mean.
Fréchet mean
The Fréchet mean gives a manner for determining the "center" of a mass distribution on a surface or, more generally, Riemannian manifold. Unlike many other means, the Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars.
It is sometimes also known as the Karcher mean (named after Hermann Karcher).
Other means
Distribution of the population mean
Using the sample mean
The arithmetic mean of a population, or population mean, is denoted μ. The sample mean (the arithmetic mean of a sample of values drawn from the population) makes a good estimator of the population mean, as its expected value is equal to the population mean (that is, it is an unbiased estimator). The sample mean is a random variable, not a constant, since its calculated value will randomly differ depending on which members of the population are sampled, and consequently it will have its own distribution. For a random sample of n observations from a normally distributed population, the sample mean distribution is normally distributed with mean and variance as follows:
- $\backslash bar\{x\}\; \backslash thicksim\; N\backslash left\backslash \{\backslash mu,\; \backslash frac\{\backslash sigma^2\}\{n\}\backslash right\backslash \}.$
Often, since the population variance is an unknown parameter, it is estimated by the mean sum of squares; when this estimated value is used, the distribution of the sample mean is no longer a normal distribution but rather a Student's t distribution with n − 1 degrees of freedom.
See also
References
External links
- MathWorld.
- MathWorld.
- Comparison between arithmetic and geometric mean of two numbers
- Some relationships involving means
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