Mean 19192 223960372 2008-07-06T18:31:00Z Mwtoews 711150 /* Other means */ + Moving average {{Otheruses}} In [[statistics]], '''''mean''''' has two related meanings: * the [[arithmetic mean]] (and is distinguished from the [[geometric mean]] or [[harmonic mean]]). * the [[expected value]] of a [[random variable]], which is also called the ''population mean''. It is sometimes stated that the 'mean' means average. This is incorrect if "mean" is taken in the specific sense of "arithmetic mean" as there are different types of averages: the mean, [[median]], and [[mode_(statistics)|mode]]. For instance, average house prices almost always use the median value for the average. For a real-valued [[random variable]] ''X'', the mean is the [[expectation operator|expectation]] of ''X''. Note that not every [[probability distribution]] has a defined mean (or [[variance]]); see the [[Cauchy distribution]] for an example. For a [[data set]], the mean is the sum of the observations divided by the number of observations. The mean is often quoted along with the [[standard deviation]]: the mean describes the central location of the data, and the standard deviation describes the spread. An alternative measure of dispersion is the mean deviation, equivalent to the average [[absolute deviation]] from the mean. It is less sensitive to outliers, but less mathematically tractable. As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. These are listed below. ==Examples of means== ===Arithmetic mean=== {{main|Arithmetic mean}} The ''arithmetic mean'' is the "standard" average, often simply called the "mean". :<math> \bar{x} = \frac{1}{n}\cdot \sum_{i=1}^n{x_i} </math> The '''mean''' may often be confused with the [[median]] or [[Mode (statistics)|mode]]. The mean is the arithmetic average of a set of values, or distribution; however, for [[skewness|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 distribution|Exponential]] and [[Poisson distribution|Poisson]] distributions. For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is: :<math>\frac{34+27+45+55+22+34}{6} = \frac{217}{6} \approx 36.167.</math> ===Geometric mean=== 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). For example rates of growth. :<math> \bar{x} = \left ( \prod_{i=1}^n{x_i} \right ) ^{1/n}</math> For example, the geometric mean of six values: 34, 27, 45, 55, 22, 34 is: :<math>(34 \cdot 27 \cdot 45 \cdot 55 \cdot 22 \cdot 34)^{1/6} = 1,699,493,400^{1/6} = 34.545.</math> ===Harmonic mean=== The [[harmonic mean]] is an average which is useful for sets of numbers which are defined in relation to some [[Unit of measurement|unit]], for example [[speed]] (distance per unit of time). :<math> \bar{x} = n \cdot \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}</math> For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is :<math>\frac{6}{\frac{1}{34}+\frac{1}{27}+\frac{1}{45} + \frac{1}{55} + \frac{1}{22}+\frac{1}{34}} = \frac{60588}{1835} \approx 33.0179836.</math> ===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''<sub>i</sub> by :<math> \bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{x_i^m} \right ) ^{1/m} </math> By choosing the appropriate value for the parameter ''m'' we get {| |<math>m\rightarrow\infty</math> || [[maximum]] |- | <math>m=2</math> || [[quadratic mean]], |- | <math>m=1</math> || [[arithmetic mean]], |- | <math>m\rightarrow0</math> || [[geometric mean]], |- | <math>m=-1</math> || [[harmonic mean]], |- | <math>m\rightarrow-\infty</math> || [[minimum]]. |} ====f-mean==== This can be generalized further as the [[generalized f-mean]] :<math> \bar{x} = f^{-1}\left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right) </math> and again a suitable choice of an invertible <math>f</math> will give {| <math>f(x) = x</math> || [[arithmetic mean]], |- | <math>f(x) = \frac{1}{x}</math> || [[harmonic mean]], |- | <math>f(x) = x^m</math> || [[power mean]], |- | <math>f(x) = \ln x</math> || [[geometric mean]]. |} ===Weighted arithmetic mean=== The [[weighted mean|weighted arithmetic mean]] is used, if one wants to combine average values from samples of the same population with different sample sizes: :<math> \bar{x} = \frac{\sum_{i=1}^n{w_i \cdot x_i}}{\sum_{i=1}^n {w_i}}. </math> The weights <math>w_i</math> represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values. ===Truncated mean=== Sometimes a set of numbers might contain outliers, i.e. a [[datum]] which is much lower or much higher than the others. Often, outliers are erroneous data caused by [[artifact (observational)|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. :<math> \bar{x} = {2 \over n} \sum_{i=(n/4)+1}^{3n/4}{x_i} </math> assuming the values have been ordered. ===Mean of a function=== In [[calculus]], and especially [[multivariable calculus]], the mean of a function is loosely defined as the average value of the function over its [[domain (mathematics)|domain]]. In one variable, the mean of a function ''f''(''x'') over the interval (''a,b'') is defined by : <math>\bar{f}=\frac{1}{b-a}\int_a^bf(x)\,dx.</math> (See also [[mean value theorem]].) In several variables, the mean over a [[relatively compact]] [[neighborhood (mathematics)|domain]] ''U'' in a [[Euclidean space]] is defined by :<math>\bar{f}=\frac{1}{\hbox{Vol}(U)}\int_U f.</math> 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 :<math>\exp\left(\frac{1}{\hbox{Vol}(U)}\int_U \log f\right).</math> 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. ===Mean of angles=== Most of the usual means fail on circular quantities, like [[angle]]s, [[daytime]]s, [[fractional part]]s of [[real number]]s. For those quantities you need a [[mean of circular quantities]]. === Other means === <div style="-moz-column-count:3; column-count:3;"> *[[Arithmetic-geometric mean]] *[[Arithmetic-harmonic mean]] *[[Cesàro mean]] *[[Chisini mean]] *[[Contraharmonic mean]] *[[Elementary symmetric mean]] *[[Geometric-harmonic mean]] *[[Heinz mean]] *[[Heronian mean]] *[[Identric mean]] *[[Least squares mean]] *[[Lehmer mean]] *[[Logarithmic mean]] *[[Median]] <!-- is a mean according to the definition below --> *[[Moving average]] *[[Root mean square]] *[[Stolarsky mean]] *[[Weighted geometric mean]] *[[Weighted harmonic mean]] *[[Rényi's entropy]] (a [[generalized f-mean]]) </div> == Properties == The most general method for defining a mean or 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 mean replacing each member of the list: g(x_1, x_2, ..., x_n) = g(y, y, ..., y). All means share some properties and additional properties are shared by the most common means. Some of these properties are collected here. === Weighted mean === A '''weighted mean''' <math>M</math> is a function which maps tuples of positive numbers to a positive number (<math>\mathbb{R}_{>0}^n\to\mathbb{R}_{>0}</math>). * "[[Fixed point (mathematics)|Fixed point]]": <math> M(1,1,\dots,1) = 1 </math> * [[Homogeneous function|Homogenity]]: <math> \forall\lambda\ \forall x\ M(\lambda\cdot x_1, \dots, \lambda\cdot x_n) = \lambda \cdot M(x_1, \dots, x_n) </math> :: (using [[coordinate vector|vector]] notation: <math> \forall\lambda\ \forall x\ M(\lambda\cdot x) = \lambda \cdot M x </math>) * [[Monotonic function|Monotony]]: <math> \forall x\ \forall y\ (\forall i\ x_i \le y_i) \Rightarrow M x \le M y </math> It follows * [[Upper bound|Boundedness]]: <math> \forall x\ M x \in [\min x, \max x] </math> * [[Continuous function|Continuity]]: <math> \lim_{x\to y} M x = M y </math> <!-- for all convergent sequences <math>x</math> it holds <math> \lim (M \circ x) = M (\lim x) </math> --> :Sketch of a proof: Because <math>\forall x\ \forall y\ \left(||x-y||_\infty\le\varepsilon\cdot\min x \Rightarrow \forall i\ |x_i-y_i|\le\varepsilon\cdot x_i\right)</math> and <math>M((1+\varepsilon)\cdot x) = (1+\varepsilon)\cdot M x</math> it follows <math>\forall x\ \forall \varepsilon>0\ \forall y\ ||x-y||_\infty\le\varepsilon\cdot\min x \Rightarrow |Mx-My|\le\varepsilon</math>. * There are means, which are not [[derivative|differentiable]]. For instance, the maximum number of a tuple is considered a mean (as an extreme case of the [[power mean]], or as a special case of a [[median]]), but is not differentiable. * All means listed above, with the exception of most of the [[Generalized f-mean]]s, satisfy the presented properties. ** If <math>f</math> is bijective, then the generalized f-mean satisfies the fixed point property. ** If <math>f</math> is strictly monotonic, then the generalized f-mean satisfy also the monotony property. ** In general a generalized f-mean will miss homogenity. The above properties imply techniques to construct more complex means: If <math>C, M_1, \dots, M_m</math> are weighted means, <math>p</math> is a positive [[real number]], then <math>A, B</math> with : <math> \forall x\ A x = C(M_1 x, \dots, M_m x) </math> : <math> \forall x\ B x = \sqrt[p]{C(x_1^p, \dots, x_n^p)} </math> are also a weighted mean. === Unweighted mean === Intuitively spoken, an '''unweighted mean''' is a weighted mean with equal weights. Since our definition of ''weighted mean'' above does not expose particular weights, equal weights must be asserted by a different way. A different view on homogeneous weighting is, that the inputs can be swapped without altering the result. Thus we define <math>M</math> being an unweighted mean if it is a weighted mean and for each [[permutation]] <math>\pi</math> of inputs, the result is the same. Let <math>P</math> be the set of permutations of <math>n</math>-tuples. : [[Symmetric function|Symmetry]]: <math> \forall x\ \forall \pi\in P \ M x = M(\pi x) </math> Analogously to the weighted means, if <math>C</math> is a weighted mean and <math>M_1, \dots, M_m</math> are unweighted means, <math>p</math> is a positive [[real number]], then <math>A, B</math> with : <math> \forall x\ A x = C(M_1 x, \dots, M_m x) </math> : <math> \forall x\ B x = \sqrt[p]{M_1(x_1^p, \dots, x_n^p)} </math> are also unweighted means. === Convert unweighted mean to weighted mean === An unweighted mean can be turned into a weighted mean by repeating elements. This connection can also be used to state that a mean is the weighted version of an unweighted mean. Say you have the unweighted mean <math>M</math> and weight the numbers by natural numbers <math>a_1,\dots,a_n</math>. (If the numbers are [[rational number|rational]], then multiply them with the [[least common denominator]].) Then the corresponding weighted mean <math>A</math> is obtained by :<math>A(x_1,\dots,x_n) = M(\underbrace{x_1,\dots,x_1}_{a_1},x_2,\dots,x_{n-1},\underbrace{x_n,\dots,x_n}_{a_n}).</math> === Means of tuples of different sizes === If a mean <math>M</math> is defined for tuples of several sizes, then one also expects that the mean of a tuple is bounded by the means of partitions. More precisely * Given an arbitrary tuple <math>x</math>, which is [[Partition of a set|partitioned]] into <math>y_1, \dots, y_k</math>, then it holds <math>M x \in \mathrm{convexhull}(M y_1, \dots, M y_k)</math>. (See [[Convex hull]]) ==Population and sample means== The mean of a [[Normal distribution|normally distributed]] [[Statistical population|population]] has an expected value of μ, known as the population mean. The sample mean makes a good [[estimator]] of the population mean, as its expected value is the same as the population mean. The sample mean of a population is a [[random variable]], not a constant, 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 : <math>\bar{x} \thicksim N\left\{\mu, \frac{\sigma^2}{n}\right\}.</math> Often, since the population variance is an unknown parameter, it is estimated by the [[sum of squares | mean sum of squares]], which changes the distribution of the sample mean from a normal distribution to a [[Student's t distribution]] with ''n''&nbsp;&minus;&nbsp;1 [[degrees of freedom]]. ==Mathematics education== {{Unreferencedsection|date=February 2008}}{{globalize/USA}} In many state and government curriculum standards, students are traditionally expected to learn either the meaning or formula for computing the mean by the fourth grade. However, in many [[standards-based mathematics]] curricula, students are encouraged to invent their own methods, and may not be taught the traditional method. Reform based texts such as [[Investigations in Numbers, Data, and Space|TERC]] in fact discourage teaching the traditional "add the numbers and divide by the number of items" method in favor of spending more time on the concept of [[median]], which does not require division. However, mean can be computed with a simple four-function calculator, while median requires a computer. The same teacher guide devotes several pages on how to find the median of a set, which is judged to be simpler than finding the mean. ==See also== *[[Average]], same as ''central tendency'' *[[Descriptive statistics]] *[[Kurtosis]] *[[Median]] *[[Mode (statistics)]] *[[Summary statistics]] *[[Law of averages]] *[[Spherical mean]] * For an independent identical distribution from the reals, the mean of a sample is an [[Bias of an estimator|unbiased]] estimator for the mean of the population. ==External links== * [http://www.stats4students.com/Essentials/Measures-Central-Tendency/Overview.php An easy-to-follow guide to understanding & calculating the mean] * [http://www.sengpielaudio.com/calculator-geommean.htm Comparison between arithmetic and geometric mean of two numbers] {{Statistics}} [[Category:Means| ]] [[Category:Mathematics education]] [[da:Gennemsnit]] [[de:Mittelwert]] [[es:Promedio]] [[eo:Meznombro]] [[fa:میانگین]] [[fi:Keskiluku]] [[fr:Moyenne]] [[he:ממוצע]] [[it:Media (statistica)]] [[ja:平均]] [[ko:평균]] [[lo:ຄ່າສະເຫຼ່ຍ]] [[nl:Gemiddelde]] [[no:Gjennomsnitt]] [[pl:Średnia]] [[pt:Média]] [[ru:Среднее значение]] [[sl:Srednja vrednost]] [[su:Mean]] [[th:มัชฌิม]] [[tr:Ortalama]] [[zh:平均数]]