This article is about mathematical mean. For a definition of the word "mean", see the Wiktionary entry mean. In statistics, mean has two related meanings: This article is about the field of statistics. ...
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. For instance, average house prices almost always use the median value for the average. 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 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. ...
In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...
This article is about the statistical concept. ...
In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...
For a realvalued random variable X, the mean is the expectation of X. Note that not every probability distribution has a defined mean (or variance); see the Cauchy distribution for an example. In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...
In probability (and especially gambling), the expected value (or expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are...
A probability distribution describes the values and probabilities that a random event can take place. ...
This article is about mathematics. ...
The CauchyLorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the halfwidth at halfmaximum (HWHM). ...
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. A data set (or dataset) is a collection of data, usually presented in tabular form. ...
In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...
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. The absolute deviation of an element of a data set is the absolute difference between that element and a given point. ...
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 
The arithmetic mean is the "standard" average, often simply called the "mean". 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 mean may often be confused with the median or mode. 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. This article is about the statistical concept. ...
In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...
Example of experimental data with nonzero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a realvalued random variable. ...
That said, many skewed distributions are best described by their mean  such as the Exponential and Poisson distributions. In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ...
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ...
For example, the arithmetic mean of 34, 27, 45, 55, 22, 34 (six values) is (34+27+45+55+22+34)/6 = 217/6 ≈ 36.167.
Geometric mean The geometric mean is an average that is useful for sets of 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. 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. ...
For example, the geometric mean of 34, 27, 45, 55, 22, 34 (six values) is (34×27×45×55×22×34)^{1/6} = 1,699,493,400^{1/6} = 34.545.
Harmonic mean 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). In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ...
Measurement is the determination of the size or magnitude of something. ...
This article does not cite any references or sources. ...
For example, the harmonic mean of the numbers 34, 27, 45, 55, 22, and 34 is 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 by A generalized mean, also known as power mean or HÃ¶lder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. ...
By choosing the appropriate value for the parameter m we get The largest and the smallest element of a set are called extreme values, or extreme records. ...
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: its specially useful when variates are positive and negative, ie the cases of waves. ...
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 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. ...
The largest and the smallest element of a set are called extreme values, or extreme records. ...
fmean This can be generalized further as the generalized fmean In mathematics and statistics, the generalised fmean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(x). ...
and again a suitable choice of an invertible f will give In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ...
A generalized mean, also known as power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means. ...
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. ...
Weighted arithmetic mean The weighted arithmetic mean is used, if one wants to combine average values from samples of the same population with different sample sizes: In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ...
The weights w_{i} 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 (the data) might be contaminated by inaccurate outliers, i.e. values which are much too low or much too high. 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. For other uses, see Data (disambiguation). ...
A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. ...
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. 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. ...
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. In one variable, the mean of a function f(x) over the interval (a,b) is defined by For other uses, see Calculus (disambiguation). ...
Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. ...
In mathematics, the domain of a function is the set of all input values to the function. ...
(See also mean value theorem.) In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...
In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ...
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
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 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. In mathematics, a measure is a function that assigns a number, e. ...
Probability theory is the branch of mathematics concerned with analysis of random phenomena. ...
In mathematics, Jensens inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. ...
Mean of angles Most of the usual means fail on circular quantities, like angles, daytimes, fractional parts of real numbers. For those quantities you need a mean of circular quantities. This article is about angles in geometry. ...
Look up daytime in Wiktionary, the free dictionary. ...
In mathematics, the floor function is the function defined as follows: for a real number x, floor(x) is the largest integer less than or equal to x. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a mean of circular quantities is a mean which suited for quantities like angles, daytimes, fractional parts of real numbers. ...
Other means In mathematics, the arithmeticgeometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a1, i. ...
The geometric mean of a set of positive data is defined as the product of all the members of the set, raised to a power equal to the reciprocal of the number of members. ...
In mathematics, the CesÃ ro means of a sequence an are the terms of the sequence cn = (a1 + a2 + ... + an)/n constructed as the arithmetic mean of the first n elements. ...
A function f of n variables xn is a Chisini mean, if and only if, for every vector <x1 . ...
Contraharmonic mean describes a mean of a set of numbers that is complementary to the harmonic mean. ...
The elementary symmetric mean is based on elementary symmetric polynomials. ...
In mathematics, the geometricharmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the geometric mean of x and y and call it g1, i. ...
The Heinz mean of two nonnegative real numbers and was defined by Bhatia[1] as: . with 0 =< x =< 1/2. ...
The Heronian mean of two nonnegative real numbers and is given by . ...
The Identric mean of two positive real numbers is defined as: . [edit] See also mean Categories:  ...
The Lehmer mean of a tuple of positive real numbers is defined as: . [edit] See also mean Categories:  ...
In mathematics, the logarithmic mean is a function of two numbers which is equal to their difference divided by the logarithm of their quotient. ...
This article is about the statistical concept. ...
In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ...
The Stolarsky mean of two positive real numbers is defined as: . [edit] See also mean Categories:  ...
This article or section does not cite its references or sources. ...
In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted geometric mean is calculated as Note that if all the weights are equal, the weighted geometric mean is the same as the geometric mean. ...
In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted harmonic mean is calculated as Note that if all the weights are equal, the weighted harmonic mean is the same as the harmonic mean. ...
In information theory, the RÃ©nyi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system. ...
In mathematics and statistics, the generalised fmean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(x). ...
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 M is a function which maps tuples of positive numbers to a positive number (). 
 (using vector notation: )
It follows In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ...
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. ...
A monotonically increasing function (it is strictly increasing on the left and just nondecreasing on the right). ...
 Sketch of a proof: Because and it follows .
 There are means, which are not 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 fmeans, satisfy the presented properties.
 If f is bijective, then the generalized fmean satisfies the fixed point property.
 If f is strictly monotonic, then the generalized fmean satisfy also the monotony property.
 In general a generalized fmean will miss homogenity.
The above properties imply techniques to construct more complex means: In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
This article is about derivatives and differentiation in mathematical calculus. ...
A generalized mean, also known as power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means. ...
This article is about the statistical concept. ...
In mathematics and statistics, the generalised fmean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(x). ...
If are weighted means, p is a positive real number, then A,B with In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
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 M being an unweighted mean if it is a weighted mean and for each permutation π of inputs, the result is the same. Let P be the set of permutations of ntuples. Permutation is the rearrangement of objects or symbols into distinguishable sequences. ...
 Symmetry:
Analogously to the weighted means, if C is a weighted mean and are unweighted means, p is a positive real number, then A,B with In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
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 M and weight the numbers by natural numbers . (If the numbers are rational, then multiply them with the least common denominator.) Then the corresponding weighted mean A is obtained by In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ...
Means of tuples of different sizes If a mean M 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 A partition of U into 6 blocks: an Euler diagram representation. ...
Convex hull: elastic band analogy In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. // For planar objects, i. ...
Population and sample means The mean of a normal distribution 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 Probability density function of Gaussian distribution (bell curve). ...
In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ...
In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...
Often, since the population variance is an unknown parameter, it is estimated by the mean sum of squares, which changes the distribution of the sample mean from a normal distribution to a Student's t distribution with n − 1 degrees of freedom. Sum of squares is a concept that permeates much of inferential statistics and descriptive statistics. ...
In probability and statistics, the tdistribution or Students tdistribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ...
The phrase degrees of freedom is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. ...
Mathematics education  This section does not cite any references or sources. (February 2008) Please improve this section by adding citations to reliable sources. Unverifiable material may be challenged and removed.  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 standardsbased mathematics curricula, students are encouraged to invent their own methods, and may not be taught the traditional method. Reform based texts such as 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 fourfunction 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. Image File history File links Question_book3. ...
Principles and Standards for School Mathematics is a document produced in 1989 by the National Council of Teachers of Mathematics [5] (NCTM) to set forth a national vision for precollege mathematics education in the US and Canada. ...
Investigations in Number, Data, and Space is a complete K5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts. ...
This article is about the statistical concept. ...
See also In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ...
Descriptive statistics are used to describe the basic features of the data in a study. ...
The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (0. ...
This article is about the statistical concept. ...
In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate as much as possible as simply as possible. ...
The law of averages is a lay term used to express the view that eventually, everything evens out. ...
The spherical mean of a function (shown in red) is the average of the values (top, in blue) with on a sphere of given radius around a given point (bottom, in blue). ...
This article is about bias of statistical estimators. ...
External links  An easytofollow guide to understanding & calculating the mean
 Comparison between arithmetic and geometric mean of two numbers
This article is about the field of statistics. ...
Descriptive statistics are used to describe the basic features of the data in a study. ...
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 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. ...
This article is about the statistical concept. ...
In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ...
Look up range in Wiktionary, the free dictionary. ...
This article is about mathematics. ...
In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...
It has been suggested that this article or section be merged with inferential statistics. ...
One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ...
In statistics, a result is significant if it is unlikely to have occurred by chance, given that a presumed null hypothesis is true. ...
The power of a statistical test is the probability that the test will reject a false null hypothesis (that it will not make a Type II error). ...
In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ...
In statistics, the Alternative Hypothesis is the hypothesis proposed to explain a statistically significant difference between results, that is if the Null Hypothesis has been rejected. ...
Type I errors (or Î± error, or false positive) and type II errors (Î² error, or a false negative) are two terms used to describe statistical errors. ...
The Ztest is a statistical test used in inference. ...
A t test is any statistical hypothesis test in which the test statistic has a Students t distribution if the null hypothesis is true. ...
Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ...
Compares the various grading methods in a normal distribution. ...
In statistical hypothesis testing, the pvalue of a random variable T used as a test statistic is the probability that T will assume a value at least as extreme as the observed value tobserved, given that a null hypothesis being considered is true. ...
In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. ...
Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. ...
The survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. ...
The KaplanMeier estimator (also known as the Product Limit Estimator) estimates the survival function from lifetime data. ...
The logrank test (sometimes called the MantelHaenszel test or the MantelCox test) [1] is a hypothesis test to compare the survival distributions of two samples. ...
Failure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. ...
// Proportional hazards models are a subclass of survival models in statistics. ...
Positive linear correlations between 1000 pairs of numbers. ...
In statistics, a spurious relationship (or, sometimes, spurious correlation) is a mathematical relationship in which two occurrences have no logical connection, yet it may be implied that they do, due to a certain third, unseen factor (referred to as a confounding factor or lurking variable). The spurious relationship gives an...
In statistics, the Pearson productmoment correlation coefficient (sometimes known as the PMCC) (r) is a measure of the correlation of two variables X and Y measured on the same object or organism, that is, a measure of the tendency of the variables to increase or decrease together. ...
In statistics, rank correlation is the study of relationships between different rankings on the same set of items. ...
In statistics, Spearmans rank correlation coefficient, named after Charles Spearman and often denoted by the Greek letter (rho) or as , is a nonparametric measure of correlation â€“ that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about...
The Kendall tau rank correlation coefficient (or simply the Kendall tau coefficient, Kendalls Ï„ or Tau test(s)) is used to measure the degree of correspondence between two rankings and assessing the significance of this correspondence. ...
In statistics, regression analysis examines the relation of a dependent variable (response variable) to specified independent variables (explanatory variables). ...
In statistics, linear regression is a regression method that models the relationship between a dependent variable Y, independent variables Xi, i = 1, ..., p, and a random term Îµ. The model can be written as Example of linear regression with one dependent and one independent variable. ...
dataset with approximating polynomials Nonlinear regression in statistics is the problem of fitting a model to multidimensional x,y data, where f is a nonlinear function of x with parameters Î¸. In general, there is no algebraic expression for the bestfitting parameters, as there is in linear regression. ...
Logistic regression is a statistical regression model for Bernoullidistributed dependent variables. ...
