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. 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 . 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 is 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 x1, x2, ..., xn is typically denoted by , 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 ) to distinguish it from the population mean (denoted or ).
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.
Types of mean 
Pythagorean means 
|It has been suggested that portions of this section be moved into Pythagorean means. (Discuss)|
Arithmetic mean (AM) 
The arithmetic mean is the "standard" average, often simply called the "mean".
For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is
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.
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.
For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:
Harmonic mean (HM) 
For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is
Relationship between AM, GM, and HM 
AM, GM, and HM satisfy these inequalities:
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 xi by
By choosing different values for the parameter m, the following types of means are obtained:
This can be generalized further as the generalized f-mean
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:
The weights 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.
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 
|It has been suggested that portions of Average#Average values of functions be moved or incorporated into this section. (Discuss)|
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
Recall that a defining property of the average value of finitely many numbers is that . In other words, is the constant value which when added to itself times equals the result of adding the terms of . By analogy, a defining property of the average value of a function over the interval is that
In other words, is the constant value which when integrated over equals the result of integrating over . But by the second fundamental theorem of calculus, the integral of a constant is just
See also the first mean value theorem for integration, which guarantees that if is continuous then there exists a point such that
The point is called the mean value of on . So we write and rearrange the preceding equation to get the above definition.
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.
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 
- Arithmetic-geometric mean
- Arithmetic-harmonic mean
- Cesàro mean
- Chisini mean
- Contraharmonic mean
- Distance-weighted estimator
- Elementary symmetric mean
- Geometric-harmonic mean
- Heinz mean
- Heronian mean
- Identric mean
- Lehmer mean
- Logarithmic mean
- Moving average
- Root mean square
- Rényi's entropy (a generalized f-mean)
- Stolarsky mean
- Weighted geometric mean
- Weighted harmonic mean
All means share some properties and additional properties are shared by the most common means. Some of these properties are collected here.
Weighted and unweighted means 
Weighted mean 
|It has been suggested that portions of this section be moved into Weighted mean. (Discuss)|
A weighted mean M is a function which maps tuples of positive numbers to a positive number
such that the following properties hold:
- "Fixed point": M(1,1,...,1) = 1
- Homogeneity: M(λ x1, ..., λ xn) = λ M(x1, ..., xn) for all λ and xi. In vector notation: M(λ x) = λ Mx for all n-vectors x.
- Monotonicity: If xi ≤ yi for each i, then Mx ≤ My
- Boundedness: min x ≤ Mx ≤ max x
- 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 f-means, satisfy the presented properties.
- If f is bijective, then the generalized f-mean satisfies the fixed point property.
- If f is strictly monotonic, then the generalized f-mean satisfy also the monotony property.
- In general a generalized f-mean will miss homogeneity.
The above properties imply techniques to construct more complex means:
If C, M1, ..., Mm are weighted means and p is a positive real number, then A and B defined by
are also weighted means.
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 to be an unweighted mean if it is a weighted mean and for each permutation π of inputs, the result is the same.
- Symmetry: Mx = M(πx) for all n-tuples x and permutations π on n-tuples.
Analogously to the weighted means, if C is a weighted mean and M1, ..., Mm are unweighted means and p is a positive real number, then A and B defined by
are also unweighted means.
Converting 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
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
- Given an arbitrary tuple x, which is partitioned into y1, ..., yk, then
- (See Convex hull.)
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:
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.
Using a very small sample 
|It has been suggested that portions of this section be moved into Standard error of the mean#Correction for finite population. (Discuss)|
Small sample sizes occur in practice and present unusually difficult problems for parameter estimation.
It is intuitive but false that from a single (n = 1) observation x, information about the variability in the population cannot be gained and consequently finite-length confidence intervals for the population mean and/or variance are impossible even in principle. Where the shape of the population distribution is known, some estimates are possible:
For a normally distributed variate, the confidence intervals for the (arithmetic) population mean at the 90% level have been shown to be x ± 5.84 |x| where |.| is the absolute value. The 95% bound for a normally distributed variate is x ± 9.68 |x| and that for a 99% confidence interval is x ± 48.39 |x|. These confidence intervals apply because for every true but unobserved parametrization of the normal distribution, the probability that the indicated confidence interval, computed from the random sample of one, encompasses the fixed true mean is at least the indicated percentage, and for the worst-case true parametrization it is exactly the indicated percentage.
The estimate derived from this method shows behavior that is atypical of more conventional methods. A value of 0 for the population mean cannot be rejected with any level of confidence. If x = 0, the confidence interval collapses to a length of 0. Finally the confidence interval is not stable under a linear transform x → ax + b where a and b are constants.
Machol has shown that given a known density symmetrical about 0 and a single sample value ( x ), the 90% confidence interval of the population mean is
where ν is the population median.
For a sample size of two ( n = 2 ), the population mean is bounded by
where x1, x2 are the variate values, μ is the population mean and k is a constant that depends on the underlying distribution. For the normal distribution, k = cotangent( π α / 2 ), in which case for α = 0.05, k = 12.71. For the rectangular distribution, k = ( 1 / α ) - 1, in which case for α = 0.05, k = 19.
For a sample size of three ( n = 3 ), the confidence intervals for the population mean are
where m is the sample mean, s is the sample standard deviation and k is a constant that depends on the distribution. For the normal distribution, k is approximately 1 / √α - 3 √α / 4 + ... When α = 0.05, k = 4.30. For the rectangular distribution with α = 0.05, k = 5.74.
The pivot depth (j) is int( ( n + 1 ) / 2 ) / 2 or int( ( n + 1 ) / 2 + 1 ) / 2 depending on which value is an integer. The lower pivot is xL = xj and the upper pivot xU is xn + 1 - j. The pivot half sum (P) is
and the pivot range (R) is
The confidence intervals for the population mean are then
where t is the value of the t test at 100( 1 - α / 2 )%.
The pivot statistic T = P / R has an approximately symmetrical distribution and its values for 4 ≤ n ≤ 20 for a number of values of 1 - α are given in Table 2 of Meloun et al.
See also 
- Algorithms for calculating variance
- Central tendency
- Descriptive statistics
- Law of averages
- Mean value theorem
- Mode (statistics)
- Spherical mean
- Summary statistics
- Taylor's law
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- Comparison between arithmetic and geometric mean of two numbers
- Some relationships involving means