In

^{0} space is not a norm, and is thus often referred to in quotes: 0-"norm".
In equations, for a given (finite) data set , thought of as a vector , the dispersion about a point is the "distance" from to the constant vector in the -norm (normalized by the number of points ):
:$f\_p(c)\; =\; \backslash left\backslash ,\; \backslash mathbf\; -\; \backslash mathbf\; \backslash right\backslash ,\; \_p\; :=\; \backslash bigg(\; \backslash frac\; \backslash sum\_^n\; \backslash left,\; x\_i\; -\; c\backslash \; ^p\; \backslash bigg)\; ^$
For and these functions are defined by taking limits, respectively as and . For the limiting values are and or , so the difference becomes simply equality, so the 0-norm counts the number of ''unequal'' points. For the largest number dominates, and thus the ∞-norm is the maximum difference.

^{2} center) and midrange (''L''^{∞} center) are unique (when they exist), while the median (''L''^{1} center) and mode (''L''^{0} center) are not in general unique. This can be understood in terms of convexity of the associated functions (

statistics
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...

, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in the Social Sciences, p.2
Colloquially, measures of central tendency are often called '' averages.'' The term ''central tendency'' dates from the late 1920s.
The most common measures of central tendency are the arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...

, the median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...

, and the mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...

. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpret ...

to cluster around some central value."Upton, G.; Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP (entry for "central tendency")Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP for International Statistical Institute
The International Statistical Institute (ISI) is a professional association of statisticians. It was founded in 1885, although there had been international statistical congresses since 1853. The institute has about 4,000 elected members from gov ...

. (entry for "central tendency")
The central tendency of a distribution is typically contrasted with its '' dispersion'' or ''variability''; dispersion and central tendency are the often characterized properties of distributions. Analysis may judge whether data has a strong or a weak central tendency based on its dispersion.
Measures

The following may be applied to one-dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be, depend heavily on the data being analyzed. ;Arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...

or simply, mean: the sum of all measurements divided by the number of observations in the data set.
; Median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...

: the middle value that separates the higher half from the lower half of the data set. The median and the mode are the only measures of central tendency that can be used for ordinal data
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described b ...

, in which values are ranked relative to each other but are not measured absolutely.
; Mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...

: the most frequent value in the data set. This is the only central tendency measure that can be used with nominal data, which have purely qualitative category assignments.
; Generalized mean: A generalization of the Pythagorean means
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians beca ...

, specified by an exponent.
; Geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...

: the ''n''th root of the product of the data values, where there are ''n'' of these. This measure is valid only for data that are measured absolutely on a strictly positive scale.
; Harmonic mean: the reciprocal of the arithmetic mean of the reciprocals of the data values. This measure too is valid only for data that are measured absolutely on a strictly positive scale.
; Weighted arithmetic mean: an arithmetic mean that incorporates weighting to certain data elements.
; Truncated mean
A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, a ...

or trimmed mean: the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.
; Interquartile mean: a truncated mean based on data within the interquartile range
In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the differenc ...

.
; Midrange: the arithmetic mean of the maximum and minimum values of a data set.
; Midhinge: the arithmetic mean of the first and third quartiles.
; Quasi-arithmetic mean: A generalization of the generalized mean, specified by a continuous injective function.
; Trimean: the weighted arithmetic mean of the median and two quartiles.
; Winsorized mean: an arithmetic mean in which extreme values are replaced by values closer to the median.
Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space.
; Geometric median
In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances ...

: the point minimizing the sum of distances to a set of sample points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions.
; Quadratic mean (often known as the root mean square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...

): useful in engineering, but not often used in statistics. This is because it is not a good indicator of the center of the distribution when the distribution includes negative values.
; Simplicial depth: the probability that a randomly chosen simplex with vertices from the given distribution will contain the given center
; Tukey median
John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distributi ...

: a point with the property that every halfspace containing it also contains many sample points
Solutions to variational problems

Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations, namely minimizing variation from the center. That is, given a measure of statistical dispersion, one asks for a measure of central tendency that minimizes variation: such that variation from the center is minimal among all choices of center. In a quip, "dispersion precedes location". These measures are initially defined in one dimension, but can be generalized to multiple dimensions. This center may or may not be unique. In the sense of spaces, the correspondence is: The associated functions are called -norms: respectively 0-"norm", 1-norm, 2-norm, and ∞-norm. The function corresponding to theUniqueness

The mean (''L''coercive function In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context
different exact definitions of this idea are in use.
Coercive vector fields
A vector field ''f'' : ...

s).
The 2-norm and ∞-norm are strictly convex, and thus (by convex optimization) the minimizer is unique (if it exists), and exists for bounded distributions. Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point.
The 1-norm is not ''strictly'' convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. Correspondingly, the median (in this sense of minimizing) is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation.
The 0-"norm" is not convex (hence not a norm). Correspondingly, the mode is not unique – for example, in a uniform distribution ''any'' point is the mode.
Clustering

Instead of a single central point, one can ask for multiple points such that the variation from these points is minimized. This leads tocluster analysis
Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of ...

, where each point in the data set is clustered with the nearest "center". Most commonly, using the 2-norm generalizes the mean to ''k''-means clustering, while using the 1-norm generalizes the (geometric) median to ''k''-medians clustering. Using the 0-norm simply generalizes the mode (most common value) to using the ''k'' most common values as centers.
Unlike the single-center statistics, this multi-center clustering cannot in general be computed in a closed-form expression, and instead must be computed or approximated by an iterative method; one general approach is expectation–maximization algorithms.
Information geometry

The notion of a "center" as minimizing variation can be generalized in information geometry as a distribution that minimizesdivergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of th ...

(a generalized distance) from a data set. The most common case is maximum likelihood estimation, where the maximum likelihood estimate (MLE) maximizes likelihood (minimizes expected surprisal), which can be interpreted geometrically by using entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

to measure variation: the MLE minimizes cross entropy (equivalently, relative entropy
Relative may refer to:
General use
*Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives''
Philosophy
*Relativism, the concept that ...

, Kullback–Leibler divergence).
A simple example of this is for the center of nominal data: instead of using the mode (the only single-valued "center"), one often uses the empirical measure (the frequency distribution
In statistics, the frequency (or absolute frequency) of an event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabular form.
Types
The cumula ...

divided by the sample size
Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populati ...

) as a "center". For example, given binary data, say heads or tails, if a data set consists of 2 heads and 1 tails, then the mode is "heads", but the empirical measure is 2/3 heads, 1/3 tails, which minimizes the cross-entropy (total surprisal) from the data set. This perspective is also used in regression analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...

, where least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...

finds the solution that minimizes the distances from it, and analogously in logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression anal ...

, a maximum likelihood estimate minimizes the surprisal (information distance).
Relationships between the mean, median and mode

Forunimodal distribution
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
Unimodal probability distribution
In statistics, a unimodal ...

s the following bounds are known and are sharp:Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". ''Annals of Mathematical Statistics'', 22 (3) 433–439
: $\backslash frac\; \backslash le\; \backslash sqrt\; ,$
: $\backslash frac\; \backslash le\; \backslash sqrt\; ,$
: $\backslash frac\; \backslash le\; \backslash sqrt\; ,$
where ''μ'' is the mean, ''ν'' is the median, ''θ'' is the mode, and ''σ'' is the standard deviation.
For every distribution,Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142
: $\backslash frac\; \backslash le\; 1.$
See also

* Central moment *Expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...

* Location parameter
*Mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the ''arith ...

* Population mean
* Sample mean
Notes

References

{{DEFAULTSORT:Central Tendency Summary statistics Probability theory de:Lagemaß