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Instatistics, acentral tendency (ormeasure of central tendency) is a central or typical value for aprobability distribution.[1]
Colloquially, measures of central tendency are often calledaverages. The termcentral tendency dates from the late 1920s.[2]
The most common measures of central tendency are thearithmetic mean, themedian, and themode. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as thenormal distribution. Occasionally authors use central tendency to denote "the tendency of quantitativedata to cluster around some central value."[2][3]
The central tendency of a distribution is typically contrasted with itsdispersion orvariability; 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.
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.
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.
Some of these measures arerobust againstoutliers, such as the median and the trimean.
Several measures of central tendency can be characterized as solving a variational problem, in the sense of thecalculus of variations, namely minimizing variation from the center. That is, given a measure ofstatistical 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 ofLp spaces, the correspondence is:
| Lp | dispersion | central tendency |
|---|---|---|
| L0 | variation ratio | mode[a] |
| L1 | average absolute deviation | median (geometric median)[b] |
| L2 | standard deviation | mean (centroid)[c] |
| L∞ | maximum deviation | midrange[d] |
The associated functions are calledp-norms: respectively 0-"norm", 1-norm, 2-norm, and ∞-norm. The function corresponding to theL0 space is not a norm, and is thus often referred to in quotes: 0-"norm".
In equations, for a given (finite) data setX, thought of as a vectorx = (x1,…,xn), the dispersion about a pointc is the "distance" fromx to the constant vectorc = (c,…,c) in thep-norm (normalized by the number of pointsn):
Forp = 0 andp = ∞ these functions are defined by taking limits, respectively asp → 0 andp → ∞. Forp = 0 the limiting values are00 = 0 anda0 = 1 fora ≠ 0, so the difference becomes simply equality, so the 0-norm counts the number ofunequal points. Forp = ∞ the largest number dominates, and thus the ∞-norm is the maximum difference.
The mean (L2 center) and midrange (L∞ center) are unique (when they exist), while the median (L1 center) and mode (L0 center) are not in general unique. This can be understood in terms ofconvexity of the associated functions (coercive functions).
The 2-norm and ∞-norm arestrictly 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 notstrictly 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 distributionany point is the mode.
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, where each point in the data set is clustered with the nearest "center". Most commonly, using the 2-norm generalizes the mean tok-means clustering, while using the 1-norm generalizes the (geometric) median tok-medians clustering. Using the 0-norm simply generalizes the mode (most common value) to using thek most common values as centers.
Unlike the single-center statistics, this multi-center clustering cannot in general be computed in aclosed-form expression, and instead must be computed or approximated by aniterative method; one general approach isexpectation–maximization algorithms.
The notion of a "center" as minimizing variation can be generalized ininformation geometry as a distribution that minimizesdivergence (a generalized distance) from a data set. The most common case ismaximum likelihood estimation, where the maximum likelihood estimate (MLE) maximizes likelihood (minimizes expectedsurprisal), which can be interpreted geometrically by usingentropy to measure variation: the MLE minimizescross-entropy (equivalently,relative entropy, 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 theempirical measure (thefrequency distribution divided by thesample size) as a "center". For example, givenbinary 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 inregression analysis, whereleast squares finds the solution that minimizes the distances from it, and analogously inlogistic regression, a maximum likelihood estimate minimizes the surprisal (information distance).
Forunimodal distributions the following bounds are known and are sharp:[4]
whereμ is the mean,ν is the median,θ is the mode, andσ is the standard deviation.
