# Central tendency

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In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution. [1] It may also be called a center or location of the distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s. [2]

Statistics is the discipline that concerns the collection, organization, displaying, analysis, interpretation and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. Examples of random phenomena can include the results of an experiment or survey.

## Contents

The most common measures of central tendency are the arithmetic mean, the median and the mode. 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 to cluster around some central value." [2] [3]

In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

The median is the value separating the higher half from the lower half of a data sample. For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.

The mode of a set of data values is the value that appears most often. If X is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled.

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.

In statistics, dispersion is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.

## 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 or simply, mean
the sum of all measurements divided by the number of observations in the data set.
Median
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, in which values are ranked relative to each other but are not measured absolutely.
Mode
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.
Geometric mean
the nth 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 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.
Midrange
the arithmetic mean of the maximum and minimum values of a data set.
Midhinge
the arithmetic mean of the first and third quartiles.
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. In addition, there are the

Geometric median
which minimizes the sum of distances to the data 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)
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
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". This center may or may not be unique. In the sense of Lp spaces, the correspondence is:

Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.

Lpdispersioncentral tendency
L0 variation ratio mode
L1 average absolute deviation median
L1 average absolute deviation geometric median
L2 standard deviation mean
L maximum deviation midrange

The associated functions are called p-norms: respectively 0-"norm", 1-norm, 2-norm, and ∞-norm. The function corresponding to the L0 space is not a norm, and is thus often referred to in quotes: 0-"norm".

In equations, for a given (finite) data set X, thought of as a vector ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$, the dispersion about a point c is the "distance" from x to the constant vector ${\displaystyle \mathbf {c} =(c,\ldots ,c)}$ in the p-norm (normalized by the number of points n):

${\displaystyle f_{p}(c)=\left\|\mathbf {x} -\mathbf {c} \right\|_{p}:={\bigg (}{\frac {1}{n}}\sum _{i=1}^{n}\left|x_{i}-c\right|^{p}{\bigg )}^{1/p}.}$

For ${\displaystyle p=0}$ and ${\displaystyle p=\infty }$ these functions are defined by taking limits, respectively as ${\displaystyle p\to 0}$ and ${\displaystyle p\to \infty }$. For ${\displaystyle p=0}$ the limiting values are ${\displaystyle 0^{0}=0}$ and ${\displaystyle a^{0}=1}$ for ${\displaystyle a\neq 0}$, so the difference becomes simply equality, so the 0-norm counts the number of unequal points. For ${\displaystyle p=\infty }$ the largest number dominates, and thus the ∞-norm is the maximum difference.

### Uniqueness

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 of convexity of the associated functions (coercive functions).

In mathematics, a real-valued function defined on an n-dimensional interval is called convex if the line segment between any two points on the graph of the function lies above or on the graph. Equivalently, a function is convex if its epigraph is a convex set. For a twice-differentiable function of a single variable, if its second derivative is always nonnegative on its entire domain, then the function is convex. In fact, if a twice-differentiable function of a single variable is convex, then its second derivative must be nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the squaring function and the exponential 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.

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.

### Information geometry

The notion of a "center" as minimizing variation can be generalized in information geometry as a distribution that minimizes divergence (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 to measure variation: the MLE minimizes cross 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 the empirical measure (the frequency distribution divided by the sample size) 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, where least squares finds the solution that minimizes the distances from it, and analogously in logistic regression, a maximum likelihood estimate minimizes the surprisal (information distance).

## Relationships between the mean, median and mode

For unimodal distributions the following bounds are known and are sharp: [4]

${\displaystyle {\frac {|\theta -\mu |}{\sigma }}\leq {\sqrt {3}},}$
${\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq {\sqrt {0.6}},}$
${\displaystyle {\frac {|\theta -\nu |}{\sigma }}\leq {\sqrt {3}},}$

where μ is the mean, ν is the median, θ is the mode, and σ is the standard deviation.

For every distribution, [5] [6]

${\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq 1.}$

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## References

1. Weisberg H.F (1992) Central Tendency and Variability, Sage University Paper Series on Quantitative Applications in the Social Sciences, ISBN   0-8039-4007-6 p.2
2. Upton, G.; Cook, I. (2008) Oxford Dictionary of Statistics, OUP ISBN   978-0-19-954145-4 (entry for "central tendency")
3. Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP for International Statistical Institute. ISBN   0-19-920613-9 (entry for "central tendency")
4. Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". Annals of Mathematical Statistics, 22 (3) 433–439
5. Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114
6. Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142