Central tendency

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]

The most common measures of central tendency are the arithmetic mean, the median and the mode. A central 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]

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. Analysts 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 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 two 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 L^p spaces, the correspondence is:

L^p	dispersion	central 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 $\mathbf {x} =(x_{1},\ldots ,x_{n})$ , the dispersion about a point c is the "distance" from x to the constant vector $\mathbf {c} =(c,\ldots ,c)$ in the p-norm (normalized by the number of points n):

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}.

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

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.

Relationships between the mean, median and mode

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

{\frac {|\theta -\mu |}{\sigma }}\leq {\sqrt {3}},

{\frac {|\nu -\mu |}{\sigma }}\leq {\sqrt {0.6}},

{\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]

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

References

↑ 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
1 2 Upton, G.; Cook, I. (2008) Oxford Dictionary of Statistics, OUP ISBN 978-0-19-954145-4 (entry for "central tendency")
↑ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP for International Statistical Institute. ISBN 0-19-920613-9 (entry for "central tendency")
↑ Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". Annals of Mathematical Statistics, 22 (3) 433–439
↑ Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114
↑ Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142

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[Weisberg-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

[Upton-2] 1 2 Upton, G.; Cook, I. (2008) Oxford Dictionary of Statistics, OUP ISBN 978-0-19-954145-4 (entry for "central tendency")

[Dodge-3] Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP for International Statistical Institute. ISBN 0-19-920613-9 (entry for "central tendency")

[Johnson1951-4] Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". Annals of Mathematical Statistics, 22 (3) 433–439

[Hotelling1932-5] Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114

[Garver1932-6] Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142