出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/05/23 08:42:34」(JST)
In statistics, a central tendency (or, more commonly, a 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." [3][2]
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.
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. In addition, there is the
The Quadratic mean (often known as the root mean square) is useful in engineering, but is 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.
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". In the sense of Lp spaces, the correspondence is:
| Lp | dispersion | central tendency |
|---|---|---|
| L1 | average absolute deviation | median |
| L2 | standard deviation | mean |
| L∞ | maximum deviation | midrange |
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 uniqueness of this characterization of mean follows from convex optimization. Indeed, for a given (fixed) data set x, the function
represents the dispersion about a constant value c relative to the L2 norm. Because the function ƒ2 is a strictly convex coercive function, the minimizer exists and is unique.
Note that the median in this sense is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation. The dispersion in the L1 norm, given by
is not strictly convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. In spite of this, the minimizer is unique for the L∞ norm.
For unimodal distributions the following bounds are known and are sharp:[4]
where μ is the mean, ν is the median, θ is the mode, and σ is the standard deviation.
For every distribution,[5][6]
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| リンク元 | 「中央値」「平均値」 |
| 関連記事 | 「central」「tendency」 |