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{{StatsPsy}}
 
{{StatsPsy}}
   
In [[descriptive statistics]], '''statistical dispersion''' (also called '''statistical variability''') is quantifiable variation of measurements of differing members of a [[statistical population|population]] within the scale on which they are measured. <!-- what exactly does this mean? -->
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In [[statistics]], '''statistical dispersion''' (also called '''statistical variability''' or '''variation''') is variability or spread in a [[variable]] or a [[probability distribution]]. Common examples of measures of statistical dispersion are the [[variance]], [[standard deviation]] and [[interquartile range]].
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Dispersion is contrasted with location or [[central tendency]], and together they are the most used properties of distributions.
   
 
==Measures of statistical dispersion==
 
==Measures of statistical dispersion==
A measure of statistical dispersion is a [[real number]] that is zero if all the data are identical, and increases as the data becomes more diverse. An important measure of dispersion is the [[standard deviation]], which is the [[square root]] of the [[variance]] (which is itself a measure of dispersion).
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A measure of statistical dispersion is a [[real number]] that is zero if all the data are identical, and increases as the data becomes more diverse. It cannot be less than zero.
   
Other such measures include the [[range (statistics)|range]], the [[interquartile range]], and the [[absolute deviation|average absolute deviation]], and, in the case of categorical random variables, the discrete [[entropy]]. None of these can be negative; their least possible value is zero.
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Most measures of dispersion have the '''same scale as the quantity being measured.''' In other words, if the measurements have [[units of measurement|unit]]s, such as metres or seconds, the measure of dispersion has the same units. Such measures of dispersion include:
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* [[Standard deviation]]
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* [[Interquartile range]]
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* [[Range (statistics)|Range]]
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* [[Mean difference]]
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* [[Median absolute deviation]]
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* [[Average absolute deviation]] (or simply average deviation)
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These are frequently used (together with [[scale factor]]s) as [[estimator]]s of [[scale parameter]]s, in which capacity they are called '''estimates of scale.'''
   
A measure of statistical dispersion is particularly useful if it is location invariant, and linear in scale. So if a random variable ''X'' has a dispersion of ''S<sub>X</sub>'' then a [[linear transformation]] ''Y''&nbsp;=&nbsp;''aX''&nbsp;+&nbsp;''b'' for [[real number|real]] ''a'' and ''b'' should have dispersion ''S<sub>Y</sub>''&nbsp;=&nbsp;''aS<sub>X</sub>''.
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All the above measures of statistical dispersion have the useful property that they are '''location-invariant''', as well as linear in scale. So if a [[random variable]] ''X'' has a dispersion of ''S<sub>X</sub>'' then a [[linear transformation]] ''Y''&nbsp;=&nbsp;''aX''&nbsp;+&nbsp;''b'' for [[real number|real]] ''a'' and ''b'' should have dispersion ''S<sub>Y</sub>''&nbsp;=&nbsp;|''a''|''S''<sub>''X''</sub>.
One of the forms in which statistical variability is realized in the empirical [[science]]s is that of differences in repeated measurements of the same quantity.
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Other measures of dispersion are '''[[dimensionless]] (scale-free)'''. In other words, they have no units even if the variable itself has units. These include:
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* [[Coefficient of variation]]
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* [[Quartile coefficient of dispersion]]
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* [[Relative mean difference]], equal to twice the [[Gini coefficient]]
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There are other measures of dispersion:
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* [[Variance]] (the square of the standard deviation) &mdash; location-invariant but not linear in scale.
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* [[Variance-to-mean ratio]] &mdash; mostly used for [[count data]] when the term [[coefficient of dispersion]] is used and when this ratio is [[dimensionless]], as count data are themselves dimensionless: otherwise this is not scale-free.
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Some measures of dispersion have specialized purposes, among them the [[Allan variance]] and the [[Hadamard variance]].
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For [[categorical variable]]s, it is less common to measure dispersion by a single number. See [[qualitative variation]]. One measure which does so is the discrete [[information entropy|entropy]].
   
 
==Sources of statistical dispersion==
 
==Sources of statistical dispersion==
In the physical sciences, such variability may result only from random measurement errors: instrument measurements are often not perfectly [[accuracy and precision|precise - i.e., reproducible]]. One may assume that the quantity being measured is unchanging and stable, and that the variation between measurements is due to [[observational error]].
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In the physical sciences, such variability may result only from random measurement errors: instrument measurements are often not perfectly [[accuracy and precision|precise, i.e., reproducible]]. One may assume that the quantity being measured is unchanging and stable, and that the variation between measurements is due to [[observational error]].
   
In the biological sciences, this assumption is false: the variation observed might be ''intrinsic'' to the phenomenon: distinct members of a population differ greatly. This is also seen in the arena of manufactured products; even there, the meticulous scientist finds idiosyncrasy of sampled items.
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In the biological sciences, this assumption is false: the variation observed might be ''intrinsic'' to the phenomenon: distinct members of a population differ greatly. This is also seen in the arena of manufactured products; even there, the meticulous scientist finds variation.
   
 
The simple model of a stable quantity is preferred when it is tenable. Each phenomenon must be examined to see if it warrants such a simplification.
 
The simple model of a stable quantity is preferred when it is tenable. Each phenomenon must be examined to see if it warrants such a simplification.
   
See also [[summary statistics]].
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== See also ==
[[Category:Statistics]]
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*[[Average]]
[[de:Streuung (Statistik)]]
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*[[Summary statistics]]
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*[[Qualitative variation]]
   
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{{Statistics}}
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[[Category:Statistical deviation and dispersion]]
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[[Category:Summary statistics]]
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[[ar:مقاييس النزعة المركزية]]
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[[bg:Статистическо разсейване]]
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[[ca:Dispersió (matemàtiques)]]
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[[de:Streuung (Statistik)]]
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[[et:Dispersioon]]
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[[es:Dispersión (matemática)]]
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[[fr:Critères de dispersion]]
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[[fi:Hajontaluku]]
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[[it:Variabilità]]
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[[is:Dreifing]]
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[[ja:統計的ばらつき]]
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[[pl:Dyspersja (matematyka)]]
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[[sl:Statistična razpršenost]]
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{{enWP|Statistical_dispersion}}
 
{{enWP|Statistical_dispersion}}

Latest revision as of 23:08, November 23, 2008

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In statistics, statistical dispersion (also called statistical variability or variation) is variability or spread in a variable or a probability distribution. Common examples of measures of statistical dispersion are the variance, standard deviation and interquartile range.

Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.

Measures of statistical dispersionEdit

A measure of statistical dispersion is a real number that is zero if all the data are identical, and increases as the data becomes more diverse. It cannot be less than zero.

Most measures of dispersion have the same scale as the quantity being measured. In other words, if the measurements have units, such as metres or seconds, the measure of dispersion has the same units. Such measures of dispersion include:

These are frequently used (together with scale factors) as estimators of scale parameters, in which capacity they are called estimates of scale.

All the above measures of statistical dispersion have the useful property that they are location-invariant, as well as linear in scale. So if a random variable X has a dispersion of SX then a linear transformation Y = aX + b for real a and b should have dispersion SY = |a|SX.

Other measures of dispersion are dimensionless (scale-free). In other words, they have no units even if the variable itself has units. These include:

There are other measures of dispersion:

Some measures of dispersion have specialized purposes, among them the Allan variance and the Hadamard variance.

For categorical variables, it is less common to measure dispersion by a single number. See qualitative variation. One measure which does so is the discrete entropy.

Sources of statistical dispersionEdit

In the physical sciences, such variability may result only from random measurement errors: instrument measurements are often not perfectly precise, i.e., reproducible. One may assume that the quantity being measured is unchanging and stable, and that the variation between measurements is due to observational error.

In the biological sciences, this assumption is false: the variation observed might be intrinsic to the phenomenon: distinct members of a population differ greatly. This is also seen in the arena of manufactured products; even there, the meticulous scientist finds variation.

The simple model of a stable quantity is preferred when it is tenable. Each phenomenon must be examined to see if it warrants such a simplification.

See also Edit



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