 Statistical dispersion

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.
Contents
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 become 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:
 Standard deviation
 Interquartile range or Interdecile range
 Range
 Mean difference
 Median absolute deviation
 Average absolute deviation (or simply called average deviation)
 Distance standard deviation
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 locationinvariant, as well as linear in scale. So if a random variable X has a dispersion of S_{X} then a linear transformation Y = aX + b for real a and b should have dispersion S_{Y} = aS_{X}.
Other measures of dispersion are dimensionless (scalefree). In other words, they have no units even if the variable itself has units. These include:
 Coefficient of variation
 Quartile coefficient of dispersion
 Relative mean difference, equal to twice the Gini coefficient
There are other measures of dispersion:
 Variance (the square of the standard deviation) — locationinvariant but not linear in scale.
 Variancetomean ratio — 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 scalefree.
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 that does so is the discrete entropy.
Sources of statistical dispersion
In the physical sciences, such variability may result from random measurement errors: instrument measurements are often not perfectly precise, i.e., reproducible, and there is additional interrater variability in interpreting and reporting the measured results. One may assume that the quantity being measured is stable, and that the variation between measurements is due to observational error. A system of a large number of particles is characterized by the mean values of a relatively few number of macroscopic quantities such as temperature, energy, and density. The standard deviation is an important measure in Fluctuation theory, which explains many physical phenomena, including why the sky is blue^{[1]} .
In the biological sciences, the quantity being measured is seldom unchanging and stable, and the variation observed might additionally be intrinsic to the phenomenon: It may be due to interindividual variability, that is, distinct members of a population differing from each other. Also, it may be due to intraindividual variability, that is, one and the same subject differing in tests taken at different times or in other differing conditions. Such types of variability are also seen in the arena of manufactured products; even there, the meticulous scientist finds variation.
In economics, finance, and other disciplines, regression analysis attempts to explain the dispersion of a dependent variable, generally measured by its variance, using one or more independent variables each of which itself has positive dispersion. The fraction of variance explained is called the coefficient of determination.
A partial ordering of dispersion
A meanpreserving spread (MPS)^{[2]} is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function while leaving the mean (the expected value) unchanged. The concept of a meanpreserving spread provides a partial ordering of probability distributions according to their dispersions: of two probability distributions, one may be ranked as having more dispersion than the other, or alternatively neither may be ranked as having more dispersion.
See also
 Average
 Summary statistics
 Qualitative variation
 Robust measures of scale
References
 ^ McQuarrie, Donald A. (1976). Statistical Mechanics. NY: Harper & Row. ISBN 060443669.
 ^ Rothschild, Michael, and Stiglitz, Joseph, "Increasing risk I: A definition," Journal of Economic Theory, 1970, 225–243.
Categories: Statistical deviation and dispersion
 Summary statistics
Wikimedia Foundation. 2010.