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Instatistics,dispersion (also calledvariability,scatter, orspread) is the extent to which adistribution is stretched or squeezed.[1] Common examples of measures of statistical dispersion are thevariance,standard deviation, andinterquartile range. For instance, when the variance of data in a set is large, the data is widely scattered. On the other hand, when the variance is small, the data in the set is clustered.
Dispersion is contrasted with location orcentral tendency, and together they are the most used properties of distributions.
Ameasure of statistical dispersion is a nonnegativereal number that is zero if all the data are the same and increases as the data become more diverse.
Most measures of dispersion have the sameunits as thequantity being measured. In other words, if the measurements are in metres or seconds, so is the measure of dispersion. Examples of dispersion measures include:
These are frequently used (together withscale factors) asestimators ofscale parameters, in which capacity they are calledestimates of scale.Robust measures of scale are those unaffected by a small number ofoutliers, and include the IQR and MAD.
All the above measures of statistical dispersion have the useful property that they arelocation-invariant andlinear in scale. This means that if arandom variable has a dispersion of then alinear transformation forreal and should have dispersion, where is theabsolute value of, that is, ignores a preceding negative sign.
Other measures of dispersion aredimensionless. 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. TheAllan variance can be used for applications where the noise disrupts convergence.[2] TheHadamard variance can be used to counteract linear frequency drift sensitivity.[3]
Forcategorical variables, it is less common to measure dispersion by a single number; seequalitative variation. One measure that does so is the discreteentropy.
In thephysical sciences, such variability may result from random measurement errors: instrument measurements are often not perfectlyprecise, i.e., reproducible, and there is additionalinter-rater 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 toobservational 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.[4]
In thebiological sciences, the quantity being measured is seldom unchanging and stable, and the variation observed might additionally beintrinsic to the phenomenon: It may be due tointer-individual variability, that is, distinct members of a population differing from each other. Also, it may be due tointra-individual 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.
Amean-preserving spread (MPS) 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.[5] The concept of a mean-preserving spread provides apartial 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.
