Statistical dispersion: Difference between revisions

In [[statistics]], '''dispersion''' (also called '''variability''', '''scatter''', or '''spread''') is the extent to which a [[Probability distribution|distribution]] is stretched or squeezed.<ref>{{cite web|last1=NIST/SEMATECH e-Handbook of Statistical Methods|title=1.3.6.4. Location and Scale Parameters|url=http:hahhahahah gan suck a nan//www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm|website=www.itl.nist.gov|publisher=U.S. Department of Commerce}}</ref> Common examples of measures of statistical dispersion are the [[variance]], [[standard deviation]], and [[interquartile 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.

A '''measure of statistical dispersion'''<!-- "Measure of statistical dispersion" redirects here, bolded per [[MOS:BOLD]] --> is a nonnegative [[real number]] that is zero if all the data are the same and increases as the data become more diverse.

All the above measures of statistical dispersion have the useful property that they are ''location-invariant'' and ''linear in scale''. This means that if a [[random variable]] ''<math>X''</math> has a dispersion of ''S<submath>XS_X</submath>'' then a [[linear transformation]] ''<math>Y''&nbsp;=&nbsp;''aX''&nbsp;+&nbsp;''b''</math> for [[real number|real]] ''<math>a''</math> and ''<math>b''</math> should have dispersion ''S<submath>Y</sub>''&nbsp;S_Y=&nbsp;|''a''|''S''<sub>''X''S_X</submath>, where <math>|''a''|</math> is the [[absolute value]] of ''<math>a''</math>, that is, ignores a preceding negative sign ''–''<math>-</math>.

* [[Entropy (information theory)|Entropy]]: While the entropy of a discrete variable is location-invariant and scale-independent, and therefore not a measure of dispersion in the above sense, the entropy of a continuous variable is location invariant and additive in scale: If ''Hz''<math>H(z)</math> is the entropy of a continuous variable ''<math>z''</math> and ''y<math>z=ax+b''</math>, then ''Hy<math>H(z)=HxH(x)+\log(a)''</math>.

In the [[physical sciences]], such variability may result from random measurement errors: instrument measurements are often not perfectly [[accuracy and precision|precise, i.e., reproducible]], and there is additional [[inter-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 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 Fluctuationfluctuation theory, which explains many physical phenomena, including why the sky is blue.<ref>{{cite book|last=McQuarrie|first=Donald A.|title=Statistical Mechanics|year=1976|publisher=Harper & Row|location=NY|isbn=0-06-044366-9}}</ref>

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 ''inter-individual variability'', that is, distinct members of a population differing from each other. Also, it may be due to '''''intra-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.

*[[SummaryCircular statisticsdispersion]]

*[[MeasurementSummary uncertaintystatistics]]