Editing Summary statistics

{{short description|Type of statistics}}
[[File:Michelsonmorley-boxplot.svg|thumb|[[Box plot]] of the [[Michelson–Morley experiment]], showing several summary statistics.]]

In [[descriptive statistics]], '''summary statistics''' are used to summarize a set of [[observation]]s, in order to communicate the largest amount of information as simply as possible. Statisticians commonly try to describe the observations in
* a measure of location, or [[central tendency]], such as the [[arithmetic mean]]
* a measure of [[statistical dispersion]] like the [[standard deviation|standard mean absolute deviation]]
* a measure of the shape of the distribution like [[skewness]] or [[kurtosis]]
* if more than one variable is measured, a measure of [[correlation and dependence|statistical dependence]] such as a [[Pearson product-moment correlation coefficient|correlation coefficient]]

A common collection of [[order statistics]] used as summary statistics are the [[five-number summary]], sometimes extended to a [[seven-number summary]], and the associated [[box plot]].

Entries in an [[analysis of variance]] table can also be regarded as summary statistics.{{r|Upton|p=378}}

==Examples==

===Location===

Common measures of location, or [[central tendency]], are the [[arithmetic mean]], [[median]], [[mode (statistics)|mode]], and [[interquartile mean]].<ref>{{cite book | last1 = Bullen | first1 = P. S. | date = 2003-08-31 | chapter =  | chapter-url =  | chapter-url-access =  | title = Handbook of Means and Their Inequalities | url =  | url-status =  | url-access =  | format =  | type =  | series = Mathematics and Its Applications | language = en | volume = 560 | edition = 2 | publisher = [[Springer Science+Business Media|Springer Dordrecht]] | doi = 10.1007/978-94-017-0399-4 | isbn = 978-1-4020-1522-9 | lccn = 2003060794 | oclc = 939214285 | ol = OL8370727M | quote =  | quote-page =  | quote-pages =  | df = dmy-all }}</ref><ref>{{cite book | last1=Grabisch | first1=Michel | last2 = Marichal | first2=Jean-Luc | last3=Mesiar | first3=Radko  | last4=Pap | first4=Endre |date=2009 |title=Aggregation Functions |publisher=Oxford University Press | isbn = 978-0521519267 }}</ref>

===Spread===
Common measures of [[statistical dispersion]] are the [[standard deviation]], [[variance]], [[range (statistics)|range]],  [[interquartile range]], [[absolute deviation]], [[mean absolute difference]] and the [[distance standard deviation]]. Measures that assess spread in comparison to the typical size of data values include the [[coefficient of variation]].

The [[Gini coefficient]] was originally developed to measure income inequality and is equivalent to one of the [[L-moment]]s.

A simple summary of a dataset is sometimes given by quoting particular [[order statistics]] as approximations to selected [[percentiles]] of a distribution.

===Shape===

Common measures of the shape of a distribution are [[skewness]] or [[kurtosis]], while alternatives can be based on [[L-moment]]s. A different measure is the [[Skewness#Distance skewness|distance skewness]], for which a value of zero implies central symmetry.

===Dependence===

The common measure of dependence between paired random variables is the [[Pearson product-moment correlation coefficient]], while a common alternative summary statistic is [[Spearman's rank correlation coefficient]]. A value of zero for the [[distance correlation]] implies independence.

==Human perception of summary statistics==

Humans efficiently use summary statistics to quickly perceive the gist of auditory and visual information.<ref>{{cite journal |doi=10.1177/0956797612473759|pmid=23761928|pmc=4381997|title=Humans Use Summary Statistics to Perceive Auditory Sequences|journal=Psychological Science|volume=24|issue=8|pages=1389–1397|year=2013|last1=Piazza|first1=Elise A.|last2=Sweeny|first2=Timothy D.|last3=Wessel|first3=David|last4=Silver|first4=Michael A.|last5=Whitney|first5=David}}</ref><ref>{{cite journal |last1=Alexander |first1=R. G. |last2=Schmidt |first2=J. |last3=Zelinsky |first3=G. Z. |title=Are summary statistics enough? Evidence for the importance of shape in guiding visual search. |journal=Visual Cognition |date=2014 |volume=22 |issue=3–4 |pages=595–609|pmid=26180505 |pmc=4500174 |doi=10.1080/13506285.2014.890989 }}</ref><ref>{{cite journal |doi=10.1167/15.4.8|pmid=26317396|title=Ensemble summary statistics as a basis for rapid visual categorization|journal=Journal of Vision|volume=15|issue=4|pages=8|year=2015|last1=Utochkin|first1=Igor S.|doi-access=free}}</ref>

==See also==
*[[Test statistic|Common test statistics]]
*[[Descriptive statistics]]
*[[Sample statistics]]
*[[Sufficient statistic]]
*[[Data processing]]

==References==
{{Reflist|30em|refs=

<ref name="Upton">{{cite book | last1 = Upton | first1 = Graham | author-link1 =  | last2 = Cook | first2 = Ian | author-link2 =  | date = 2008-10-02 | chapter = Dictionary (S) | chapter-url = https://archive.org/details/dictionaryofstat0000upto/page/378/mode/1up | chapter-url-access = registration | title = A Dictionary of Statistics | url = https://archive.org/details/dictionaryofstat0000upto | url-access = registration | language = en-gb | edition = Second (revised) | publisher = [[Oxford University Press]] | isbn = 978-0199541454 | lccn = 2008300706 | oclc = 935100347 | ol = OL23145891M | via = [[Internet Archive]] | quote = summary statistics [...] *ANOVA table might be referred to as summary statistics | quote-page = 378 | df = dmy-all }}</ref>

}}

==External links==
*{{Commonscatinline}}

{{Statistics}}

{{DEFAULTSORT:Summary Statistics}}
[[Category:Summary statistics| ]]

[[ja:要約統計量]]