In descriptive statistics, summary statistics are used to summarize a set of observations, 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 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 statistical dependence such as a 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.: 378
Examples
Location
Common measures of location, or central tendency, are the arithmetic mean, median, mode, and interquartile mean.
Spread
Common measures of statistical dispersion are the standard deviation, variance, 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-moments.
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-moments. A different measure is the 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.
See also
- Common test statistics
- Descriptive statistics
- Sample statistics
- Sufficient statistic
- Data processing
References
- Upton, Graham; Cook, Ian (2 October 2008). "Dictionary (S)". A Dictionary of Statistics (Second (revised) ed.). Oxford University Press. ISBN 978-0199541454. LCCN 2008300706. OCLC 935100347. OL 23145891M – via Internet Archive. p. 378:
summary statistics [...] *ANOVA table might be referred to as summary statistics
- Bullen, P. S. (31 August 2003). Handbook of Means and Their Inequalities. Mathematics and Its Applications. Vol. 560 (2 ed.). Springer Dordrecht. doi:10.1007/978-94-017-0399-4. ISBN 978-1-4020-1522-9. LCCN 2003060794. OCLC 939214285. OL 8370727M.
- Grabisch, Michel; Marichal, Jean-Luc; Mesiar, Radko; Pap, Endre (2009). Aggregation Functions. Oxford University Press. ISBN 978-0521519267.
- Piazza, Elise A.; Sweeny, Timothy D.; Wessel, David; Silver, Michael A.; Whitney, David (2013). "Humans Use Summary Statistics to Perceive Auditory Sequences". Psychological Science. 24 (8): 1389–1397. doi:10.1177/0956797612473759. PMC 4381997. PMID 23761928.
- Alexander, R. G.; Schmidt, J.; Zelinsky, G. Z. (2014). "Are summary statistics enough? Evidence for the importance of shape in guiding visual search". Visual Cognition. 22 (3–4): 595–609. doi:10.1080/13506285.2014.890989. PMC 4500174. PMID 26180505.
- Utochkin, Igor S. (2015). "Ensemble summary statistics as a basis for rapid visual categorization". Journal of Vision. 15 (4): 8. doi:10.1167/15.4.8. PMID 26317396.
External links
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Media related to Summary statistics at Wikimedia Commons
In descriptive statistics summary statistics are used to summarize a set of observations in order to communicate the largest amount of information as simply as possible Statisticians commonly try to describe the observations ina measure of location or central tendency such as the arithmetic mean a measure of statistical dispersion like the 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 statistical dependence such as a correlation coefficientBox plot of the Michelson Morley experiment showing several summary statistics 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 378 ExamplesLocation Common measures of location or central tendency are the arithmetic mean median mode and interquartile mean Spread Common measures of statistical dispersion are the standard deviation variance 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 moments 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 moments A different measure is the 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 statisticsHumans efficiently use summary statistics to quickly perceive the gist of auditory and visual information See alsoCommon test statistics Descriptive statistics Sample statistics Sufficient statistic Data processingReferencesUpton Graham Cook Ian 2 October 2008 Dictionary S A Dictionary of Statistics Second revised ed Oxford University Press ISBN 978 0199541454 LCCN 2008300706 OCLC 935100347 OL 23145891M via Internet Archive p 378 summary statistics ANOVA table might be referred to as summary statistics Bullen P S 31 August 2003 Handbook of Means and Their Inequalities Mathematics and Its Applications Vol 560 2 ed Springer Dordrecht doi 10 1007 978 94 017 0399 4 ISBN 978 1 4020 1522 9 LCCN 2003060794 OCLC 939214285 OL 8370727M Grabisch Michel Marichal Jean Luc Mesiar Radko Pap Endre 2009 Aggregation Functions Oxford University Press ISBN 978 0521519267 Piazza Elise A Sweeny Timothy D Wessel David Silver Michael A Whitney David 2013 Humans Use Summary Statistics to Perceive Auditory Sequences Psychological Science 24 8 1389 1397 doi 10 1177 0956797612473759 PMC 4381997 PMID 23761928 Alexander R G Schmidt J Zelinsky G Z 2014 Are summary statistics enough Evidence for the importance of shape in guiding visual search Visual Cognition 22 3 4 595 609 doi 10 1080 13506285 2014 890989 PMC 4500174 PMID 26180505 Utochkin Igor S 2015 Ensemble summary statistics as a basis for rapid visual categorization Journal of Vision 15 4 8 doi 10 1167 15 4 8 PMID 26317396 External linksMedia related to Summary statistics at Wikimedia Commons
