View source for Anscombe's quartet

{{Short description|Four data sets with the same descriptive statistics, yet very different distributions}}
[[File:Anscombe's quartet 3.svg|right|425px|thumb|The four [[dataset]]s composing Anscombe's quartet.  All four sets have identical statistical parameters, but the graphs show them to be considerably different]]

'''Anscombe's quartet''' comprises four [[dataset]]s that have nearly identical simple [[descriptive statistics]], yet have very different [[probability distribution|distributions]] and appear very different when [[Plot (graphics)|graphed]]. Each dataset consists of eleven [[Cartesian coordinate system|(''x'',&nbsp;''y'') points]]. They were constructed in 1973 by the [[statistician]] [[Francis Anscombe]] to demonstrate both the importance of graphing data when analyzing it, and the effect of [[outlier]]s and other [[influential observations]] on statistical properties. He described the article as being intended to counter the impression among statisticians that "numerical calculations are exact, but graphs are rough".<ref name="Anscombe">{{cite journal |last=Anscombe |first=F. J. |authorlink=Frank Anscombe |title=Graphs in Statistical Analysis |journal=[[American Statistician]] |volume=27 |year=1973 |issue=1 |pages=17–21 |jstor=2682899 |doi=10.1080/00031305.1973.10478966}}</ref>

== Data ==

For all four datasets:
{| class="wikitable"
! Property
! Value
! Accuracy
|-
| [[Mean]] of ''x''
| 9
| exact
|-
| Sample [[variance]] of ''x'': ''s''{{supsub|2|''x''}}
| 11
| exact
|-
| Mean of ''y''
| 7.50
| to 2 decimal places
|-
| Sample variance of ''y'': ''s''{{supsub|2|''y''}}
| 4.125
| ±0.003
|-
| [[Correlation]] between ''x'' and ''y''
| 0.816
| to 3 decimal places
|-
| [[Linear regression]] line
| ''y''&nbsp;=&nbsp;3.00&nbsp;+&nbsp;0.500''x''
| to 2 and 3 decimal places, respectively
|-
| [[Coefficient of determination]] of the linear regression: <math>R^2</math>
| 0.67
| to 2 decimal places
|}
<!-- to be added to table above:
sums of squared errors (about the mean) = 110.0
regression sums of squared errors (variance accounted for by ''x'') = 27.5
residual sums of squared errors (about the regression line) = 13.75
-->
* The first [[scatter plot]] (top left) appears to be a simple [[linear relationship]], corresponding to two correlated [[variable (mathematics)|variables]], where ''y'' could be modelled as [[normal distribution|gaussian]] with mean linearly dependent on&nbsp;''x''. 
* For the second graph (top right), while a relationship between the two variables is obvious, it is not linear, and the [[Pearson correlation coefficient]] is not relevant. A more general regression and the corresponding [[coefficient of determination]] would be more appropriate. 
* In the third graph (bottom left), the modelled relationship is linear, but should have a different [[regression line]] (a [[robust regression]] would have been called for). The calculated regression is offset by the one [[outlier]], which exerts enough influence to lower the correlation coefficient from 1 to 0.816. 
* Finally, the fourth graph (bottom right) shows an example when one [[high-leverage point]] is enough to produce a high correlation coefficient, even though the other data points do not indicate any relationship between the variables.

The quartet is still often used to illustrate the importance of looking at a set of data graphically before starting to analyze according to a particular type of relationship, and the inadequacy of basic statistic properties for describing realistic datasets.<ref>{{cite journal |url=http://physics.info/linear-regression/practice.shtml#4 |title=Linear Regression |journal=The Physics Hypertextbook |last=Elert |first=Glenn |year=2021 |access-date=2017-02-23 |archive-date=2020-10-01 |archive-url=https://web.archive.org/web/20201001193224/http://physics.info/linear-regression/practice.shtml#4 |url-status=live }}</ref><ref>{{cite book |last=Janert |first=Philipp K. |title=Data Analysis with Open Source Tools |year=2010 |publisher=[[O'Reilly Media]] |pages=[https://archive.org/details/isbn_9780596802356/page/65 65–66] |isbn=978-0-596-80235-6 |url=https://archive.org/details/isbn_9780596802356/page/65 }}</ref><ref>{{cite book |last1=Chatterjee |first1=Samprit |last2=Hadi |first2=Ali S. |year=2006 |title=Regression Analysis by Example |publisher=John Wiley and Sons |page=91 |isbn=0-471-74696-7}}</ref><ref>{{cite book |last1=Saville |first1=David J. |last2=Wood |first2=Graham R. |year=1991 |title=Statistical Methods: The geometric approach |publisher=[[Springer Science+Business Media|Springer]] |page=418 |isbn=0-387-97517-9}}</ref><ref>{{cite book |last=Tufte |first=Edward R. |authorlink=Edward Tufte |year=2001 |title=The Visual Display of Quantitative Information |edition=2nd |location=Cheshire, CT |publisher=Graphics Press |isbn=0-9613921-4-2 |url=https://archive.org/details/visualdisplayofq00tuft }}</ref>

The datasets are as follows. The ''x'' values are the same for the first three datasets.<ref name="Anscombe"/>
<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
{| class="wikitable" style="text-align: center"
|+ Anscombe's quartet
|-
! colspan="2"| Dataset I
! colspan="2"| Dataset II
! colspan="2"| Dataset III
! colspan="2"| Dataset IV
|-
| ''x'' 
| ''y'' 
| ''x'' 
| ''y'' 
| ''x'' 
| ''y'' 
| ''x'' 
| ''y''   
|-
| 10.0 ||  8.04 || 10.0 ||  9.14 || 10.0 ||  7.46 ||  8.0 ||  6.58
|-
|  8.0 ||  6.95 ||  8.0 ||  8.14 ||  8.0 ||  6.77 ||  8.0 ||  5.76
|-
| 13.0 ||  7.58 || 13.0 ||  8.74 || 13.0 || 12.74 ||  8.0 ||  7.71
|-
|  9.0 ||  8.81 ||  9.0 ||  8.77 ||  9.0 ||  7.11 ||  8.0 ||  8.84
|-
| 11.0 ||  8.33 || 11.0 ||  9.26 || 11.0 ||  7.81 ||  8.0 ||  8.47
|-
| 14.0 ||  9.96 || 14.0 ||  8.10 || 14.0 ||  8.84 ||  8.0 ||  7.04
|-
|  6.0 ||  7.24 ||  6.0 ||  6.13 ||  6.0 ||  6.08 ||  8.0 ||  5.25
|-
|  4.0 ||  4.26 ||  4.0 ||  3.10 ||  4.0 ||  5.39 || 19.0 || 12.50
|-
| 12.0 || 10.84 || 12.0 ||  9.13 || 12.0 ||  8.15 ||  8.0 ||  5.56
|-
|  7.0 ||  4.82 ||  7.0 ||  7.26 ||  7.0 ||  6.42 ||  8.0 ||  7.91
|-
|  5.0 ||  5.68 ||  5.0 ||  4.74 ||  5.0 ||  5.73 ||  8.0 ||  6.89
|}
</div>

It is not known how Anscombe created his datasets.<ref name="ChatterjeeFirat">{{cite journal |last1=Chatterjee |first1=Sangit |last2=Firat |first2=Aykut |year=2007 |title=Generating Data with Identical Statistics but Dissimilar Graphics: A follow up to the Anscombe dataset |journal=[[The American Statistician]] |volume=61 |issue=3 |pages=248–254 |doi=10.1198/000313007X220057| jstor=27643902|s2cid=121163371 }}</ref> Since its publication, several methods to generate similar datasets with identical statistics and dissimilar graphics have been developed.<ref name="ChatterjeeFirat"/><ref>{{cite book |last1=Matejka |first1=Justin |last2=Fitzmaurice |first2=George |title=Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems |chapter=Same Stats, Different Graphs: Generating Datasets with Varied Appearance and Identical Statistics through Simulated Annealing |year=2017 |pages=1290–1294 |doi=10.1145/3025453.3025912|isbn=9781450346559 |s2cid=9247543 }}</ref>
One of these, the ''[[Datasaurus dozen]]'', consists of points tracing out the outline of a dinosaur, plus twelve other datasets that have the same summary statistics.<ref>{{Cite web |last1=Matejka |first1=Justin |last2=Fitzmaurice |first2=George |date=2017 |title=Same Stats, Different Graphs: Generating Datasets with Varied Appearance and Identical Statistics through Simulated Annealing |url=https://www.autodesk.com/research/publications/same-stats-different-graphs |url-status=live |access-date=2021-04-20 |website=Autodesk Research |language=en-US |archive-url=https://web.archive.org/web/20201004003855/https://www.autodesk.com/research/publications/same-stats-different-graphs |archive-date=2020-10-04 }}</ref><ref>{{Cite journal |last1=Murray |first1=Lori L. |last2=Wilson |first2=John G. |date=April 2021 |title=Generating data sets for teaching the importance of regression analysis |url=https://onlinelibrary.wiley.com/doi/10.1111/dsji.12233 |journal=Decision Sciences Journal of Innovative Education |language=en |volume=19 |issue=2 |pages=157–166 |doi=10.1111/dsji.12233 |s2cid=233609149 |issn=1540-4595 |access-date=2021-04-20 |archive-date=2021-04-23 |archive-url=https://web.archive.org/web/20210423155254/https://onlinelibrary.wiley.com/doi/10.1111/dsji.12233 |url-status=live }}</ref><ref>{{Citation |last1=Andrienko |first1=Natalia |author1-link=Natalia Andrienko |title=Visual Analytics for Investigating and Processing Data |date=2020 |url=http://link.springer.com/10.1007/978-3-030-56146-8_5 |work=Visual Analytics for Data Scientists |pages=151–180 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-56146-8_5 |isbn=978-3-030-56145-1 |access-date=2021-04-20 |last2=Andrienko |first2=Gennady |last3=Fuchs |first3=Georg |last4=Slingsby |first4=Aidan |last5=Turkay |first5=Cagatay |last6=Wrobel |first6=Stefan |s2cid=226648414 |postscript=. |archive-date=2024-10-03 |archive-url=https://web.archive.org/web/20241003162552/https://link.springer.com/chapter/10.1007/978-3-030-56146-8_5 |url-status=live }}</ref>

== See also ==
* [[Datasaurus dozen]]
* [[Exploratory data analysis]]
* [[Goodness of fit]]
* [[Regression validation]]
* [[Simpson's paradox]]
* [[Statistical model validation]]

== References ==
{{Reflist}}

== External links ==
* [http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Visualisation/Visualisation.html Department of Physics, University of Toronto]
* [https://www.geogebra.org/m/tbwXxySn Dynamic Applet] made in [[GeoGebra]] showing the data & statistics and also allowing the points to be dragged (Set 5).
* [https://www.autodeskresearch.com/publications/samestats Animated examples from Autodesk] called the "Datasaurus Dozen".
* [https://stat.ethz.ch/R-manual/R-devel/library/datasets/html/anscombe.html Documentation] for the datasets in [[R (programming language)|R]].

[[Category:Misuse of statistics]]
[[Category:Statistical charts and diagrams]]
[[Category:Statistical data sets]]
[[Category:1973 introductions]]
[[Category:1973 in science]]