Anscombe's quartet

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Short description: Four data sets with the same descriptive statistics, yet very different distributions
All four sets are identical when examined using simple summary statistics, but vary considerably when graphed

Anscombe's quartet comprises four data sets that have nearly identical simple descriptive statistics, yet have very different distributions and appear very different when graphed. Each dataset consists of eleven (x,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 outliers 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."[1]

Data

For all four datasets:

Property Value Accuracy
Mean of x 9 exact
Sample variance of x : s2x 11 exact
Mean of y 7.50 to 2 decimal places
Sample variance of y : s2y 4.125 ±0.003
Correlation between x and y 0.816 to 3 decimal places
Linear regression line y = 3.00 + 0.500x to 2 and 3 decimal places, respectively
Coefficient of determination of the linear regression : [math]\displaystyle{ R^2 }[/math] 0.67 to 2 decimal places
  • The first scatter plot (top left) appears to be a simple linear relationship, corresponding to two variables correlated where y could be modelled as gaussian with mean linearly dependent on x.
  • 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.[2][3][4][5][6]

The datasets are as follows. The x values are the same for the first three datasets.[1]

Anscombe's quartet
I II III 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

It is not known how Anscombe created his datasets.[7] Since its publication, several methods to generate similar data sets with identical statistics and dissimilar graphics have been developed.[7][8] One of these, the Datasaurus Dozen, consists of points tracing out the outline of a dinosaur, plus twelve other data sets that have the same summary statistics.[9][10][11] Datasaurus Dozen was created by Justin Matejka and George Fitzmaurice. The process is described in their paper “Same stats, different graphs: generating datasets with varied appearance and identical statistics through simulated annealing“.

The Datasaurus Dozen proves us as much as Anscombe Quartet why visualizing our data is important as summary statistics can be the same, while data distributions can be very different.

See also

References

  1. 1.0 1.1 Anscombe, F. J. (1973). "Graphs in Statistical Analysis". American Statistician 27 (1): 17–21. doi:10.1080/00031305.1973.10478966. 
  2. Elert, Glenn (2021). "Linear Regression". The Physics Hypertextbook. http://physics.info/linear-regression/practice.shtml#4. 
  3. Janert, Philipp K. (2010). Data Analysis with Open Source Tools. O'Reilly Media. pp. 65–66. ISBN 978-0-596-80235-6. https://archive.org/details/isbn_9780596802356/page/65. 
  4. Chatterjee, Samprit; Hadi, Ali S. (2006). Regression Analysis by Example. John Wiley and Sons. p. 91. ISBN 0-471-74696-7. 
  5. Saville, David J.; Wood, Graham R. (1991). Statistical Methods: The geometric approach. Springer. p. 418. ISBN 0-387-97517-9. 
  6. Tufte, Edward R. (2001). The Visual Display of Quantitative Information (2nd ed.). Cheshire, CT: Graphics Press. ISBN 0-9613921-4-2. https://archive.org/details/visualdisplayofq00tuft. 
  7. 7.0 7.1 Chatterjee, Sangit; Firat, Aykut (2007). "Generating Data with Identical Statistics but Dissimilar Graphics: A follow up to the Anscombe dataset". The American Statistician 61 (3): 248–254. doi:10.1198/000313007X220057. 
  8. Matejka, Justin; Fitzmaurice, George (2017). "Same Stats, Different Graphs: Generating Datasets with Varied Appearance and Identical Statistics through Simulated Annealing". Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems: 1290–1294. doi:10.1145/3025453.3025912. 
  9. Murray, Lori L.; Wilson, John G. (April 2021). "Generating data sets for teaching the importance of regression analysis" (in en). Decision Sciences Journal of Innovative Education 19 (2): 157–166. doi:10.1111/dsji.12233. ISSN 1540-4595. https://onlinelibrary.wiley.com/doi/10.1111/dsji.12233. 
  10. Andrienko, Natalia; Andrienko, Gennady; Fuchs, Georg; Slingsby, Aidan; Turkay, Cagatay; Wrobel, Stefan (2020), "Visual Analytics for Investigating and Processing Data" (in en), Visual Analytics for Data Scientists (Cham: Springer International Publishing): pp. 151–180, doi:10.1007/978-3-030-56146-8_5, ISBN 978-3-030-56145-1, http://link.springer.com/10.1007/978-3-030-56146-8_5, retrieved 2021-04-20 
  11. Matejka, Justin; Fitzmaurice, George (2017). "Same Stats, Different Graphs: Generating Datasets with Varied Appearance and Identical Statistics through Simulated Annealing" (in en-US). https://www.autodesk.com/research/publications/same-stats-different-graphs. 

External links