Humans like to count things. We count the hours, minutes, and seconds in each day. We count people, antelopes, and internet page hits. After we have counted stuff, we like to try and make some sense out of the numbers, so we use statistics. Statistics help us “see” patterns in the numbers, and since numbers can be tricky, the example below shows us why we should take the time to graphically represent our data.

Here’s to a happy new year and another exciting year of Lean Learning!

Bob Hubbard, January 2, 2013

All four sets are identical when examined using simple summary statistics, but vary considerably when graphed.

**Anscombe’s quartet** comprises four datasets that have nearly identical simple statistical properties, yet 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 before analysing it and the effect of outliers on statistical properties.^{[1]}

## For all four datasets:

Property | Value |
---|---|

Mean of x in each case: |
9 (exact) |

Variance of x in each case: |
11 (exact) |

Mean of y in each case: |
7.50 (to 2 decimal places) |

Variance of y in each case: |
4.122 or 4.127 (to 3 decimal places) |

Correlation between x and y in each case: |
0.816 (to 3 decimal places) |

Linear regression line in each case: | y = 3.00 + 0.500x (to 2 and 3 decimal places, respectively) |

The first scatter plot (top left) appears to be a simple linear relationship, corresponding to two variables correlated and following the assumption of normality. The second graph (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear, and the Pearson correlation coefficient is not relevant. In the third graph (bottom left), the distribution is linear, but with a different regression line, which is offset by the one outlier which exerts enough influence to alter the regression line and lower the correlation coefficient from 1 to 0.816. Finally, the fourth graph (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

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]}

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 |

A procedure to generate similar data sets with identical statistics and dissimilar graphics has since been developed.^{[7]}

## References

- ^
^{a}^{b}Anscombe, F. J. (1973). “Graphs in Statistical Analysis”.*American Statistician***27**(1): 17–21. JSTOR 2682899. **^**Elert, Glenn. “Linear Regression”.*The Physics Hypertextbook*.**^**Janert, Philipp K. (2010).*Data Analysis with Open Source Tools*. O’Reilly Media, Inc.. pp. 65–66. ISBN 0-596-80235-8.**^**Chatterjee, Samprit & Hadi, Ali S. (2006).*Regression analysis by example*. John Wiley and Sons. pp. 91. ISBN 0-471-74696-7.**^**Saville, David J. & Wood, Graham R. (1991).*Statistical methods: the geometric approach*. Springer. pp. 418. ISBN 0-387-97517-9.**^**Tufte, Edward R. (2001).*The Visual Display of Quantitative Information*(2nd ed.). Cheshire, CT: Graphics Press. ISBN 0-9613921-4-2.**^**Chatterjee, Sangit; Firat, Aykut (2007). “Generating Data with Identical Statistics but Dissimilar Graphics: A Follow up to the Anscombe Dataset”.*American Statistician***61**(3): 248–254.doi:10.1198/000313007X220057.

## See also

- Exploratory data analysis

## External links

- Department of Physics, University of Toronto
- Curve fitting, Central Queensland University, Australia
- Practice Problems, Linear Regression, The Physics Hypertextbook (See practice problem 4.)

Anscombe’s quartet on Wikipedia.com