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Basic Statistics Lecture #8: Anscombe's Quartet

"There are three kinds of lies: lies, damned lies, and statistics."

As promised last time, I will be covering Anscombe's Quartet.  It is an idea where people may use statistics to lie about a data set.  It is a series of data sets developed by statistician Francis Anscombe and published in the journal American Statistician in 1973.

I'm going to provide you with four sets of data.  Do me a favor and apply what you know of statistical analysis to them.  If the statistical data look weird to you, don't be scared; you may have done the analysis perfectly.  Here's the four sets:

I II III IV
x y x y x y x y
10 8.04 10 9.14 10 7.46 8 6.58
8 6.95 8 8.14 8 6.77 8 5.76
13 7.58 13 8.74 13 12.74 8 7.71
9 8.81 9 8.77 9 7.11 8 8.84
11 8.33 11 9.26 11 7.81 8 8.47
14 9.96 14 8.1 14 8.84 8 7.04
6 7.24 6 6.13 6 6.08 8 5.25
4 4.26 4 3.1 4 5.39 19 12.5
12 10.84 12 9.13 12 8.15 8 5.56
7 4.82 7 7.26 7 6.42 8 7.91
5 5.68 5 4.74 5 5.73 8 6.89
If you've calculated the statistical data I've covered so far of these four separate sets, you'll notice that they are all identical.  This seems weird at first, especially with the y values, which are different for all four sets and the x values of the fourth set has one 19 and the other values being 8.  There are other statistical values which I will cover a little later called the correlation coefficient, coefficient of determination, and the linear regression equation, which are all the same for these data sets.

That's the point Francis was trying to make with these two data sets; even though multiple data sets may have the same statistical quantities, they are not equal by any stretch of the imagination.  To emphasize this point, he has corresponding graphs with these data sets:

All of these graphs have the same mean, standard deviation, and other statistical quantities.

These graphs immediately show what the problems are.  In the last two, we need to remove the extreme values which obviously don't fit the data, than recalculate.  That will be shown two posts after this one.  This is a visual representation of how statistics can be used as a lie.  There are other sets of sets which show this concept, but this is the first and most famous to show the point.

Don't get me wrong; statistics is a good tool to describe a series of data points if done properly; the problem lies with when statistical methods are done either poorly or with the intention of deception. With that in mind, it would be a good idea to look at the graph of the data before taking the statistics to heart.  If you want to take a look at the original paper concerning this phenomena, I'll link it here.

If you have any questions, please leave them in the comments. Next time, I'll begin the process of hypothesis testing. Until then, stay curious.
K. "Alan" Eister has his bacholers of Science in Chemistry. He is also a tutor for Varsity Tutors.  If you feel you need any further help on any of the topics covered, you can sign up for tutoring sessions here.
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