Anscombe's Quartet (1973)

Question

A problem involving this dataset appeared recently in one of my student's assignments. You can try it too.

1.) Google "Anscombe", "quartet", and "kaggle." This will lead you to a site where you can download a CSV of the dataset. The variable labeled, 'x123' serves as the "x' variable for the first 3 examples.

2.) Run descriptive stats, correlations, and bivariate regressions for the 4 variables. (y1 = x1, y2 = x2, y3 = x3, y4 = x2)

3.) You will find that the descriptive stats, correlations, and regression equations for the 4 problems are nearly identicial. Let's dig deeper.

4.) Make scatterplots for each of the 4 problems. What do you see?

5.) Hints:

1. problem 1. The only OK one in the set
2. problem 2. Add a particular transformation of x2 to your original equation and you'll perfectly explain the data. new question y = x2 + transformed_x2
3. problem 3. Remove one x3,y3 pair and obtain a perfect prediction
4. problem 4. Remove on x4, y4 pair and discover the case is hopeless

Answer 1

Descriptive statistics

Mean x= 9

variance of x : s2 x 11

Mean of y 7.50

Sample variance of y : s2

y 4.125

Correlation

between x and y 0.82

Linear regression line y = 3.00 + 0.500x

R2= 0.67

The first scatter plot 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 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 the modelled relationship is linear, but should have a different regression line 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 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.

Answer 2

and the moral of the story is, always have a look at your data via scatterplots and histograms.