{"id":1252,"date":"2013-01-16T20:06:29","date_gmt":"2013-01-16T20:06:29","guid":{"rendered":"http:\/\/johncanning.net\/wordpress\/?p=1252"},"modified":"2016-03-10T14:43:27","modified_gmt":"2016-03-10T14:43:27","slug":"1252","status":"publish","type":"post","link":"http:\/\/johncanning.net\/wp\/?p=1252","title":{"rendered":"Why you should graph data"},"content":{"rendered":"

Amended 10 March 2016 (corrections\/ update made)<\/span><\/p>\n

\"Anscombe's<\/a>
Click to enlarge<\/figcaption><\/figure>\n

I came across Anscombe\u2019s Quartet on Wikipedia recently. I must confess to not having seen it before and don\u2019t recall seeing it in any introductory statistics books.<\/p>\n

The Anscombe\u2019s Quartet is a conceptually and graphically clear way of showing the importance of graphs in statistical analysis. Each of the 11 pairs of observations have the same, x mean, y mean, x variance, y variance, correlation co-efficient and regression equation, though each have very different distributions. They clearly demonstrate the impact of outliers and how non-linear relationships can be identified.<\/p>\n

Citation:<\/p>\n

F. J. Anscombe (1973) Graphs in Statistical Analysis The American Statistician<\/i> , Vol. 27, No. 1 (Feb., 1973), pp. 17-21<\/p>\n

Article Stable URL: http:\/\/www.jstor.org\/stable\/2682899<\/a> (Not open access)<\/p>\n

 <\/p>\n

LaTeX code below.
\n
\n\\documentclass{article}
\n\\usepackage{pgfplots}
\n\\usepackage{pgfplotstable}
\n\\pgfplotsset{compat=1.7}
\n\\usepackage{amssymb, amsmath}
\n\\usepackage{subcaption}
\n\\begin{document}
\n\\begin{figure}
\n\\caption{Anscombe's quartet is a good demonstration why a scatterplot is so valuable, prior to calculating regression equations and correlation co-efficients. In all four cases the $x's$ have a mean of 9, and variance of 11. The mean of all the $y's$ is 7.5, and a variance 4.125. The correlation co-efficient of each is 0.816 and the linear regression line is $y=3+0.5x $}
\n\\begin{subfigure}{.45 \\textwidth}
\n\\centering
\n\\caption{Normal linear relationship}
\n\\begin{tikzpicture}
\n\\begin{axis} [width=5cm, height=5cm, xlabel=X1, ylabel=Y1]
\n\\addplot[scatter, only marks, mark=x, mark size=4pt]
\ncoordinates
\n{
\n(10, 8.04)
\n(8.0, 6.95)
\n(13, 7.58)
\n(9, 8.81)
\n(11, 8.33)
\n(14, 9.96)
\n(6, 7.24)
\n(4, 4.26)
\n(12, 10.84)
\n(7, 4.82)
\n(5, 5.68)
\n};
\n\\addplot[scatter, mark=.]
\ncoordinates
\n{
\n(0, 4.1)
\n(20, 12.5)
\n};
\n\\end{axis}
\n\\end{tikzpicture}
\n\\end{subfigure}
\n\\begin{subfigure}{.45 \\textwidth}
\n\\centering
\n\\caption{Relationship clear, but not linear}
\n\\begin{tikzpicture}
\n\\begin{axis}[width=5cm, height=5cm, xlabel=X2, ylabel=Y2]
\n\\addplot[scatter, only marks, mark=x, mark size=4pt]
\ncoordinates
\n{
\n(10, 9.14)
\n(8.0, 8.14)
\n(13, 8.74)
\n(9, 8.77)
\n(11, 9.26)
\n(14, 8.10)
\n(6, 6.13)
\n(4, 3.1)
\n(12, 9.13)
\n(7, 7.26)
\n(5, 4.74)
\n};
\n\\addplot[scatter, mark=.]
\ncoordinates
\n{
\n(0, 4.1)
\n(20, 12.5)
\n};
\n\\end{axis}
\n\\end{tikzpicture}
\n\\end{subfigure}
\n\\
\n\\begin{subfigure}{.45 \\textwidth}
\n\\centering
\n\\caption{Clear linear relationship, but one outlier offsets the regression line}
\n\\begin{tikzpicture}
\n\\begin{axis} [width=5cm, height=5cm, xlabel=X3, ylabel=Y3]
\n\\addplot[scatter, only marks, mark=x, mark size=4pt]
\ncoordinates
\n{
\n(10, 7.46)
\n(8.0, 6.77)
\n(13, 12.74)
\n(9, 7.11)
\n(11, 7.81)
\n(14, 8.84)
\n(6, 6.08)
\n(4, 5.39)
\n(12, 8.15)
\n(7, 6.42)
\n(5, 5.73)
\n};
\n\\addplot[scatter, mark=.]
\ncoordinates
\n{
\n(0, 4.1)
\n(20, 12.5)
\n};
\n\\end{axis}
\n\\end{tikzpicture}
\n\\end{subfigure}
\n\\begin{subfigure}{.45 \\textwidth}
\n\\centering
\n\\caption{Clear relationship, but one outlier puts the regression line at 45 degrees to the other 10 observations}
\n\\begin{tikzpicture}
\n\\begin{axis} [width=5cm, height=5cm, xlabel=X4, ylabel=Y4]
\n\\addplot[scatter, only marks, mark=x, mark size=4pt]
\ncoordinates
\n{
\n(8, 6.58)
\n(8.0, 5.76)
\n(8, 7.71)
\n(8, 8.84)
\n(8, 7.04)
\n(8, 5.26)
\n(19, 12.5)
\n(8, 5.56)
\n(8, 7.91)
\n(8, 6.89)
\n(8, 6.89)
\n};
\n\\addplot[scatter, mark=.]
\ncoordinates
\n{
\n(0, 4.1)
\n(20, 12.5)
\n};
\n\\end{axis}
\n\\end{tikzpicture}
\n\\end{subfigure}
\n\\end{figure}
\n\\end{document}
\n<\/code><\/p>\n\n\n