11.2: The Need for Data Visualization
- Page ID
- 66337
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)While it may seem obvious that visual methods for summarizing data can be helpful, a simple example can demonstrate just how usefulness of visual representations of data. To do this we will consider four sets of paired data that are known collectively as "Ascombe's quartet" (Anscombe 1973). The data are given in Table 11.1.
| \(X\) | \(Y_1\) | \(Y_2\) |
\(Y_3\) |
\(X_4\) | \(Y_4\) |
| 10.0 | 8.04 | 9.14 | 7.46 | 8.0 | 6.58 |
| 8.0 | 6.95 | 8.14 | 6.77 | 8.0 | 5.76 |
| 13.0 | 7.58 | 8.74 | 12.74 | 8.0 | 7.71 |
| 9.0 | 8.81 | 8.77 | 7.11 | 8.0 | 8.84 |
| 11.0 | 8.33 | 9.26 | 7.81 | 8.0 | 8.47 |
| 14.0 | 9.96 | 8.10 | 8.84 | 8.0 | 7.04 |
| 6.0 | 7.24 | 6.13 | 6.08 | 8.0 | 5.25 |
| 4.0 | 4.26 | 3.10 | 5.39 | 19.0 | 12.50 |
| 12.0 | 10.84 | 9.13 | 8.15 | 8.0 | 5.56 |
| 7.0 | 4.82 | 7.26 | 6.42 | 8.0 | 7.91 |
| 5.0 | 5.68 | 4.74 | 5.73 | 8.0 | 6.89 |
This is paired data. The first set of the data consists of the values of \(X\) paired with \(Y_1\). The second set of data uses the same values for \(X\) but is paired with \(Y_2\) instead of \(Y_1\). Similarly, the third set of data consists of the values of \(X\) paired with \(Y_3\). The fourth set of data consists of the values in \(X_4\) paired with \(Y_4\).
Suppose that we would like to compare these four sets of paired data. From the previous chapters we know that this appears to be quantitative data on at least an interval scale and therefore we can use numerical measures like the mean, median, range, and standard deviation to describe the location and variation of each of these variables. Starting with the variables \(X\) and \(X_4\) it can be shown that the mean of both of these variables is equal to 9. Similarly, it can be shown that the mean of \(Y_1\), \(Y_2\), \(Y_3\), and \(Y_4\) are all equal to 7.5. For this second result the means are equal up to the fourth decimal place. There are slight differences in some of the medians, though they are all quite similar. The median of \(X\) and \(X_4\) are 9 and 8, respectively. The medians for \(Y_1\),...,\(Y_4\) are 7.58, 8.14, 7.11, and 7.04, respectively.