11.2: The Need for Data Visualization

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. The standard deviations of \(X\) and \(X_4\) are both 3.32 while the standard deviations of \(Y_1\),...,\(Y_4\) are all equal to 2.03. Finally, the range of \(X\) and \(X_4\) are 10 and 11, respectively, while the ranges of \(Y_1\),...,\(Y_4\) are 6.58, 6.16, 7.35, and 7.25, respectively.

A simple question that one may ask is how the sets of data for \(X\) and \(X_4\) differ from one another and how the sets of data \(Y_1\),...,\(Y_4\) differ from one another. By considering the numerical summaries of location and variation one can observe that the locations and amount of variation are similar when comparing \(X\) and \(X_4\) to one another and when separately comparing \(Y_1\),...,\(Y_4\) to one another. Does this mean that the data sets are similar to one another. A quick look at Table 11.1 will quickly show that is not the case, and that there may be fundamental differences between the sets of data even though the numerical measures of location and variation are similar. Let us begin by considering the sets of data observed for \(X\) and \(X_4\). It is a simple matter to note that the values of \(X\) are all different whereas the values of \(X_4\) are all the same except for one. Therefore, it follows that even though the numerical measures of location and variation are very similar for \(X\) and \(X_4\), the actual structure of the data is quite different. It is much more difficult to try to discern the structural differences between \(Y_1\),...,\(Y_4\). While we may be able to pick out some differences, such as the fact that \(Y_4\) has a maximum of 12.50, which is larger than the maximum of each of \(Y_1\),...,\(Y_3\), the majority of the structure of these sets of data is lost in the lists of numbers. So are there differences between these sets of data?

This chapter will consider the use of visual representations of data in the form of mathematical graphs and pictures. The advantage of these visual summaries of the data allow individuals to quickly comprehend the broad structure of a set of data without any further calculations. The disadvantage is that it may be difficult to discern detailed ideas about the structure of data. For example, while a visual summary of a set of data may show that two sets of data have different structures, we may not be able to tell whether the location of one set of data is different than the the other set of data. To determine that type of difference we will need to use numerical data summaries. Hence, visual and numerical summaries of data are usually used together to get a complete conception of what a set of data is trying to tell us.

To demonstrate the usefulness of visually summarizing data we will construct a scatterplot of each of the four sets of data given in Table 11.1. While scatterplots will be considered in greater detail later in this chapter, we will briefly describe them for the purpose of the current example. A scatterplot is a plot produced on a set of axes where the position of a point on horizontal axis represents the value of one of the variables and the position of the same point on th vertical axis represents the value of the paired value of the other variable. A scatterplot of \(Y_1\) plotted against \(X_1\) is showin in Figure X.  

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