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U large values from one variable occur with small values for the other variable, s12 will be negative. If there is no particular association between the values for the two variables, s 12 will be approximately zero. This measure of the linear association between two variables does not depend on the units of measurement. The sample correlation coefficient for the ith and kth variables is defined as. The sample correlation coefficient is a standardized version of the sample covariance, where the product of the square roots of the sample variances provides the standardization.

Notiee that r;k has the same value whether nor n - 1 is chosen as the common divisor for s;;, skk, and s;k The sample correlation coefficient r;k can also be viewed as a sample covariance. Thestandardizedvaluesarecornmensurablebe cause both sets are centered at zero and expressed in standard deviation units. The sample correlation coefficient is just the sample covariance of the standardized observations. Although the signs of the sample correlation and the sample covariance are the same, the correlation is ordinarily easier to interpret because its magnitude is bounded.

To summarize, the sample correlation r has the following properties:. Here r measures the strength of the linear association. The quantities sik and r;k do not, in general, convey all there is to know about the association between two variables. Nonlinear associations can exist that are not revealed by these descriptive statistics. Covariance and correlation provide measures of linear association, or association along a line.

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Their values are less informative for other kinds of association. On the other hand, these quantities can be very sensitive to "wild" observations "outliers" and may indicate association when, in fact, little exists. In spite of these shortcomings, covariance and correlation coefficients are routinely calculated and analyzed.

They provide cogent numerical summaries of association when the data do not exhibit obvious nonlinear patterns of association and when wild observations are not present. Suspect observations must be accounted for by correcting obvious recording mistakes and by taking actions consistent with the identified causes. The values of s;k and r;k should be quoted both with and without these observations. The sum of squares of the deviations from the mean and the sum of crossproduct deviations are often of interest themselves.

These quantities are wkk. The descriptive statistics computed from n measurements on p variables can also be organized into arrays. The subscript n on the array Sn is a mnemonic device used to remind you that n is employed as a divisor for the elements s;k The size of all of the arrays is determined by the number of variables, p. The arrays Sn and R consist of p rows and p columns. The array i is a single column with p rows. The first subscript on an entry in arrays Sn and R indicates the row; the second subscript indicates the column.

Each receipt yields a pair of measurements, total dollar sales, and number of books sold. Find the arrays i, Sn, and R. Since there are four receipts, we have a total of four measurements observations on each variable. The-sample means are. Graphical Techniques Plots are important, but frequently neglected, aids in data analysis.

Although it is impossible to simultaneously plot all the measurements made on several variables and study the configurations, plots of individual variables and plots of pairs of variables can still be very informative.

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Sophisticated computer programs and display equipment allow one the luxury of visually examining data in one, two, or three dimensions with relative ease. On the other hand, many valuable insights can be obtained from the data by constructing plots with paper and pencil. Simple, yet elegant and effective, methods for displaying data are available in 29]. It is good statistical practice to plot pairs of variables and visually inspect the pattern of association. Consider, then, the following seven pairs of measurements on two variables: Variable 1 x1 : Variable 2 x 2 : 3 4.

These data are plotted as seven points in two dimensions each axis representing a variable in Figure 1. The coordinates of the points are determined by the paired measurements: 3, 5 , 4, 5. The resulting two-dimensional plot is known as a scatter diagram or scatter plot.

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These plots are called marginal dot diagrams. They can be obtained from the original observations or by projecting the points in the scatter diagram onto each coordinate axis.

The information contained in the single-variable dot diagrams can be used to calculate the sample means xi and x and the sample variances si I and s See Ex2 ercise 1. The scatter diagram indicates the orientation of the points, and their coordinates can be used to calculate the sample covariance Siz In the scatter diagram of Figure 1.

Dot diagrams and scatter plots contain different kinds of information. The information in the marginal dot diagrams is not sufficient for constructing the scatter plot. As an illustration, suppose the data preceding Figure 1. We have simply rearranged the values of variable 1. The scatter and dot diagrams for the "new" data are shown in Figure 1. Comparing Figures 1. In Figure 1. Consequently, the descriptive statistics for the individual variables xi, x , sii, and s22 remain unchanged, but the sample covari2 ance si 2 , which measures the association between pairs of variables, will now be negative.

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The different orientations of the data in Figures 1. At the same time, the fact that the marginal dot diagrams are the same in the two cases is not immediately apparent from the scatter plots. The two types of graphical procedures complement one another; they are not competitors. The next two examples further illustrate the information that can be conveyed by a graphic display.

We have labeled two "unusual" observations. Time Warner has a "typical" number of employees, but comparatively small negative profits per employee. The sample correlation coefficient computed from the values of x 1 and x 2 is -. It is clear that atypical observations can have a considerable effect on the sample correlation coefficient. The results are given in Thble 1. The scatter plot in Figure 1. Of course, this cause-effect relationship cannot be substantiated, because the experiment did not include a random assignment of payrolls.

To construct the scatter plot in Figure 1. The figure allows us to examine visually the grouping of teams with respect to the vari ables total payroll and won-lost percentage. Example I. S Multiple scatter plots for paper strength measurements Paper is manufactured in continuous sheets several feet wide. Because of the orientation of fibers within the paper, it has a different strength when measured in the direction produced by the machine than when measured across, or at right angles to, the machine direction.

Table 1. A novel graphic presentation of these data appears in Figure 1. The scatter plots are arranged as the off-diagonal elements of a covariance array and box plots as the diagonal elements. The latter are on a different scale with this. Machine direction Cross direction The scatter plots can be inspected for patterns and unusual observations. Some of the scatter plots have patterns suggesting that there are two separate clumps of observations.

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These scatter plot arrays are further pursued in our discussion of new software graphics in the next section. In the general multiresponse situation, p variables are simultaneously recorded on n items. Scatter plots should be made for pairs of important variables and, if the task is not too great to warrant the effort, for all pairs. However, two further geometric representations of the data provide an important conceptual framework for viewing multi variable statistical methods.

In cases where it is possible to capture the essence of the data in three dimensions, these representations can actually be graphed. Consider the natural extension of the scatter plot top dimensions, where the p measurements.

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The coordinate axes are taken to correspond to the variables, so that the jth point is xi! The resulting plot with n points not only will exhibit the overall pattern of variability, but also will show similarities and differences among then items. Groupings of items will manifest themselves in this representation.

The next example illustrates a three-dimensional scatter plot. The weight, or mass, is given in grams while the snout-vent length SVL and hind limb span HLS are given in millimeters. The data are displayed in Table 1. Although there are three size measurements, we can ask whether or not most of the variation is primarily restricted to two dimensions or even to one dimension. To help answer questions regarding reduced dimensionality, we construct the three-dimensional scatter plot in Figure 1.

Clearly most of the variation is scatter about a one-dimensional straight line. Knowing the position on a line along the major axes of the cloud of points would be almost as good as knowing the three measurements Mass, SVL, and HLS. However, this kind of analysis can be misleading if one variable has a much larger variance than the others.

Figure 1. Most of the vanatwn can be explamed by a smgle vanable de-. A three-dimensional scatter plot can often reveal group structure. The gender, by row, for the lizard data in Table 1. Clearly, males are typically larger than females. The n observations of the p variables can also be regarded as p points in n-dimensional space. Each column of X determines one of the points.