512 CHAPTER 10 Correlation and Regression Properties of the Linear Correlation Coefficient r 1. The value of r is always between -1 and 1 inclusive. That is, -1 … r … 1. 2. If all values of either variable are converted to a different scale, the value of r does not change. 3. The value of r is not affected by the choice of x or y. Interchange all x values and y values, and the value of r will not change. 4. r measures the strength of a linear relationship. It is not designed to measure the strength of a relationship that is not linear [as in Figure 10-2(d)]. 5. r is very sensitive to outliers in the sense that a single outlier could dramatically affect its value. Calculating the Linear Correlation Coefficient r The following three examples illustrate three different methods for finding the value of the linear correlation coefficient r, but you need to use only one method. The use of technology (as in Example 1) is strongly recommended. If manual calculations are absolutely necessary, Formula 10-1 is recommended (as in Example 2). If a better understanding of r is desired, Formula 10-2 is recommended (as in Example 3). TABLE 10-1 Powerball Tickets Sold and Jackpot Amounts Jackpot 334 127 300 227 202 180 164 145 255 Tickets 54 16 41 27 23 18 18 16 26 Statdisk Minitab StatCrunch XLSTAT TI-83, 84 Plus JMP Use technology to find the value of the linear correlation coefficient r for the Powerball jackpot amounts and numbers of tickets listed in Table 10-1. CP EXAMPLE 1 Finding r Using Technology SOLUTION The value of r will be automatically calculated with software or a calculator. See the accompanying technology displays showing that r = 0.947 (rounded to three decimal places).
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