2-4 Scatterplots, Correlation, and Regression 75 FIGURE 2-16 Weights of Pennies and Years of Production and 3% zinc, but post-1983 pennies are 2.5% copper and 97.5% zinc. If we ignored the characteristic of the clusters, we might incorrectly think that there is a relationship between the weight of a penny and the year it was made. If we examine the two groups separately, we see that there does not appear to be a relationship between the weights of pennies and the years they were produced. The preceding three examples involve making decisions about a correlation based on subjective judgments of scatterplots, but Part 2 introduces the linear correlation coefficient as a numerical measure that can help us make such decisions more objectively. Using paired data, we can calculate the value of the linear correlation coefficient r. PART 2 Linear Correlation Coefficient r DEFINITION The linear correlation coefficient is denoted by r, and it measures the strength of the linear association between two variables. The value of a linear correlation coefficient r can be manually computed by applying Formula 10-1 or Formula 10-2 found in Section 10-1 on page 511, but in practice, r is almost always found by using statistics software or a suitable calculator. Using r for Determining Correlation The computed value of the linear correlation coefficient is always between -1 and 1. If r is close to -1 or close to 1, there appears to be a correlation, but if r is close to 0, there does not appear to be a linear correlation. For the data depicted in the scatterplot of Figure 2-14, r = 0.948 (close to 1), and the data in the scatterplot of Figure 2-15 result in r = -0.144 (pretty close to 0). These descriptions of “close to” -1 or 1 or 0 are vague, but there are other objective criteria. For now we will use a table of special values (Table 2-11 on page 77) for deciding whether there is a linear correlation. See the following example illustrating the interpretation of the linear correlation coefficient r. Go Figure No more than 1.5 tons: The future weight of a single computer forecast by Popular Mechanics in 1949
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