474 CHAPTER 9 Correlation and Regression Calculating a Correlation Coefficient In Words In Symbols 1. Find the sum of the x@values. Σx 2. Find the sum of the y@values. Σy 3. Multiply each x@value by its corresponding Σxy y@value and find the sum. 4. Square each x@value and find the sum. Σx2 5. Square each y@value and find the sum. Σy2 6. Use these five sums to calculate the correlation coefficient. r = nΣxy - 1Σx21Σy2 2 nΣx2 - 1Σx222nΣy2 - 1Σy22 GUIDELINES Calculating a Correlation Coefficient Calculate the correlation coefficient for the gross domestic products and carbon dioxide emissions data in Example 1. Interpret the result in the context of the data. SOLUTION Use a table to help calculate the correlation coefficient. With these sums and n = 10, the correlation coefficient is r = nΣxy - 1Σx21Σy2 2 nΣx2 - 1Σx222nΣy2 - 1Σy22 = 10116,687.992 - 125.7215269.52 2 10182.812 - 125.72221013,548,633.252 - 15269.522 = 31,453.75 2 167.6127,718,702.25 ≈ 0.874. Round to three decimal places. The result r ≈ 0.874 suggests a strong positive linear correlation. Interpretation As the gross domestic product increases, the carbon dioxide emissions tend to increase. EXAMPLE 4 GDP (in trillions of dollars), x CO2 emissions (in millions of metric tons), y xy x2 y2 1.7 620.1 1054.17 2.89 384,524.01 2.4 475.2 1140.48 5.76 225,815.04 3.0 457.6 1372.8 9 209,397.76 1.2 389.7 467.64 1.44 151,866.09 4.1 810.8 3324.28 16.81 657,396.64 2.3 352.9 811.67 5.29 124,538.41 0.9 235.0 211.5 0.81 55,225 1.8 297.8 536.04 3.24 88,684.84 2.9 413.9 1200.31 8.41 171,313.21 5.4 1216.5 6569.1 29.16 1,479,872.25 Σx = 25.7 Σy = 5269.5 Σxy = 16,687.99 Σx2 = 82.81 Σy2 = 3,548,633.25 Study Tip Notice that the correlation coefficient r in Example 4 is rounded to three decimal places. This round-off rule will be used throughout the text. Study Tip The symbol Σx2 means square each value and add the squares. The symbol 1Σx22 means add the values and square the sum.
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