SECTION 9.1 Correlation 473 Correlation Coefficient Interpreting correlation using a scatter plot is subjective. A precise measure of the type and strength of a linear correlation between two variables is to calculate the correlation coefficient. A formula for the sample correlation coefficient is given, but it is more convenient to use technology to calculate this value. The correlation coefficient is a measure of the strength and the direction of a linear relationship between two variables. The symbol r represents the sample correlation coefficient. A formula for r is r = nΣxy - 1Σx21Σy2 2 nΣx2 - 1Σx222nΣy2 - 1Σy22 Sample correlation coefficient where n is the number of pairs of data. The population correlation coefficient is represented by r (the lowercase Greek letter rho, pronounced “row”). DEFINITION The range of the correlation coefficient is -1 to 1, inclusive. When x and y have a strong positive linear correlation, r is close to 1. When x and y have a strong negative linear correlation, r is close to -1. When x and y have perfect positive linear correlation or perfect negative linear correlation, r is equal to 1 or -1, respectively. When there is no linear correlation, r is close to 0. It is important to remember that when r is close to 0, it does not mean that there is no relation between x and y, just that there is no linear relation. Several examples are shown below. 10 1 2 3 4 5 6 7 8 Total cost (in dollars) Number of adult movie tickets x 20 30 40 50 60 y 60 7 8 9 10 11 12 13 62 64 66 68 70 72 Shoe size Height (in inches) x y 10 40 60 80 100 120 140 160 20 30 40 50 60 70 Amount spent on milk per year (in dollars) Income per year (in thousands of dollars) x y Perfect positive correlation Strong positive correlation Weak positive correlation r = 1 r = 0.81 r = 0.45 1 2 3 4 5 6 7 8 Exam score Number incorrect x 50 60 70 80 90 100 y 50 60 70 80 90 100 1 2 3 4 5 6 7 8 Test grade Number of absences x y 60 62 64 66 68 70 72 98 102 106 Height (in inches) IQ score x y Perfect negative correlation Strong negative correlation No correlation r = -1 r = -0.92 r = 0.04 To use a correlation coefficient r to make an inference about a population, it is required that (1) the sample paired data 1x, y2 are random and (2) x and y have a bivariate normal distribution (you will learn more about this distribution in Section 9.3). In this text, unless stated otherwise, you can assume that these requirements are met. Study Tip The formal name for r is the Pearson product moment correlation coefficient. It is named after the English statistician Karl Pearson (1857–1936). (See page 35.)
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