Step 4: Determine the critical values, or cutoffs . Now we can look up the critical values in the r table in Appendix B. Like the z table and the t table, the r table includes only positive values. For a two-tailed test, we take the negative and positive versions of the critical test statistic indicated in the table. So the critical values for an r distribution with 8 degrees of freedom for a two-tailed test with a p level of 0.05 are -0.632 and 0.632. Step 5: Calculate the test statistic. We already calculated the test statistic, r , in the preceding section. It is – 0.85. Step 6: Make a decision . The test statistic, r = - 0.85, is larger in magnitude than the critical value of – 0.632. We can reject the null hypothesis and conclude that number of absences and exam grade seem to be negatively correlated. Reviewing the Concepts : The Pearson correlation coefficient allows us to quantify the relations that we observe. Before we calculate a Pearson correlation coefficient, we must always construct a scatterplot to be sure the two variables are linearly related. The Pearson correlation coefficient is calculated in three basic ways: 1) We calculate the deviation of each score from its mean, multiply the two deviations for each person, and sum the products of the deviations. 2) We calculate a sum of squares for each variable, multiply the sums of squares, and take the square root. 3) We divide the sum from step 1 by the square root in step 2. We use the six steps of hypothesis testing to determine whether the correlation coefficient is statistically significantly different from 0 on an r distribution. (Page 355)
Applying Correlation in Psychometrics Here’s an in-demand career available to students of the behavioral sciences: Psychometrics is the branch of statistics used in the development of tests and measures . Not surprisingly, the statisticians and psychologists who develop tests and measures are called psychometricians . Psychometricians use the statistical procedures referred to in this textbook, particularly those for which correlation forms the mathematical backbone. Psychometricians make sure that elections are fair, test for cultural biases in standardized tests, identify high-achieving employees, and make a wide range of social contributions – and we don’t have nearly enough of them. The New York Times reported a “critical shortage” of such experts and intense competition for the few who are available – offered U.S. salaries as high as $200,000 a year! Psychometricians use correlation to examine two important aspects of the development of measures – reliability and validity . Reliability In Chapter 1, we defined a reliable measure as one that is consistent. For example, if we measure shyness, then a reliable measure leads to nearly the same score every time a person takes the shyness test.