1. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 1 What is a Perfect Positive Linear Correlation? –It occurs when everyone has the same exact score on two different variables. –It also occurs when everyone’s score on one variable differs by a constant from their score on the other variable (e.g., everyone’s Final exam score is exactly 10 points higher than their midterm score, or exactly twice as much). –It will occur if everyone occupies the same position in a normal distribution for one variable that they occupy for the other variable (i.e., if everyone has the same z score on both variables). –Perfect negative correlation occurs when everyone has the same z score on both variables, but with opposite signs (e.g., if z = +1.5 for one variable, that person’s z will be –1.5 for the other variable. Chapter 9: Linear Correlation

2. Chapter 9For Explaining Psychological Statistics, 4th ed. by B. Cohen. 2 The Pearson Correlation Coefficient (r) –Pearson’s r ranges from -1.0 to +1.0, where +1.00 = perfect positive correlation –1.00 = perfect negative correlation 0 = a total lack of correlation –A formula that illustrates the relationship between z-scores and Pearson’s r is the following: –The magnitude of the number represents the amount of correlation, while its sign represents the direction of the correlation.

3. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 3 Graphing a Correlation: The Scatterplot

4. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 4 The Graph of a Correlation That Is Less Than Perfect

5. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 5 Calculating Pearson’s r Computing formula in terms of population standard deviations: –The numerator is the biased estimate of the covariance. Dividing the covariance by the product of the biased standard deviations ensures that r will never be greater than +/– 1.0. Computing formula in terms of the unbiased covariance estimate divided by the unbiased standard deviations: This formula always yields the same value for r as the preceding formula based on the biased covariance.

6. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 6 Try this example… Is education about other ethnicities correlated with tolerant attitudes towards others?

7. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 7 Testing Pearson’s r for Significance –H 0 : ρ = 0; df = N – 2 –Using the t distribution: –t.05 (8) = 2.306 < 3.07, so we can reject the null hypothesis for the correlation example from the previous slide. –Using the table of critical values: df = N – 2 (where N is the number of pairs of scores). Critical r for df = 8 is.632, for a.05, two- tailed test..632 <.735, so H 0 can be rejected (consistent with t test). –Effects of N on Critical r For small N, you need a fairly large r to find significance. For very large N, even tiny sample rs can attain statistical significance. As N increases, r is a better estimate of ρ.

8. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 8 Assumptions of the Test of Significance for Pearson’s r –The sample has been obtained by independent random sampling. –Both variables have been measured on interval or ratio scales. –Both variables exhibit normal distributions in the population. –The two variables jointly follow a bivariate normal distribution (see next slide). –If one or both variables has been measured on an ordinal scale, or the distribution assumptions have been severely violated, consider calculating the Spearman rank-order correlation coefficient (r S ) as an alternative (a special table must be used for its critical values when N is small).

9. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 9

10. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 10 Limitations and Cautions Pearson’s r measures only the degree of linear correlation (r can be small even though there is a very close curvilinear relationship between the two variables). Pearson’s r can underestimate the population correlation (ρ) if there are: –Restricted (truncated) ranges on one or both variables. –Bivariate outliers. Correlation does not imply causation!

11. Chapter 9 For Explaining Psychological Statistics, 4th ed. by B. Cohen 11 Uses of Pearson’s r –Reliability Test-retest reliability Split-half reliability Inter-rater reliability –Criterion validity (e.g., of a self- report measure). –To measure the degree of linear association between two variables that are not obviously related, but are predicted by some theory or past research to have an important connection. –To evaluate the results of experimental studies in which both the DV, and the levels of the manipulated variable (IV), have been measured on interval or ratio scales.