Statistics for Psychology Sixth Edition Chapter 11 Correlation Copyright © 2013 Pearson Education, Inc. All Rights Reserved Correlation • Association between scores on two variables – Ex.: age and coordination skills in children, price and quality Copyright © 2013 Pearson Education, Inc. All Rights Reserved Graphing Correlations: The Scatter Diagram • Steps for making a scatter diagram 1. Draw axes and assign variables to them 2. Determine range of values for each variable and mark on axes 3. Mark a dot for each person’s pair of scores Copyright © 2013 Pearson Education, Inc. All Rights Reserved Table 11-1 Hours Slept Last Night and Happy Mood Example (Fictional Data) Hours Slept Happy Mood 5 2 7 4 8 7 6 2 6 3 10 6 Copyright © 2013 Pearson Education, Inc. All Rights Reserved Figure 11-2 Steps for Making a Scatter Diagram. (a) (1) Draw the Axes and Decide Which Variable Goes on Which Axis—the Predictor Variable (Hours Slept Last Night) On the Horizontal Axis, the Other (Happy Mood) On the Vertical Axis. (b) (2) Determine the Range of Values to Use for Each Variable and Mark Them on the Axes. (c) (3) Mark a Dot for the Pair of Scores for the First Student. (d) (3) Continued: Mark Dots for the Remaining Pairs of Scores Copyright © 2013 Pearson Education, Inc. All Rights Reserved Patterns of Correlation • Linear correlation • Curvilinear correlation • No correlation • Positive correlation • Negative correlation Copyright © 2013 Pearson Education, Inc. All Rights Reserved Linear Correlation • Describes a situation where the pattern of dots falls roughly in a straight line Copyright © 2013 Pearson Education, Inc. All Rights Reserved Curvilinear Correlation • Pattern of dots is curved, not in a straight line Copyright © 2013 Pearson Education, Inc. All Rights Reserved Positive Correlation • Pattern of dots goes up from left to right – High scores on one variable go with high scores on the other variable – Low scores on one variable go with low scores on the other variable – Middle scores go with middle scores Copyright © 2013 Pearson Education, Inc. All Rights Reserved Negative Correlation • Pattern of dots goes down from left to right – High scores on one variable go with low scores on the other variable – Low scores on one variable go with high scores on the other variable – Middle scores go with middle scores Copyright © 2013 Pearson Education, Inc. All Rights Reserved Correlation Coefficients (1 of 2) • Describe the degree and direction of linear correlation • Such a measure would be especially useful if it has the following properties: 1. Numeric on a standard scale for all uses 2. A positive number for positive correlations and a negative number for negative correlations Copyright © 2013 Pearson Education, Inc. All Rights Reserved Correlation Coefficients (2 of 2) • Such a measure would be especially useful if it has the following properties: 3. A zero for no correlation Copyright © 2013 Pearson Education, Inc. All Rights Reserved Degree of Linear Correlation: The Correlation Coefficient (1 of 2) • Figure correlation using Z scores • Cross-product of Z scores – Multiply Z score on one variable by Z score on the other variable • Correlation coefficient – Average of the cross-products of Z scores Copyright © 2013 Pearson Education, Inc. All Rights Reserved Degree of Linear Correlation: The Correlation Coefficient (2 of 2) • General formula for the correlation coefficient: Z å r = X ZY N • Positive perfect correlation: r = +1 • No correlation: r = 0 • Negative perfect correlation: r = -1 Copyright © 2013 Pearson Education, Inc. All Rights Reserved Interpreting a Correlation • A correlation is strong and positive if highs on one variable go with highs on the other, and lows with lows • A correlation is strong and negative if lows go with highs, and highs with lows • There is no correlation if sometimes highs go with highs and sometimes with lows Copyright © 2013 Pearson Education, Inc. All Rights Reserved Correlation and Causality (1 of 2) Copyright © 2013 Pearson Education, Inc. All Rights Reserved Correlation and Causality (2 of 2) • Correlational research design – Correlation as a statistical procedure – Correlation as a research design Copyright © 2013 Pearson Education, Inc. All Rights Reserved Issues in Interpreting Correlation Coefficients • Statistical significance • Proportionate reduction in error – r2 – Used to compare correlations • Restriction in range • Unreliability of measurement • Curvilinearity – Spearman’s rho Copyright © 2013 Pearson Education, Inc. All Rights Reserved Significance of the Correlation Coefficient • t is used to determine the significance of a correlation • coefficient t= r (I - r 2 ) / (N - 2) • with df = N - 2 Copyright © 2013 Pearson Education, Inc. All Rights Reserved Table 11-6 Approximate Power of Studies Using the Correlation Coefficient (r) for Testing Hypotheses at the .05 Level of Significance Blank Blank Effect Size Small (r = .10) Effect Size Medium (r = .30) Effect Size Large (r = .50) Two-tailed Total N: 10 .06 .13 .33 Two-tailed Total N: 20 .07 .25 .64 Two-tailed Total N: 30 .08 .37 .83 Two-tailed Total N: 40 .09 .48 .92 Two-tailed Total N: 50 .11 .57 .97 Two-tailed Total N: 100 .17 .86 * One-tailed Total N: 10 .08 .22 .46 One-tailed Total N: 20 .11 .37 .75 One-tailed Total N: 30 .13 .50 .90 One-tailed Total N: 40 .15 .60 .96 One-tailed Total N: 50 .17 .69 .98 One-tailed Total N: 100 .26 .92 * * Power is nearly 1. Copyright © 2013 Pearson Education, Inc. All Rights Reserved Table 11-7 Approximate Number of Participants Needed for 80% Power for a Study Using the Correlation Coefficient (r) for Testing a Hypothesis at the .05 Significance Level Blank Effect Size Small (r = .10) Effect Size Medium (r = .30) Effect Size Large (r = .50) Two-tailed 783 85 28 One-tailed 617 68 22 Copyright © 2013 Pearson Education, Inc. All Rights Reserved Correlation in Research Articles • Scatter diagrams occasionally shown • Correlation matrix Copyright © 2013 Pearson Education, Inc. All Rights Reserved Table 11-9 Pearson Correlations Between Temperature and Recorded Behaviors Variable Temperature Panting Stretching Wing venting Yawning Blank Blank Blank Blank Blank Blank Blank Blank Blank Blank Blank Blank Blank Blank Temperature Panting .740** Stretching .069 Wing venting .214* .166 .182* Yawning .333** .365** .175 -.059 negative .059 .312* Blank Note: Yawning and stretching were measured as the number of yawns and stretches recorded for all four birds in a group during each 2-min interval; panting and wing venting were measured as the number of birds observed engaging in these behaviors during each trial. * p < .05. ** p < .01. Source: Gallup, A. C., Miller, M. L., & Clark, A. B. (2010). The direction and range of ambient temperature change influences yawning in budgerigars (Melopsittacus undulatus). Journal of Comparative Psychology, 124, 133-138. Published by the American Psychological Association. Reprinted with permission. Copyright © 2013 Pearson Education, Inc. All Rights Reserved Copyright This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. Copyright © 2013 Pearson Education, Inc. All Rights Reserved
