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Correlations

This article provides an overview of correlations in inferential statistics, including calculating Pearson's correlation coefficient, interpreting correlation coefficients, determining statistical significance, and calculating Spearman's correlation coefficient. Examples and explanations are provided throughout.

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Correlations

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  1. Correlations Inferential Statistics

  2. Overview • Correlation coefficients • Scatterplots • Calculating Pearson’s r • Interpreting correlation coefficients • Calculating & interpreting coefficient of determination • Determining statistical significance • Calculating Spearman’s correlation coefficient

  3. Correlation • Reflects the degree of relation between variables • Calculation of correlation coefficient • Direction • + (positive) or – (negative) • Strength (i.e., magnitude) • Further away from zero, the stronger the relation • Form of the relationship

  4. Check yourself • Indicate whether the following statements suggest a positive or negative relationship: • High school students with lower IQs have lower GPAs • More densely populated areas have higher crime rates • Heavier automobiles yield poorer gas mileage • More anxious people willingly spend more time performing a simple repetitive task

  5. Scatterplots

  6. Correlation & Scatterplots r= .91 • Benefits of scatterplot • Form of relation • Any possible outliers? • Rough guess of r

  7. Correlation & Scatterplots

  8. Correlation & Scatterplots # of times arrested

  9. Pearson’s r • Formula • SP = Sum of products (of deviations) • SSx = Sum of Squares of X • SSy = Sum of Squares of Y

  10. Pearson’s r • Calculating SP • Definitional formula • Computational formula • Find X & Y deviations for each individual • Find product of deviations for each individual • Sum the products

  11. Example #1Calculating SP – Definitional Formula Step 2: Multiply the deviations from the mean Step 3: Sum the products SX = 12 SY = 16 MX = SX/n = 12/4 = 3 MY = SY/n = 16/4 = 4 Step 1: Find deviations for X and Y separately

  12. Example #1Calculating SP – Computational Formula SX = 12 SY = 16 MX = SX/n = 12/4 = 3 MY = SY/n = 16/4 = 4

  13. Calculating Pearson’s r • Calculate SP • Calculate SS for X • Calculate SS for Y • Plug numbers into formula

  14. Calculating Pearson’s r • Calculate SP • Calculate SS for X • Calculate SS for Y • Plug numbers into formula

  15. Example #1 - AnswersCalculating Pearson’s r SX = 12 SY = 16 MX = SX/n = 12/4 = 3 MY = SY/n = 16/4 = 4

  16. Pearson’s r • r = covariability of X and Y variability of X and Y separately

  17. Using Pearson’s r • Prediction • Validity • Reliability

  18. Verbal Descriptions • 1) r = -.84 between total mileage & auto resale value • 2) r = -.35 between the number of days absent from school & performance on a math test • 3) r = -.05 between height & IQ • 4) r = .03 between anxiety level & college GPA • 5) r = .56 between age of schoolchildren & reading comprehension level

  19. Interpreting correlations • Describe a relationship between 2 vars • Correlation does not equal causation • Directionality Problem • Third-variable Problem • Restricted range • Obscures relationship

  20. Interpreting correlations • Outliers • Can have BIG impact on correlation coefficient

  21. Interpreting correlations • Strength & Prediction • Coefficient of determination r2 • Proportion of variability in one variable that can be determined from the relationship w/ the other variable • r = .60, then r2 = .36 or 36% • 36% of the total variability in X is consistently associated with variability in Y • “predicted” and “accounted for” variability

  22. Mini-Review • Correlations • Calculation of Pearson’s r • Sum of product deviations • Using Pearson’s r • Verbal descriptions • Interpretation of Pearson’s r

  23. Example #2Practice – Calculate Pearson’s r • Calculate SP • Calculate SS for X • Calculate SS for Y • Plug numbers into formula

  24. Example #2 SP = S(X-MX)(Y-MY) SSY SX = 10 SY = 35 MX = SX/n = 10/5 = 2 MY = SY/n = 35/5 = 7 SSX

  25. Hypothesis Testing • Making inferences based on sample information • Is it a statistically significant relationship? Or simply chance?

  26. = 3 With n=5, there can be only 4 df Conceptually - Degrees of freedom • Knowing M (the mean) restricts variability in sample • 1 score will be dependent on others • n = 5, SX = 20 • If we know first 3 scores • If we know first 4 scores Score X1 = 6 X2 = 4 X3 = 2 X4 X5 Σx= 20 = 5

  27. Correlations – Degrees of freedom • There are no degrees of freedom when our sample size is 2. When there are only two points on a scatterplot, they will fit perfectly on a straight line. • Thus, for correlations df= n – 2

  28. Using table to determine significance • Find degrees of freedom • Correlations: df= n – 2 • Use level of significance (e.g., a = .05) for two-tailed test to find column in Table • Determine critical value • Value calculated r must equal or exceed to be significant • Compare calculated r w/ critical value • If calculated r less than critical value = not significant • APA • The correlation between hours watching television and amount of aggression is not significant, r (3) = -.80, p > .05. Think about sample size

  29. Spearman correlation • Used when: • Ordinal data • If 1 variable is on ratio scale, then change scores for that variable into ranks Difference between pair of ranks

  30. Example #3: Spearman correlation

  31. Example #3: Spearman - Answers

  32. Example #4: Spearman Correlation • Two movie raters each watched the same six movies. Is there are relationship between the raters’ rankings?

  33. Example #4: Spearman - answers

  34. Pearson r (from SPSS) Spearman rs (from SPSS)

  35. Example #5: Pearson’s r Participant Motivation (X)Depression (Y) 1 3 8 2 6 4 3 9 2 4 2 2 • Sketch a scatterplot. • Calculate the correlation coefficient. • Determine if it is statistically significant at the .05 level for a 2-tailed test. • Write an APA format conclusion.

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