1. Statistics for the Social Sciences Psychology 340 Fall 2006 Relationships between variables

2. Statistics for the Social Sciences Correlation Write down what (you think) a correlation is. Write down an example of a correlation Association between scores on two variables –Age and coordination skills in children, as kids get older their motor coordination tends to improve –Price and quality, generally the more expensive something is the higher in quality it is

3. Statistics for the Social Sciences Correlation and Causality Correlational research design –Correlation as a kind of research design (observational designs) –Correlation as a statistical procedure

4. Statistics for the Social Sciences Another thing to consider about correlation Correlations describe relationships between two variables, but DO NOT explain why the variables are related Suppose that Dr. Steward finds that rates of spilled coffee and severity of plane turbulents are strongly positively correlated. One might argue that turbulents cause coffee spills One might argue that spilling coffee causes turbulents

5. Statistics for the Social Sciences Another thing to consider about correlation Correlations describe relationships between two variables, but DO NOT explain why the variables are related Suppose that Dr. Cranium finds a positive correlation between head size and digit span (roughly the number of digits you can remember). One might argue that bigger your head, the larger your digit span 1 21 24 15 37 One might argue that head size and digit span both increase with age (but head size and digit span aren’t directly related)

6. Statistics for the Social Sciences Another thing to consider about correlation Correlations describe relationships between two variables, but DO NOT explain why the variables are related For many years instructors have noted that the reported fatality rate of grandparents increases during midterm and final exam periods. One might argue that college exams cause grandparent death

7. Statistics for the Social Sciences Relationships between variables Properties of a correlation –Form (linear or non-linear) –Direction (positive or negative) –Strength (none, weak, strong, perfect) To examine this relationship you should : –Make a scatterplot - a picture of the relationship –Compute the Correlation Coefficient - a numerical description of the relationship

8. Statistics for the Social Sciences Graphing Correlations Steps for making a scatterplot (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

9. Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 XY

10. Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 XY

11. Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 C 5 6 XY

12. Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 C 5 6 D 3 4 XY

13. Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 XY

14. Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Imagine a line through the data points Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 XY Useful for “seeing” the relationship –Form, Direction, and Strength

15. Statistics for the Social Sciences Form Non-linearLinear

16. Statistics for the Social Sciences NegativePositive Direction X & Y vary in the same direction As X goes up, Y goes up Positive Pearson’s r X & Y vary in opposite directions As X goes up, Y goes down Negative Pearson’s r Y X Y X

17. Statistics for the Social Sciences Strength The strength of the relationship –Spread around the line (note the axis scales) –Correlation coefficient will range from -1 to +1 Zero means “no relationship” The farther the r is from zero, the stronger the relationship

18. Statistics for the Social Sciences Strength r = 1.0 “perfect positive corr.” r 2 = 100% r = -1.0 “perfect negative corr.” r 2 = 100% r = 0.0 “no relationship” r 2 = 0.0 0.0+1.0 The farther from zero, the stronger the relationship

19. Statistics for the Social Sciences The Correlation Coefficient Formulas for the correlation coefficient: Used this one in PSY138Common alternative

20. Statistics for the Social Sciences The Correlation Coefficient Formulas for the correlation coefficient: Used this one in PSY138Common alternative

21. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 1: SP (Sum of the Products) mean 3.64.0 6 1 2 5 6 3 4 3 2 X Y

22. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 1: SP (Sum of the Products) mean 3.64.0 2.4 0.0 6 1 2 5 6 3 4 3 2 X Y = 6 - 3.6 -2.6 = 1 - 3.6 1.4 = 5 - 3.6 -0.6 = 3 - 3.6 -0.6= 3 - 3.6 Quick check

23. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 1: SP (Sum of the Products) mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 6 1 2 5 6 3 4 3 2 X Y 2.0= 6 - 4.0 -2.0 = 2 - 4.0 2.0= 6 - 4.0 0.0 = 4 - 4.0 -2.0 = 2 - 4.0 Quick check

24. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 1: SP (Sum of the Products) mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.014.0SP 6 1 2 5 6 3 4 3 2 X Y 4.8 * = 5.2 * = 2.8 * = 0.0 * = 1.2 * =

25. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 2: SS X & SS Y

26. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 2: SS X & SS Y mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.014.0 6 1 2 5 6 3 4 3 2 X Y 4.8 5.2 2.8 0.0 1.2 5.76 15.20 SS X 2 =6.76 2 =1.96 2 =0.36 2 = 2 =

27. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 2: SS X & SS Y mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.014.0 6 1 2 5 6 3 4 3 2 X Y 4.8 5.2 2.8 0.0 1.2 5.76 6.76 1.96 0.36 15.20 2 =4.0 2 = 2 = 2 =0.0 2 = 4.0 16.0 SS Y

28. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 3: compute r

29. Statistics for the Social Sciences Computing Pearson’s r (using SP) Step 3: compute r mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.014.0 6 1 2 5 6 3 4 3 2 X Y 4.8 5.2 2.8 0.0 1.2 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 SS Y SS X SP

30. Statistics for the Social Sciences Computing Pearson’s r Step 3: compute r 14.015.2016.0 SS Y SS X SP

31. Statistics for the Social Sciences Computing Pearson’s r Step 3: compute r 15.2016.0 SS Y SS X

32. Statistics for the Social Sciences Computing Pearson’s r Step 3: compute r 15.20 SS X

33. Statistics for the Social Sciences Computing Pearson’s r Step 3: compute r

34. Statistics for the Social Sciences Computing Pearson’s r Step 3: compute r Y X 1 2 3 4 5 6 123 456 Appears linear Positive relationship Fairly strong relationship.89 is far from 0, near +1

35. Statistics for the Social Sciences The Correlation Coefficient Formulas for the correlation coefficient: Used this one in PSY138Common alternative

36. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 1: compute standard deviation for X and Y (note: keep track of sample or population) 6 1 2 5 6 3 4 3 2 X Y For this example we will assume the data is from a population

37. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 1: compute standard deviation for X and Y (note: keep track of sample or population) Mean 3.6 2.4 -2.6 1.4 -0.6 0.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 SS X Std dev 1.74 For this example we will assume the data is from a population

38. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 1: compute standard deviation for X and Y (note: keep track of sample or population) Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 0.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 SS Y Std dev 1.741.79 For this example we will assume the data is from a population

39. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 2: compute z-scores Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 1.38

40. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 2: compute z-scores Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 1.38 -1.49 0.8 - 0.34 0.0 Quick check

41. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 2: compute z-scores Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 1.11.38 -1.49 0.8 - 0.34

42. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 2: compute z-scores Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 1.1 -1.1 0.0 -1.1 1.1 0.0 1.38 -1.49 0.8 - 0.34 Quick check

43. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 3: compute r Mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 0.0 1.1 -1.1 0.0 -1.1 1.1 0.0 1.521.38 -1.49 0.8 - 0.34 * =

44. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 3: compute r Mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 0.0 1.1 -1.1 0.0 -1.1 1.1 0.0 1.52 1.64 0.88 0.0 0.37 1.38 -1.49 0.8 - 0.34 4.41

45. Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 3: compute r Y X 1 2 3 4 5 6 123 456 Appears linear Positive relationship Fairly strong relationship.89 is far from 0, near +1

46. Statistics for the Social Sciences A few more things to consider about correlation Correlations are greatly affected by the range of scores in the data –Consider height and age relationship Extreme scores can have dramatic effects on correlations –A single extreme score can radically change r When considering "how good" a relationship is, we really should consider r 2 (coefficient of determination), not just r.

47. Statistics for the Social Sciences Correlation in Research Articles Correlation matrix –A display of the correlations between more than two variables Acculturation Why have a “-”? Why only half the table filled with numbers?

48. Statistics for the Social Sciences Next time Predicting a variable based on other variables