Summarizing Data

1. Summarizing Data Graphical Methods

2. Histogram Grouped Freq Table Stem-Leaf Diagram Box-whisker Plot

3. Measure of Central Location • Mean • Median

4. Measure of Variability (Dispersion, Spread) • Range • Inter-Quartile Range • Variance, standard deviation • Pseudo-standard deviation

5. Descriptive techniques for Multivariate data In most research situations data is collected on more than one variable (usually many variables)

6. Graphical Techniques • The scatter plot • The two dimensional Histogram

7. The Scatter Plot For two variables X and Y we will have a measurements for each variable on each case: xi, yi xi = the value of X for case i and yi = the value of Y for case i.

8. To Construct a scatter plot we plot the points: (xi, yi) for each case on the X-Y plane. (xi, yi) yi xi

9. Data Set #3 The following table gives data on Verbal IQ, Math IQ, Initial Reading Acheivement Score, and Final Reading Acheivement Score for 23 students who have recently completed a reading improvement program Initial Final Verbal Math Reading Reading Student IQ IQ Acheivement Acheivement 1 86 94 1.1 1.7 2 104 103 1.5 1.7 3 86 92 1.5 1.9 4 105 100 2.0 2.0 5 118 115 1.9 3.5 6 96 102 1.4 2.4 7 90 87 1.5 1.8 8 95 100 1.4 2.0 9 105 96 1.7 1.7 10 84 80 1.6 1.7 11 94 87 1.6 1.7 12 119 116 1.7 3.1 13 82 91 1.2 1.8 14 80 93 1.0 1.7 15 109 124 1.8 2.5 16 111 119 1.4 3.0 17 89 94 1.6 1.8 18 99 117 1.6 2.6 19 94 93 1.4 1.4 20 99 110 1.4 2.0 21 95 97 1.5 1.3 22 102 104 1.7 3.1 23 102 93 1.6 1.9

11. (84,80)

13. Some Scatter Patterns

16. Circular • No relationship between X and Y • Unable to predict Y from X

19. Ellipsoidal • Positive relationship between X and Y • Increases in X correspond to increases in Y (but not always) • Major axis of the ellipse has positive slope

21. Example Verbal IQ, MathIQ

23. Some More Patterns

26. Ellipsoidal (thinner ellipse) • Stronger positive relationship between X and Y • Increases in X correspond to increases in Y (more freqequently) • Major axis of the ellipse has positive slope • Minor axis of the ellipse much smaller

28. Increased strength in the positive relationship between X and Y • Increases in X correspond to increases in Y (almost always) • Minor axis of the ellipse extremely small in relationship to the Major axis of the ellipse.

31. Perfect positive relationship between X and Y • Y perfectly predictable from X • Data falls exactly along a straight line with positive slope

34. Ellipsoidal • Negative relationship between X and Y • Increases in X correspond to decreases in Y (but not always) • Major axis of the ellipse has negative slope slope

36. The strength of the relationship can increase until changes in Y can be perfectly predicted from X

42. Some Non-Linear Patterns

45. In a Linear pattern Y increase with respect to X at a constant rate • In a Non-linear pattern the rate that Y increases with respect to X is variable

46. Growth Patterns

49. Growth patterns frequently follow a sigmoid curve • Growth at the start is slow • It then speeds up • Slows down again as it reaches it limiting size

50. Measures of strength of a relationship (Correlation) • Pearson’s correlation coefficient (r) • Spearman’s rank correlation coefficient (rho, r)