Chapter 2 Statistical Tools in Evaluation

1. Chapter 2Statistical Tools in Evaluation

2. Chapter 2 Outline • Types of Scores • Organizing and Graphing Test Scores • Descriptive Statistics • Percentile Rank • Standard Scores • Normal Curve • Determining Relationships between Scores • Regression Analysis • Additional Statistical Tests

3. Types of Scores • Continuous Scores – scores with a potentially infinite number of values. • Discrete Scores – scores limited to a specific number of values.

4. Levels of Measurement • Nominal • Ordinal • Interval • Ratio

5. Scales of Measurement • Nominal • Set of mutually exclusive categories. • Classify or categorize subject. • No meaningful order to categorization.

6. Scales of Measurement • Ordinal • Order to scores so that one can be classified as higher or lower. • No common unit of measurement between numbers. • Numbers cannot be averaged or used in any way except to indicate “better than”.

7. Scales of Measurement • Interval • Have meaningful order and common unit of measurement between scores. • Arbitrary zero point.

8. Scales of Measurement • Ratio • Common unit of measurement and absolute zero point. • A score of zero indicates lack of value.

9. Organizing and Graphing • Simple frequency distribution – listing of a distribution of scores in order. • Easy to construct using a data analysis program (e.g., SPSS).

10. Helps organize and interpret data. Frequency Distribution

11. Graphing • Frequency Polygon • Histogram

12. For Frequency Polygon or Histogram: • Similar scores are grouped together in an interval. • Midpoint of interval is plotted on X-axis. • Frequency is plotted on Y-axis.

13. SPSS Sample Frequency Polygon

14. SPSS Sample Histogram

15. Skewness • An asymmetrical distribution. • Normal Curve - no skewness. • Positive Skew - tail of curve on right, few high scores. • Negative Skew - tail of curve on left, few low scores.

16. Measurement • - process of obtaining test scores. Statistics - methodology for analyzing the scores to enhance interpretation.

17. In this course, we use statistics: • To describe a set of scores. • To standardize scores. • To estimate validity and reliability.

18. Descriptive Statistics • Central Tendency • (how data cluster around the center) • Variability (how data spread around the center)

19. Mode • Most frequently occurring score.

20. Median • 50th percentile • Middle score • Need to order scores in a frequency distribution • Found from cumulative percent column

21. Mean • Mean = X ÷ N

22. Calculate the Mean, Median, and Mode for Three Distributions 1 2 3 100 75 51 50 50 50 50 50 50 0 25 49 Mean: Median: Mode:

23. Calculate the Mean, Median, and Mode for Three Distributions 1 2 3 100 75 51 50 50 50 50 50 50 0 25 49 Mean: 50 Median: 50 Mode: 50

24. Calculate the Mean, Median, and Mode for Three Distributions 1 2 3 100 75 51 50 50 50 50 50 50 0 25 49 Mean: 50 50 Median: 50 50 Mode: 50 50

25. Calculate the Mean, Median, and Mode for Three Distributions 1 2 3 100 75 51 50 50 50 50 50 50 0 25 49 Mean: 50 50 50 Median: 50 50 50 Mode: 50 50 50

26. So these three distributions are all the same, right? No What makes them different? Measure of Variability

27. Range = High score - Low score 1 2 3 100 75 51 50 50 50 50 50 50 0 25 49 Range: 100 50 2

28. Variability • A second type of descriptive statistic. • Describes spread or heterogeneity of scores.

29. Measures of Variability • Range • Standard Deviation • Variance

30. Range • Range = high score - low score. • Unstable because it depends on only two scores.

31. Standard Deviation (s) • Average deviation of each score from the mean. • Minimum value of s = 0. • Larger s, more heterogeneous the group.  = standard deviation of population s = standard deviation of sample

32. Standard Deviation (s) • Definitional Formula: s = (X - X)2 ÷ (n - 1)

33. Calculate the Standard Deviation s = (X - X)2 ÷ (N - 1) X (X - X) (X - X)2 5 0 0 4 -1 1 2 -3 9 9 4 16

34. Standard Deviation X (X - X) (X - X)2 5 0 0 4 -1 1 2 -3 9 9 4 16  X=20 (X-X)=0 (X-X)2=26 X = 5 s =  26 ÷ (4 -1) =  8.67 = 2.94

35. Standard Deviation • Calculational Formula: s = [ X2 - ( X)2 / n] ÷ (n - 1) X X2 5 25 4 16 2 4 981 X=20 X2=126

36. Standard Deviation X X2 s = [126-((20)2/4)]÷(4-1) 5 25 4 16 s = [126 - 100] ÷ 3 2 4 s = 8.667 981 s = 2.94 X=20 X2=126

37. Variance (s2) • Average squared deviation from the mean. • Standard deviation squared. • Not used for description. • Used with higher level statistics like regression analysis or analysis of variance. s2 = (X - X)2 ÷ (n - 1) s2 = [X2 - (X)2 / n] ÷ (n - 1)

38. Percentile Rank • Percentage of subjects that scored below a given score. • Read from cumulative percent column in a simple frequency distribution. • Percentile ranks are ordinal data.

39. Standard Scores • Change variables to a constant mean and standard deviation. • Different units of measurement are converted to the same unit (standardized) and can then be averaged.

40. Z - score • standard score with a mean = 0 and standard deviation = 1. • Z = (X - X) ÷ S

41. T -score • standard score with a mean = 50 and standard deviation = 10. • T = 10(Z) + 50

42. Z-scores • Provide descriptions of relative performance on one or more tests.

43. Example use of Z-scores Student A Subject Raw Score Math 30 English 70 Science 120 On which test did Student A perform best?

44. Don’t know

45. Example use of Z-scores Student A Subject Raw Score Mean Math 30 25 English 70 65 Science 120 140 On which test did Student A perform best?

46. Still Don’t Know

47. Example use of Z-scores Student A Subject Raw Score Mean SD Math 30 25 5 English 70 65 10 Science 120 140 10 On which test did Student A perform best?

48. Now we know

49. Example use of Z-scores Student A Subject Raw Score Mean SD Z-score Math 30 25 5 1.00 English 70 65 10 0.50 Science 120 140 10 -2.00 On which test did Student A perform best? Math The test with the highest standard score.

50. Why use standard scores? • To combine different units of measurement. • To assign different weights to each score.