1. STATISTICS Educational Research

2. OBJECTIVES ä THE Normal Curve ä Skewed Distributions ä Standard Scores: z and t ä Test score interpretations

3. CHARACTERISTICS OF THE NORMAL CURVE  symmetrical ä central tendencies ä dispersion/variabilities

4. ASSUMPTIONS OF THE NORMAL CURVE ä Traits normally distributed ä Moderate amount--most of us ä Extremely little or lots--few of us ä Hypothetical--large #s

5. Percentiles ä %age of people who fall at or below a given raw score ä person’s relative position ä 50%ile = median ä half below & half above ä ordinal data--cannot +, -, X, divide

6. AREAS OF THE NORMAL CURVE ä Ary et al. (1996) text ä Table A.1 pages 546-550 ä pages 152-153 in Chapter 5

7. Areas of the Normal Curve ä + / - 1 SD = 68.26% ä 34.13 + 34.13 ä + / - 2 SD = 95.44% ä 47.72 + 47.72 ä + / - 3 SD = 99.74% ä 49.87 + 49.87

8. Areas of the Normal Curve ä - 3 SD = 0.13 %ile ä column 3 on page 550 Ary ä - 2 SD = 2.28 %ile ä column 3 on page 548 ä - 1 SD = 15.87 %ile ä column 3 on page 547

9. Areas of the Normal Curve ä +1 SD = 84.13 %ile ä column 2 on page 547 ä +2 SD = 97.72 %ile ä column 2 on page 548 ä +3 SD = 99.87 %ile ä column 2 on page 550

10. Z-scores ä When z-score is positive, use column 2 to find percentile score Add 50 % ä use column 2, add 50 ä When z-score is negative, use column 3 to find percentile score ä negative z, use 3

11. ä A percentile score of 74.52 means that a person did as good as or better than 74.52 percent of people who took a test. ä It also means that 25.48% of people who took the test did as good as or better than the person who scored at the 74.52%ile.

12. Skewed Distributions ä mean pulled toward tail ä why--outliers ä - skewed = tail to left ä lots of high scores ä + skewed = tail to right  lots of low scores ä pages 142-144

13. Standard Scores ä based on normal curve ä fixed ways of reporting information ä compare scores from different tests--Rdg, Math ä compare scores from different people

14. Standard Scores ä Z-SCORES ä M = 0 SD = 1 ä z = # of SDs from mean ä negative #s, decimals

15. ä To Convert to a Z-score ä Z = X - M SD SD ä X = raw score ä M = mean of raw score distribution ä SD = SD of raw score distribution

16. Example of History test ä Sam = 55 Sue = 60  M = 45SD = 5 ä Sam z = 55 - 45 5z = +2 5z = +2 ä Sue z = 60 - 45 5z = +3 5z = +3

17. T-scores ä M = 50 ä SD = 10 ä preferred over z-scores--no negative #s

18. ä To Convert to a T-score ä First, convert to a z-score ä Z = X - M SD SD ä Then, T = 10 (z) + 50

19. Example of History test ä History test: M = 45 SD = 5 ä Sam = 55 Sue = 60 ä Sam z = +2 Sue z = +3 ä Sam t = 10 (+2) + 50 = 70 ä Sue t = 10 (+3) + 50 = 80

20. STUDY GUIDE ä Calculate z-score for Reading ä M = 75SD = 12.91 ä raw score = 95 ä z = 95 - 75 12.91 12.91 ä z = +1.549 = +1.55

21. STUDY GUIDE ä Calculate z-score for Math ä M = 75SD = 2.58 ä raw score = 76 ä z = 76 - 75 2.58 2.58 ä z = +0.3876 = +0.39

22. Conversion to T-scores Calculate T-score for Reading Calculate T-score for Reading ä z = +1.55 ä T = 10 (+1.55) + 50 ä T = 15.50 + 50 ä T = 65.50

23. Conversion to T-scores Calculate T-score for Math Calculate T-score for Math ä z = +0.39 ä T = 10 (+.39) + 50 ä T = 3.90 + 50 ä T = 53.90

24. Conversion to Percentiles Reading z = +1.55 Reading z = +1.55 ä page 548 in Ary et al. (1996) ä %ile =.4394 +.5000 = 93.94 ä good as/better than 93.94% of people ä 6.06% of people did better than s/he did

25. Conversion to Percentiles ä Math z = +0.39 ä page 546 in Ary et al. (1996) ä %ile =.1517 +.5000 = 65.17 ä good as/better than 65.17% of people ä 34.83% did as good as/better than

26. COMPARISONS OF %ILES ä %ile in Reading = 93.94 ä %ile in Math = 65.17 ä Reading performance in top 6/7% ä Math performance average ä Substantially better Reading than Math

27. ä 6.06 % did as good as or better in Rdg ä 34.83% did as good as or better in Math

28. ä THE ä LONG AWAITED ä END