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STATISTICS

STATISTICS. Educational Research. OBJECTIVES. THE Normal Curve Skewed Distributions Standard Scores: z and t Test score interpretations. CHARACTERISTICS OF THE NORMAL CURVE. symmetrical central tendencies dispersion/variabilities. ASSUMPTIONS OF THE NORMAL CURVE.

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STATISTICS

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  1. STATISTICS Educational Research

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

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

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

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

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

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

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

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

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

  11. 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. • 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 • mean pulled toward tail • why--outliers • - skewed = tail to left • lots of high scores • + skewed = tail to right • lots of low scores • pages 142-144

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  28. THE • LONG AWAITED • END

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