STATISTICS

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STATISTICS

• Educational Research

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OBJECTIVES

• THE Normal Curve
• Skewed Distributions
• Standard Scores z and t
• Test score interpretations

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CHARACTERISTICS OF THE NORMAL CURVE

• symmetrical
• central tendencies
• dispersion/variabilities

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ASSUMPTIONS OF THE NORMAL CURVE

• Traits normally distributed
• Moderate amount--most of us
• Extremely little or lots--few of us
• Hypothetical--large s

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Percentiles

• age of people who fall at or below a given raw
  score
• persons relative position
• 50ile median
• half below half above
• ordinal data--cannot , -, X, divide

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AREAS OF THE NORMAL CURVE

• Ary et al. (1996) text
• Table A.1 pages 546-550
• pages 152-153 in Chapter 5

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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

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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

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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

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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

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• 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 74.52ile.

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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

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Standard Scores

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

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Standard Scores

• Z-SCORES
• M 0 SD 1
• z of SDs from mean
• negative s, decimals

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• 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

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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

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T-scores

• M 50
• SD 10
• preferred over z-scores--no negative s

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

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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

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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

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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

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Conversion to T-scores

• Calculate T-score for Reading
• z 1.55
• T 10 (1.55) 50
• T 15.50 50
• T 65.50

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Conversion to T-scores

• Calculate T-score for Math
• z 0.39
• T 10 (.39) 50
• T 3.90 50
• T 53.90

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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

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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

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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

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• 6.06 did as good as or better in Rdg
• 34.83 did as good as or better in Math

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• THE
• LONG AWAITED
• END

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