STATISTICSSTATISTICS

Educational ResearchEducational Research

OBJECTIVESOBJECTIVES

THE Normal CurveTHE Normal CurveSkewed DistributionsSkewed DistributionsStandard Scores: z and tStandard Scores: z and tTest score interpretationsTest score interpretations

CHARACTERISTICS OF THE NORMAL CURVE CHARACTERISTICS OF THE NORMAL CURVE

symmetricalsymmetrical central tendenciescentral tendencies dispersion/variabilitiesdispersion/variabilities

ASSUMPTIONS OF THE NORMAL CURVEASSUMPTIONS OF THE NORMAL CURVE

Traits normally distributedTraits normally distributedModerate amount--most of usModerate amount--most of usExtremely little or lots--few of usExtremely little or lots--few of usHypothetical--large #sHypothetical--large #s

Percentiles Percentiles

%age of people who fall at or below %age of people who fall at or below a given raw scorea given raw score

person’s relative positionperson’s relative position50%ile = median50%ile = medianhalf below & half abovehalf below & half aboveordinal data--cannot +, -, X, divideordinal data--cannot +, -, X, divide

AREAS OF THE NORMAL CURVEAREAS OF THE NORMAL CURVE

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

Areas of the Normal CurveAreas of the Normal Curve

+ / - 1 SD = 68.26%+ / - 1 SD = 68.26% 34.13 + 34.1334.13 + 34.13+ / - 2 SD = 95.44%+ / - 2 SD = 95.44% 47.72 + 47.7247.72 + 47.72+ / - 3 SD = 99.74%+ / - 3 SD = 99.74% 49.87 + 49.8749.87 + 49.87

Areas of the Normal CurveAreas of the Normal Curve

- 3 SD = 0.13 %ile- 3 SD = 0.13 %ile column 3 on page 550 Arycolumn 3 on page 550 Ary - 2 SD = 2.28 %ile- 2 SD = 2.28 %ile column 3 on page 548column 3 on page 548 - 1 SD = 15.87 %ile- 1 SD = 15.87 %ile column 3 on page 547column 3 on page 547

Areas of the Normal CurveAreas of the Normal Curve

+1 SD = 84.13 %ile+1 SD = 84.13 %ile column 2 on page 547column 2 on page 547+2 SD = 97.72 %ile+2 SD = 97.72 %ile column 2 on page 548column 2 on page 548+3 SD = 99.87 %ile+3 SD = 99.87 %ile column 2 on page 550column 2 on page 550

Z-scoresZ-scores

When z-score is positive, use column 2 to When z-score is positive, use column 2 to find percentile score Add 50 %find percentile score Add 50 %

use column 2, add 50use column 2, add 50 When z-score is negative, use column 3 to When z-score is negative, use column 3 to

find percentile scorefind percentile score negative z, use 3negative z, use 3

A percentile score of 74.52 means that a A percentile score of 74.52 means that a person did as good as or better than person did as good as or better than 74.52 percent of people who took a test.74.52 percent of people who took a test.

It also means that 25.48% of people who It also means that 25.48% of people who took the test did as good as or better than took the test did as good as or better than the person who scored at the 74.52%ilethe person who scored at the 74.52%ile ..

Skewed DistributionsSkewed Distributions

mean pulled toward tailmean pulled toward tailwhy--outlierswhy--outliers- skewed = tail to left- skewed = tail to left

lots of lots of highhigh scores scores

+ skewed = tail to right + skewed = tail to right lots of lots of lowlow scores scores

pages 142-144pages 142-144

Standard ScoresStandard Scores

based on normal curvebased on normal curve fixed ways of reporting informationfixed ways of reporting information compare scores from different compare scores from different

tests--Rdg, Mathtests--Rdg, Math compare scores from different compare scores from different

peoplepeople

Standard ScoresStandard Scores

Z-SCORESZ-SCORESMM = 0 = 0 SDSD = 1 = 1 z = # of SDs from meanz = # of SDs from meannegative #s, decimalsnegative #s, decimals

To Convert to a Z-scoreTo Convert to a Z-scoreZ = Z = X - M X - M

SDSDX = raw scoreX = raw scoreMM = mean of raw score distribution = mean of raw score distributionSDSD = SD of raw score distribution = SD of raw score distribution

Example of History testExample of History test

Sam = 55 Sam = 55 Sue = 60 Sue = 60 MM = 45 = 45 SDSD = 5 = 5Sam z = Sam z = 55 - 45 55 - 45

55 z = +2z = +2Sue z = Sue z = 60 - 45 60 - 45

55 z = +3z = +3

T-scoresT-scores

MM = 50 = 50SDSD = 10 = 10preferred over z-scores--no preferred over z-scores--no

negative #snegative #s

To Convert to a T-scoreTo Convert to a T-scoreFirst, convert to a z-scoreFirst, convert to a z-scoreZ = Z = X - M X - M

SDSDThen, T = 10 (z) + 50 Then, T = 10 (z) + 50

Example of History testExample of History test

History test: History test: MM = 45 = 45 SDSD = 5 = 5Sam = 55 Sam = 55 Sue = 60Sue = 60Sam z = +2Sam z = +2 Sue z = +3 Sue z = +3Sam t = 10 (+2) + 50 = 70Sam t = 10 (+2) + 50 = 70Sue t = 10 (+3) + 50 = 80Sue t = 10 (+3) + 50 = 80

STUDY GUIDESTUDY GUIDE

Calculate z-score for ReadingCalculate z-score for Reading MM = 75 = 75 SDSD = 12.91 = 12.91 raw score = 95raw score = 95 z = z = 95 - 75 95 - 75

12.9112.91 z = +1.549 = +1.55z = +1.549 = +1.55

STUDY GUIDESTUDY GUIDE

Calculate z-score for MathCalculate z-score for MathMM = 75 = 75 SDSD = 2.58 = 2.58 raw score = 76raw score = 76 z = z = 76 - 75 76 - 75

2.582.58 z = +0.3876 = +0.39z = +0.3876 = +0.39

Conversion to T-scoresConversion to T-scores

Calculate T-score for ReadingCalculate T-score for Reading z = +1.55z = +1.55T = 10 (+1.55) + 50T = 10 (+1.55) + 50T = 15.50 + 50 T = 15.50 + 50 T = 65.50T = 65.50

Conversion to T-scoresConversion to T-scores

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

Conversion to PercentilesConversion to Percentiles

Reading z = +1.55Reading z = +1.55page 548 in Ary et al. (1996)page 548 in Ary et al. (1996)%ile = .4394 + .5000 = 93.94%ile = .4394 + .5000 = 93.94good as/better than 93.94% of peoplegood as/better than 93.94% of people6.06% of people did better than s/he 6.06% of people did better than s/he

diddid

Conversion to PercentilesConversion to Percentiles

Math z = +0.39Math z = +0.39page 546 in Ary et al. (1996)page 546 in Ary et al. (1996)%ile = .1517 + .5000 = 65.17%ile = .1517 + .5000 = 65.17good as/better than 65.17% of peoplegood as/better than 65.17% of people34.83% did as good as/better than34.83% did as good as/better than

COMPARISONS OF %ILESCOMPARISONS OF %ILES

%ile in Reading = 93.94%ile in Reading = 93.94%ile in Math = 65.17%ile in Math = 65.17Reading performance in top 6/7%Reading performance in top 6/7%Math performance averageMath performance averageSubstantially better Reading Substantially better Reading

than Maththan Math

6.06 % did as good as or better in 6.06 % did as good as or better in RdgRdg

34.83% did as good as or better in 34.83% did as good as or better in MathMath

THETHELONG AWAITEDLONG AWAITED

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