Scores & Norms

1. Scores & Norms Derived Scores, scales, variability, correlation, & percentiles

2. Variability (Dispersion) • Measures of Central Tendency • Mean, Median, & Mode • Variance and Standard Deviation • Descriptive Statistics

3. Relationship of Derived Scores Normal Curve z -2 -1 0 1 2 T 30 40 50 60 70 IQ 70 85 100 115 130 Percentiles 1 5 10 20 30 40 50 60 70 80 90 95 99

4. Scales • Nominal • Ordinal • Interval • Ratio

5. Nominal Categorical Example: LD, EB/D, MMR Ordinal Sequential: positional from 1st to last or vice versa Example: Winners and place finishers in a race No assumption about relative distance Nominal & Ordinal These scales are difficult to manipulate mathematically

6. Interval • Equal units of measure • Ranking and relative distance matter • No absolute zero • Therefore cannot multiply and divide • Despite problems, useful in many educational measures

7. Ratio • Equal units of measure with an absolute zero • Can be multiplied and divided • Useful in measuring physical properties

8. Norms and Standardization • Two purposes for standardizaed assessment • Determine individual performance to a group • Norm-referenced testing • Determine group performance compared to a curriculum goal • Criterion-referenced testing

9. Norm group factors • Age, gender, grade • Sampling • Representation • Size • Recency

10. Criterion-Referenced Testing • Used to determine if specific skills/content have been mastered • Can also be standardized

11. Factors in C-RTs • Represent a curriculum • may or may not be what was taught • Represent a standard of skill • may or may not represent student’s present skill level

12. Derived Scores of NR Testing • Developmental Scores • Scores of Relative Standing

13. Developmental Scores • Grade and Age equivalents • Defined as the average performance of the norm group at the grade or age level.

14. Difficulties with Developmental Scores • Based on group average performance • Extrapolations from the group D scores do not really exist • D scores are ordinal with curvilinear progression

15. Additional Problems with D scores • Highly correlated: not independent measures • Exhibit non-homogenous variance Violate statistical assumptions • normality and independence

16. Decision Rule for Developmental Scores • Do not use these scores for eligibility decisions (APA, CEC, and virtually every major educational/psychological/assessment organization)

17. Scores of Relative Standing • Purpose: to derive a comparable unit of measure across different tests. • Include standard scores and percentile rankings.

18. Derived Scores: Measures of Relative Position • z-scores • T-scores • IQ scores

19. X - X Z = SD Z-scores • Defined as a mean of 0 and a SD of 1

20. T-Scores • Derived score with a mean of 50 and SD of 10 T = 50 + 10(z)

21. IQ Scores • Derived score with a mean of 100 and SD of 15 • In some cases SD = 16 • IQ = 100 + 15(z) More Broadly: SS = lss + (sss) (z)

22. Percentiles • Derived score indicated the percentage of scores that fall below a given score. • Distribution is based on the median of scores • %ile = %below score + (0.5)(% getting a score)

23. Calculating a percentile • order all scores highest to lowest • place equal scores one above the other • take a targeted score and calculate percent all those geting the score • multiply target score percentage by 0.5 • calculate percentage of all scores below the target score • add 0.5*%getting the score with % below the score.

24. Other Important Standard Scores • Normal Curve Equivalents (NCE) • Mean of 50, SD of 21.06 (divides normal curve into exactly 100 parts) • Stanine scores • Divides the distribution in into nine parts of .5 SD (z score) width • S1 & S9 represent distribution beyond ±z = 1.75

25. Important Notes on Standard Scores • SS allow comparison across different standard and non-standardized scores • Percentiles can be compared with SS when distribution is normal (e.g., within and between standardized tests)

26. Correlation • Relationship between variables • High correlations predict behavior among variables • Low correlation indicates less relationship

27. Relationships among tests • A correlation quantifies the relationship between two items • A correlation coefficient, r, is calculated • indicates the magnitude of the relationship • r is a number between -1.0 and +1.0 • r = 0, indicates no correlation • r = 1.0 indicates a high positive correlation • r = -1.0 indicates a high negative correlation

28. Basic Rule of Correlation • A correlation does not imply causality • prediction is not the same as precipitation

29. (E T1) (ET2) E T1T2 - N r = S2X S2Y S2Y Measures of Correlation • Pearson product moment, r

30. Measures of Correlation • Coefficient of Determination • Adjusts r to determine relative usefulness of the relationship. • Corrects r for determining strength of related variance between the two variables. Coeff. Of Det. = r2

31. 6 6 4.66 2.16 6 17 15 15 2.66 25% 5% 20% Descriptive Statistics • What is the mean of 3, 4, 5, 6, 7, 8, 9? • What is the median of 3, 4, 5, 6, 7, 8, 9? • What is the variance of 3, 4, 5, 6, 7, 8, 9? • What is the standard deviation of 3, 4, 5, 6, 7, 8, 9? • What is the range of 3, 4, 5, 6, 7, 8, 9? • What is the mean of 10, 13, 13, 15, 15, 15, 17, 17, 38? • What is the median of 10, 13, 13, 15, 15, 15, 17, 17, 20? • What is the mode of 10, 13, 13, 15, 15, 15, 17, 17,20? • What is the variance of 1, 3, 3, 5? • The area of a z-score (SD) of 0.67 is about 25% and the area for a z-score (SD) of 1.64 is about 45%. • What proportion falls below a z-score of -.67? • What proportion falls below a z-score of –1.64? • What proportion falls between z s of +.67 and +1.64?