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

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

Relationship of Derived Scores Percentiles z IQ T

Scales Nominal Ordinal Interval Ratio

Nominal & Ordinal 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 These scales are difficult to manipulate mathematically

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

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

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

Norm group factors Age, gender, grade Sampling Representation Size Recency

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

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

Derived Scores of NR Testing Developmental Scores Scores of Relative Standing

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

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

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

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

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

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

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

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

IQ Scores Derived score with a mean of 100 and SD of 15 In some cases SD = 16 IQ = (z) More Broadly: SS = l ss + (s ss ) (z)

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)

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.

Other Important Standard Scores Normal Curve Equivalents (NCE) Mean of 50, SD of (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

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)

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

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

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

Measures of Correlation Pearson product moment, r r = E T 1 T 2 - ( E T 1 ) ( E T 2 ) N S2XS2X S2YS2Y S2YS2Y

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. = r 2

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? % 5% 20%