scores & norms derived scores, scales, variability, correlation, & percentiles

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-2 -1 0 1 2

30 40 50 60 70

70 85 100 115 130IQ

T

1 5 10 20 30 40 50 60 70 80 90 95 99

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 = 50 + 10(z)

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)

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 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

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 T1T2 -

(E T1) (ET2)N

S2X S2

YS2Y

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

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?

66

4.662.16

617

15

15

2.66

25% 5%

20%