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 varianceViolate 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
61715
152.66
25% 5%
20%
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