This document discusses different types of scores used to evaluate performance. It describes objective versus subjective scoring and two families of derived scores: scores of relative standing (e.g. percentiles, standard scores) and developmental scores (e.g. age equivalents, grade equivalents). Percentiles express performance as a percentage of others scored lower, while standard scores use deviations from the mean. Developmental scores define performance based on average scores of specific age or grade groups. The document cautions against using developmental scores.

1. Scoring Performance
• Objective v. Subjective Scoring. Objective
scoring is based on observable qualities and
not affected by other things. Subjective
scoring relies on private and personal criteria
that are not open to observation.

2. 2 Families of Derived Scores
• Scores of Relative Standing that define a
person’s performance in terms of the
performances of other similar individuals.
(e.g., percentiles and standard scores)
• Developmental Scores that define a person’s
performance in terms of the average of
particular group’s performance. (e.g., age
equivalents and grade equivalents).

3. Scores of Relative Standing
• Advantages
– They mean the same thing regardless of a
student's age or the content being tested.
– They allow us to compare one person's
performance on several tests, for example Jerry's
scores on math, science, and language subtests.
– They also allow us to compare several people on
the same test, for example Jerry, Mary, and Tony
on a math subtest.

4. . . . Percentiles
• Percentiles. Express a person’s performance
in terms of the percent of people who score
as well or worse than the test taker
• Used with ordinal and equal interval scales.

5. . . . 4 Steps in Percentile Calculation
1. Arrange scores from best to worst.
2. Compute the percentage of people with
scores below the score to which you want to
assign a percentile.
3. Compute the percentage of people earning
the score to which you want to assign a
percentile.
4. Add the percentage of people below the
score to ½ the people earning the score.

6. Calculating percentiles, An Example

7. . . . Percentiles, Interpretation
• In the preceding table
– A raw score of 50 was at the 92nd percentile. Both
students who got 50 correct get a percentile of 92. A
percentile of 92 means that each of the two did as
well or better than 92 percent of the test takers.
Eight percent is equal or better.
– A raw score of of 45 was at the 46th percentile. Each
of the five students who got a 46 scored equal to or
better than 46 percent of the test takers.
– Note: is computationally impossible to earn a
percentile of 100 or 0.

8. . . . Percentile Family
• Deciles divide the percentile range into bands of
10 percentiles. The first decile goes from 0.1 to
9.9. The second decile goes from 10 to 19.9. etc.
• Quartiles divide the percentile range into four
bands, each 25 percentiles wide. The first
quartile goes from 0.1 to 24.9; the second, from
25 to 49.9. etc.
• Other bands are possible – tertiles, quintiles
dodecadiles, etc.

9. . . . Percentiles, concluding comments
• Percentiles are ordinal. The difference
between adjacent values are not the same
across the score.
• Because the are ordinal you cannot add,
subtract, multiple, divide, or average them
• The 50th percentile is the median. If the
distribution is normal, it is also the mean and
mode.

10. . . . .Standard Scores
• Standard Scores express a person’s score in
terms of standard deviations above or below
the mean
• Used with equal interval scales
• Common Standard Scores
– z-score ( X
= 0, S = 1)
– T-score ( = 50, S = 10)
– Deviation IQ ( = 100, S = 15)
– Normal
Curve Equivalent ( = 50, S = 21.06)
X
X
X

11. . . . Calculating z-scores
• z = (raw score – mean)/ standard deviation
– Example: raw score = 31, mean =27, S = 6.
• z = (31 – 27) / 6
• = 4/6 or .67
• Interpretation: the person scored .67 standard deviations above
the mean.
– Example 2: raw score = 18, mean = 27, S = 6
• z = (18 - 27) / 6
• = -9 / 6 or -1.5
• Interpretation: the person scored 1.5 standard deviations below
the mean.

12. . . . Calculating any other Standard Score
• z-scores can be changed to any other standard
score (SS) if the mean and standard deviation
of that SS are known with the following
formula.
– SS = meanss + (z) (Sss)
– An Example: suppose we wanted to convert a z-score
to a deviation IQ. (Recall μ = 100 and S =
15.) A student with a z-score of .6 would earn a
standard score of 109, that is, 100 + (15)(0.6.

13. . . . Stanines
• Stanines. Short for standard nines are standard
score bands that divide a distribution into nine
parts. The second through eighth stanines are 0.5
standard deviations wide. The fifth stanine is
centered on the mean and goes from -0.25 to
+0.25 S. The first and ninth stanines are each
1.75 standard deviations wide.
• Stanines were once popular, but are seldom used
today.

14. Standard Scores, Advantages
• Standard scores are equal interval scales so they
can be added, subtracted, multiplied, divided,
and averaged.
• They allow two important comparisons.
– Several scores for the same student can be directly
compared.
– Several individuals can be directly compared on the
same score.
• Warning: Standard scores are not ratio scales.
Thus, a student with an IQ of 150 is not twice as
smart as a student with a 75 IQ.

15. Standard Scores, Disadvantages
• Standard scores are not readily interpreted by
individuals without some measurement
background (e.g., students and their parents).
• Standard scores often imply greater precision
than a test may allow.

16. Developmental Scores
• Developmental scores are of two types:
equivalents and quotients.
• A developmental equivalent is the age or
grade at which a raw score on a test is the
average (mean or median) scores

17. Developmental Equivalent, an example
• Suppose you tested 50 students in each of five age groups:
6, 7, 8, 9, and 10 years of age. Assume the mean of 6 year
olds was 15, of 7 year olds was 20, of 8 year olds was 30, of
9 year olds was 45, and of 10 year olds was 55.
– If Sam earns a score of 30, he would have earned the same score as the
mean of 8 year olds. Thus Sam would earn an age equivalent of 8 – 0.
– If Sam earned a score of 45, his age equivalent would have been 9 – 0.
• Formats: Age equivalents always contain an hyphen to
indicate years and months. Grade equivalents always contain
a decimal to indicate tenths of a year.
– 7 – 6 is seven years, six months
– 7.6 is the 7 and six tenths grade or 6/10 through seventh grade.

18. Developmental Scores, Quotients
• Developmental quotients (DQs) are the ratio
of developmental age divided by
chronological age (CA), with the quotient
multiplied by 100. DQs are always expressed
as whole numbers.
– Social Quotient = (social age/CA)(100)
– Intelligence Quotient = (mental age/CA)(100)
• Note: IQs are no longer calculated this way.

19. Avoid Using Developmental Scores
• These scores are fatally flawed.
– Systematic misinterpretation
– Dubious interpolation and extrapolation
– Promotion of typological thinking
– False standard of performance
– Scales are ordinal, not equal interval
– Variance differ at different ages so quotients are
uninterpretible.

20. Developmental Scores Conclusion
• The International Reading Association, the
American Psychological Association, the
National Council on Measurement in
Education, and the Council for Exceptional
Children do not support the use of
developmental scores.

21. Scores in Criterion-Referenced Assessments
• Single-Item Scores
– Dichotomous scores (pass/fail, right/wrong)
– Continuum scores
• partial credit (e.g., number of correct letters or digits)
• how the score was obtained (e.g., amount of assistance
needed to perform)
• Quality of the performance (e.g., novice to expert)

22. Criterion-Referenced Scores
• Multiple Item Scores
– Percent Correct (correct responses ÷ number of
questions)
– Accuracy (correct responses ÷ attempted responses)
– Labels for Accuracy
• Frustration level (<85%)
• Instructional level (between 85% and 95%)
• Independent level (>95%)
– Retention (responses correctly recalled ÷ number of
responses learned)
– Rate (correct responses per minute)