Standardized Tests (Continued)
Table of Contents
- More on Percentile rank
- Transformations of z-scores
- Summary Figure
- Grade and Age Equivalents
More on Percentile rank
Mathematically, the percentile rank is defined as:
p = [cfi + .5(fi)]/N * 100%
where cfi stands for the cumulative frequency, the counts accumulated by the current count and all previous ones, for all scores below the score of interest, fi is the frequency of scores in the interval of interest, and N is the sample size.
Steps to compute a percentile rank
- Construct a frequency distribution for the raw scores.
- For a given score, determine the cumulative frequency for all scores below the point of interest.
- Add half the frequency for the score of interest to the cf value yielded from step 2.
- Divide the total by N and multiply by 100%.
Table 1 Raw Scores, Frequency Distribution,
& Percentile Rank
Raw Score frequency cumulative freq. Percentile rank 11 2 2 01 12 1 3 02 13 6 9 04 14 5 14 08 15 12 26 13 16 17 43 23 17 21 64 36 18 28 92 52 19 19 111 67 20 15 126 79 21 10 136 87 22 5 141 92 23 3 144 95 24 4 148 97 25 2 150 99
For example, based on the data in Table 1, the percentile rank of a raw score of 17 is:
P = 43 + (.5)(21)/150 * 100% = 36
- The scale of the percentile rank is a non-linear transformation of that of the raw score, meaning that at different regions on the raw score scale, a gain of 1 point may not correspond to a gain of one unit or the same magnitude on the percentile rank scale.
- Percentile rank scores are less stable or less reliable for scores in the central part of the distribution than for those in the polars, the two tails of the distribution.
Transformations of z-scoresZ scores center on zero. The values below the avaerage (center) are negatives. It may be difficult to explain to test users. How can you explain an examinee that his z score is -1.5? If you tell an average student that his score is zero, he may take it literally ("I didn't earn any points!") To avoid confusion, it is often convenient to perform a linear transformation on z-scores to convert them to values that are easier to record or explain. The general form of such a transformation is:
y = m + k(z)
where Y is the derived score, and m and k are constant values arbitrarily chosen for convenience. The value chosen for the m will be the new center (mean) of the new distribution after the score transformation, and the value chosen for the k will be its new standard deviation.
Teacher scores (T scores)
T = 50 + 10(z)
College Entrance Examination Board (CEEB)
y = 500 + 100(z)
Intelligence Quotient (IQ)
IQ = 100 + 15(z)
Stanines are ranges, intervals, or bands within fixed percentages that bracket or include the score.
- They divide the normal curve into nine portions
- Each portion is 1.5 standard deviation wide.
- Mean = 5
- Standard deviation = 2
Table 2 Stanines Stanine Percentage of cases 1 4% (lowest) 2 7% 3 12% 4 17% 5 20% 6 17% 7 12% 8 7% 9 4% (highest)
Grade and Age Equivalents
When children are tested on aptitude or achievement measures, the test user often wants a normative score which will indicate how a given child's performance compares with that of others at a particular age or grade level.
Misinterpretations of grade equivalent scores
- "Alex obtained a math computation grade-equivalent (GE) score of 7.6 (seventh grade, sixth month) on the CAT. That means that even though hhe's a fourth grader, he has reached the level of seventh-grade-level math." The preceding interpretation is incorrect. Alex's obtained GE score is the estimate made by the test developer indicating the average seventh-grader during the sixth month of the school. It just means that Alex's math is above the average.
- Comparisons among GE scores across areas for the same student also may be misleading. e.g. a GE of 3.5 in reading and 3.5 in science would appear to imply equal proficiency in the two subjects, but this is not necessarily true. Although the two scores are equal, the percentile ranks in the two subjects could be, for example, 69 in reading and 86 in science.
Questions1. If the z score is 1.5, what are the following scores?
- T score
2. Given the following data, compute the percentile rank.
- cfi = 40
- fi = 20
- N = 100
3. My percentile rank in a test is 52 and John's rank in the same test is 30. "Ha! I am doing much better than John." Am I right? Why or why not?
4. Dr. Who developed a test and collected data on its reliability. These are what he got:
- Internal consistency: 1.10
- Test-retest reliability: .99
- Equivalency : .99
Because the coefficients exceed the standards, he concluded that his test is highly reliable. Is he right? Why or why not?
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