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Classroom Assessment Standard Test Interpretation (This presentation concerns the interpretation of scores from NormReferenced Tests {NRTs}) CRTs vs NRTs But first, what makes a test a criterionreferenced test? We’ve talked about this several times in this class. Does saying, for instance, that the passing criterion on this test is 80% of the items correct make it a CRT? To be a CRT, the items in a test must be representative of a well-defined content domain. Interpret the following statements Marigold reports that she got a 32 on her history test. She says she got 80% of the items correct. Her teacher told her the average scores was 34. You learn that the standard deviation on the test was 5, and that Her %tile rank (PR) is 34. Who had the best performance on a 25-item test? (The test has a mean of 20 and a standard deviation of 3) Lars, who’s raw score was 20. Laura, who got 80% of the items correct. René, who’s standard score was 1. Hildebrand, who’s PR was 84. Manuel, who scored at the 7th stanine. Standardized Test Interpretation Standardized Tests: What are they? – Tests that are administered, scored, and interpreted in a standard manner. – Any type of assessment can be standardized: Achievement tests. Aptitude tests. Diagnostic tests. Performance assessments. Portfolios. – Even classroom tests can be standardized. Individual Scores Commonly Reported on Standardized Tests Raw scores: – number of items correct. – percent of items correct. Derived scores (all require a comparison or NORM group. – – – – Percentile ranks. Standard Scores. Grade equivalent scores. Scale scores: Arbitrary mean and standard deviation. E.g., EOG developmental scales. – Lexiles (for standardized reading tests). Percentile Ranks (PRs) Scores on different tests, taken by different groups, can have widely different means and standard deviations. – Eg. A reading test with a Mean of 55 and a standard deviation of 5; and a math test with a mean of 35 and a standard deviation of 3. Percentile ranks (PRs) provide a useful scale for comparing an individual’s performance across different tests. – PRs tell us on which test an individual performed best relative to other individuals who have taken the same tests (i.e., a norm group). Percentile Ranks vs Percentiles An individual’s PR tells us what percent of examinees had lower scores. I.e. an individual with a PR of 63 scored better than 63% of the examinees in the comparison (norm) group. A percentile is the raw score equivalent of the percentile rank. I.e. if an individual who’s raw score in 48 has a PR of 65 then 48 is the 65th percentile. Comprehension Check Roberto’s percentile rank, on a math test, is 34. If there are 50 students in his class, how many students obtained scores lower than Roberto’s? Percentile Ranks NEVER confuse PRs with percent correct. – Unlike raw scores (and other types of scores) that tend to approximate a normal distribution PRs are uniformly (or rectangularly) distributed. PRs exaggerate raw score differences near the middle of the distribution (the differences seem larger than they really are), but Reduce differences toward the extremes (the differences seem smaller than then they really are). Percentile Ranks Relation of Percentile Ranks and Raw Scores (Normal Distribution) RS PR PR RS 100 100 100 100 80 96 80 79 60 73 60 63 40 27 40 47 20 4 20 26 0 0 0 0 At the extremes, large score differences give smaller PR differences. Normal Distribution Standard (z) Scores: Computation RS - Mean z = —————————— Standard deviation = distance from mean in standard deviation units. You should REMEMBER this equation! So…what is a standard deviation? The standard deviation (S.D.) provides an index of how far, on average, a score is likely to be from the mean of a set of scores. E.g., if a set (distribution) of scores has a mean of 74 and a standard deviation of 6, then the average distance between scores in the set and the mean will be 6 points. More about the standard deviation The standard deviation (SD) tells something about the variability of the scores in a distribution (or set) of scores. A set of scores with a standard deviation of 15 is more variable (spread out) than is a set of scores with a standard deviation of 5. z-scores give the distance between a score and the mean of the scores in standard deviation units. A z score of 1.2 means that it’s corresponding raw score is 1.2 SD units from the mean score. An Example Suppose, on a classroom, 20-item math test, Lamont gets a score of 18 correct. If the set of scores has a mean of 16 and a standard deviation of 2, then… Lamont was (18-16)/2 = 1 standard deviation above the mean for the class. What else can we tell about Lamont’s performance in class? More about Standard Scores Standard scores, by definition, have a mean of 0 and a standard deviation of 1. Hence, we can easily determine the following: About 68 percent of the scores in a distribution are between -1 and +1 standard deviations (between -1z and +1z,) About 95 percent of the scores are between -2z and +2z, and Nearly all the scores (99.7%) are found between -3z and +3z. All test scores can be converted to z scores. Click the box for a graphic of the Normal Curve Normal Distribution Interpreting Standard Scores Indicate relative standing (relative to the comparison, or norm, group taking the same test). Converting from raw scores does not change meaning of performance. Can be used to compare student’s performance on different tests. Can be transformed without changing their interpretation. Converting Standard Scores Recall that z-scores have a mean of 0 and an SD of 1. We can convert (transform) a set of z-scores to a new scale having a mean of m by adding m to all the scores. For instance, suppose we want a new scale with a mean of 100. All we have to do is add 100 to all the z-scores. Now the set of scores will have a mean of 100 and an SD of 1. Converting Standard Scores Now, suppose we want the set of scores to have a a standard deviation of 10. To accomplish this we multiply all the zscores by 10. This gives us a new set of scale scores with a mean of 100 and an SD of 10. EOG and EOC scale scores are nothing more than transformed standard (z) scores. The Linear Equation for Converting Standard Scores to a New Scale New Score =(New Mean) + (New SD) x z Using the Standard Scores to Interpret Test Scores Example: Mortimer attained a raw score of 68 on a test having a mean of 62 and a standard deviation of 4. Approximately what percent of those tested attained a raw score lower than Mortimer’s? Normal Distribution Using the Standard Scores to Interpret Test Scores Buffy scored 75% items correct on both a math test containing 20 items and a spelling test containing 40 items. Assume the math test has a mean of 14 and an S.D of 2, and the spelling test, a mean of 25 and an S.D. of 5. Relative to others taking the same tests, in which area, math or spelling, did Buffy exhibit the strongest performance? What are their respective PRs? Normal Distribution Using Percentile Ranks and Standard Scores to Interpret Test Scores James scored at the 16th percentile on a history test. His sister, Maggie scored 75% items correct on the same test. Assuming the test has 40 items, a mean of 36, and a standard deviation of 3, who exhibited the strongest performance? Normal Distribution Using Percentile Ranks and Standard Scores to Interpret Test Scores On a 100-item test, having a mean of 60 and a standard deviation of 10, – Willie obtained a raw score of 75. – His friend, Waylon, scored at the 75th percentile on the same test. Who test? had the better score on the Normal Distribution Stanine Scores Much underutilized scores. Stanines split the distribution of raw scores into nine intervals. 1 2 3 4 5 6 7 8 9 The middle seven intervals (2 thru 8) have equal widths in terms of standard deviation units (each unit is ½ standard deviation wide). The two extreme intervals (Stanine 1 and Stanine 2) have open intervals. . Some facts about Stanines The middle three stanines contain a little more than 50 percent of the scores in a normal distribution of scores. 1 2 3 4 5 6 7 8 9 Shouldn’t any interval of scores that includes HALF the population of scores be considered AVERAGE? The lower boundary of the 4th stanine has a percentile rank of 23. The upper boundary of the 6th stanine has a percentile rank of 77. Grade Equivalent Scores Give the median test score for a particular grade level. Do not convey achievement in terms of years and months of schooling. Do not allow comparisons across content areas. Do indicate level of performance relative to the group being tested. Lexiles The Lexile Framework® for Reading matches reader ability and text difficulty. It includes the Lexile® measure and the Lexile scale. – The Lexile measure is a reading ability or text difficulty score followed by an “L” (e.g., “850L”). – The Lexile scale is a developmental scale for reading ranging from 200L for beginning readers to above 1700L for advanced text. Lexiles Tens of thousands of books and Tens of millions of newspaper and magazine articles have Lexile measures. More than 450 publishers Lexile their titles. All major standardized reading tests and many popular reading programs can report student reading scores in Lexiles. Lexiles To determine the Lexile level of a book or article, text is split into 125-word slices. Each slice is compared to a list of nearly 600million words taken from a variety of sources and genres — and words in each sentence are counted. These calculations are put into a psychometric equation. From this, the Lexile measure for the entire text is determined. More informaiton on Lexiles For general information about Lexiles, go to: http://www.lexile.com/ To find the Lexile measure for a title to to: http://www.lexile.com/DesktopDefault.aspx ?view=ed&tabindex=5&tabid=67 Recap:Interpret the following statements Marigold reports that she got a 32 on her history test. She says she got 80% of the items correct. Her teacher told her the average scores was 34. You learn that the standard deviation on the test was 5. Her %tile rank (PR) is 34. Who had the best performance on a 25-item test? (The test has a mean of 20 and a standard deviation of 3) Larrs, who’s raw score was 20. Laura, who got 80% of the items correct. René, who’s standard score was 1. Hildebrand, who’s PR was 84. Manuel, who scored at the 7th stanine. END