Interpreting Standardized Achievement Test Scores

Standardized achievement tests have been widely used in the schools as a means of determining how well schools are doing. These tests have been primarly norm-referenced tests that compared local student performance to the performance of a representative sample of students in a norm group (e.g., a group of students at the national, regional, or state level). In the past, the test items were selection-type items, primarily multiple choice. In recent years, the tests have been modified to provide for criterion-referenced interpretations as well (e.g., by including more items per task, using open-ended tasks, and providing for interpretation by clusters of tasks). Both types of interpretation will be discussed in turn.

            Being able to interpret the various types of norm-referenced test scores and understand how criterion-referenced interpretations are used in standardized tests is important, if the tests are to play a role used in standardized tests is important, of course, to understand them well enough to be able to explain them to students and parents. To start with, it is important to keep in mind that norm-referenced interpretation indcates a student’s relative level of performance in comparison to others and criterion-referenced interpretation describes the task a student can perform.

Features of standardized Achievement Test

Standardized achievement test are designed to determine how well students are achieving a common set of broadly based goals. Well-constructed standardized achievement test typically have the following features.

1.      The content of the test is based on widely used textbooks and curriculum guides.

2.      The test item are writtten by test experts in consultation with subject-matter experts and are based on a clear set of specifications.

3.      The test items are tried out, reviewed, analyzed for difficulty and discriminating power, and either revised or eliminated.

4.      The final set of items is selected on the basis of the test specifications.

5.      Directions for administering and scoring the test are rigidly prescribed.

6.      The test is administering to select groups of students to establish national, regional, or statewide norms for interpretation of the test scores.

7.      The final version of the test is published along with a test manual that describes the test’s technical qualities and the procedures for administering, scoring, interpreting, and using the results.

            Thus, a standardized test measures a standard set of broadly based educational outcomes, uses standard directions and standard scoring procedures, and provides for a comparison of a student’s score to that of similar students who have taken the same test under the same conditions. If a battery of tests is used, and all tests have been standardized on the same norm group, a student’s performance on the different tests can also be compared. On a basic skill battery, for example, we can determine a student’s relative level of performance in reading, language, and mathematics. With comparable froms of the test, we can also examine learning progress over a series of grade levels. All of these norm-referenced interpretations and comparison of standardized test scores requires an understanding of the various types of test scores that are used in describing students test performance.

Interpreting Norm-Referenced Scores

The scores a student receives when a test has been scored according to the directions is called the raw score. On a classroom test, this is typically the number of items a student answers correctly. Although raw scores are used in classroom testing, the interpretations and comparisons made with standardized test require that the raw scores be converted to some type of derived score. Comparison of performance on two different test (e.g., reading and math), for example, require that both tests be on the same scale. Raw scores won’t work because the two tests may differ in the number of items in the tests same derived score scale, we provide a common basis for comparing relative performance. Although there are may different types of derived scores, the most common types used in school achievement  testing are

1.      Percentile ranks.

2.      Grade equivalent scores.

3.      Standard scores

            The raw scores on a standardized test are converted to derived scores during the norming of the test. Attempts are made to obtain norm groups that contain a sample of students like those for whom the test is intended. National norm, for example, typically include students from the various geographic regions of the United States, urban and rural schools, and schools of different size. A balance of boys and girls, socioeconomic levels, and ethnic groups is also sought. Thus, national norm should apporoximate as closely as possible the student population throughout the United State. The same care is typically also followed in obtaining regional, state, and special groups norm (e.g., private schools). Despite the care in obtaining norm groups, however, the obtained sample of students only apporoximates the ideal sample, due to such constraints as the needed cooperation of selected schools to administer the tests and the time limits for obtaining the norm sample.

            After the norm groups have been selected and the tests administered and scored, the raw scores are converted to derived scores and presented in the test manual in tables of norms. These tables present the raw scores and derived scores in columns so that a raw scores can be converted into a derived scores by going across the parallel columns from the raw score to the derived score. Of course, the printout from machine scoring will give both the raw score and the derived score.

            Before using the derived scores from a standardized test, it is wise to consider the nature of  the norm group. Does the norm group provide a relevant basis for interpreting student performance? How was the norm group obtained? When were the norm obtained? We can obtain the most meaningful norm-refernced interpretation of test scores when the norms are relevent, representative, and up to date.

            A final caution. The scores in the norm group should not be  viewed  as goals or standards. They are simply the scores that a representative group of students have earned on the test. They aid in interpreting and comparing test performance, but they do not represent levels of performance to strive for. They are average or typical scores obtained in average or typical schools.

Percentile Ranks

The percentile rank is one of the easiest scores to understand and to interpret to presents. A percentile rank indicates a student’s relative position in a group in terms of the percentage of group members scoring at or below the student’s raw score. For example, if a raw score of 33 equals a percentile rank of 80, it means 80 percent of the group members had raw score equal to or lower than 33. By converting raw scores meaning with different size groups and for different length tests.

            To further clarify the meaning of percentile ranks, Table 13.1 illustrates how raw scores are coonverted to percentile ranks. The following steps illustrate the procedure.

1.      The raw scores are ranked from high to low (column 1).

2.      The number of students obtaining each scoe is listed in the frequency column (column 2).

3.      The score frequencies are added from the bottom up (i.e., adding each score frequency to the total frequency of all lower scores) to obtain the cumulative frequency (column 3).

4.      Apply the following formula at each score level to get the percentile rank for that raw score (column 4).

Where PR        = percentile rank

            CF       = cumulative frequency

Table 13.1       frequency distribution and percentile rank for an objective test of 40 items

 

To illustrate the computation, let’s compute the percentile ranks for two scores.

Score 33          

Score 30  

            Precentile ranks are rounded to the nearest whole number, so the percentile rank for the raw score of 30 is listed in table 13.1 as 58. To be sure you understans this procedure, you can compute the percentile ranks of other raw scores in the table and check your answers.

            When interpreting percentile ranks, there are a number of cautions to be kept in mind. (1) Percentile ranks describe test performance in terms of the percentage of persons earning a lower score and not the percentage of items answered correctly. The percentage correct score is a criterion-referenced interpretation; percentile rank indicates relative standing and, therefore, is a norm-referenced score. (2) Percentile ranks are always specific to a particular group. For example, a percentile rank of 90 in a gifted group represents higher test performance than a percentile rank of 90 in an average group. Thus, whenever we are describing a student’s relative performance, knowing the nature of the group is just as important as knowing the student’s relative standing. (3) Percentile ranks are not equally spaced on the scale. A difference of 5 percentile ranks near the middle of the distribution of scores represents a smaller difference in test performance than a 5-percentile-rank difference at the ends of the distribution. This is because percentile ranks are based on the percentage of persons being surpassed and there is a larger percentage of persons in the middle of a score distribution to surpass than at the ends of the distribution. For example, at the high end of the distribution, a raw score difference of several points will make little difference in percentile rank because there are so few high scores. Although this limits some uses of percentile ranks (e.g., they can’t be directly averaged), they remain one of the most useful and easiest to interpret types of derived scores.

Percentile Bands.

Some test manuals use percentile bands in presenting test norms. Instead of a specific percentile rank for each raw score, a range of percentile ranks is presented. For example, a table of norms may show that a raw score of 52 has a percentile band of 60-64. This allows for the possible error in the test score. The band tells us that we can be fairly certain that a student who earns a raw score of 52 on the test has arelative standing that falls somewhere between the 60th percentile rank. We cannot be more precise than this because our estimates of test performance (i.e., raw scores) always contain some error, due to such factors a flucttuations in attentioon, memory, efforrt, and luck in guessing during testing.

            The width of the percentile band is determined by the reliability of the test. With a highly reliable test, the band is narrow. With a test of low reliability, the band is wide. The width of the band is computed by using the standard error of measurement. This is a statistic computed from the realiability coefficient and used to estimate the amount of error in an individual test score (Chapter 4). These error bands can, of cousre, be computed for raw score or for any type of derived score. They are sometimes called confidence bands because they indicate how much confidence we can have in the score representing a person’s test performance. With a narrow band, we are more confident-that the score represents the respon’s “true” or “real” level of achievement.

            In addition to using percentile bands to interpret an individual’s test performance, percentile brands can also be used to interpret differences in test performance on a battery of tests. In comparing percentile bands for thr different tests, we cwn conclude that where the bands do not overlap there is probably a “real” differnece in test performance and where they do overlap, the differences are likely to be due to error. For example, he following percentile bands from a test battery for Mario indicate that there is no “real” difference in her performance in reading and lenguage, but she is lower in math.

Maria’s percentile bands

Reading           : 70-75

Lenguage        : 74-79

Math                : 63-68

            The use of percentile bands prevents us from overinterpreting small differnces in test scores. Test publisher that use percentile bands typically plot them as bars on student’s test profiles, making it eassy to determine when the ends of bands overlap and when they don’t.

Grade Equivalent Scores

Garde equivalent scores provide another widely used method of describing test performance. They are used primarily at the elementary school level. With these scores, a student’s raw score on the test is converted to get grade level at which the score matches the average raw score of students in the norm group. As with other derived score, tables in the test manual present parallel column pf raw scores and grade equivalents. Thus, all we need to do is consult the table and obtain the grade equivalent for any give raw score. Although it is easy to obtain and is apparently easy to interpret, it is probably one of the most misinterpreted types of score. Let’s take a look at what the grade equivalent score means and what it doesn’tmean.

            To clarify the meaning of grade equivalent, let’s assume that wa obtained the following grade equivalent score from a test battery for Dave,

            Reading           4.5

            Language        6.5

            Math                7.8

            First note that the gradde equivalent score is expressed in terms of the grade level and the month in that school year. Thus, Dave’s score in reading is equal to the average score earned by student (in the norm group) in the middle of the fourth grade. Because Dave is in the middle of the fourth grade, we interpret his performance in reading as average. In language, Dave is two years advanced, and in math, he is more than three years advanced. Does that mean that Dave can do the work at these levels? No, it most likely means that he does fourth-grade work in these areas faster and more efficiently than other fourth-graders. The test probably did not include sixth- and seventh-grade material. The same misinterpretations can occur with low grade equivalent. If Dave hade a math score of 2.0, for example, it wouldn’t mean he could only do second-grade math problems. It would more likely mean that he did fourth-grade problems slower and with more errors than other fourth-graders. High and low grade equivalent scores are typically obtained by extrapolation and do not represent average score earned by those goups. This is often necessary because students at lower grade levels may not have the knowledge and skill needed to take the test, and students at higher grade levels may have moved beynod the types of skill measured by the test.

            Grade equivalent score provide a simple method of interpreting test performance, but when using them and interpreting them to parents, the following common misinterpretations should be avoided.

1.      They are not standards to be achieved but simply the average scores of students in the norm group.

2.      They do not indicate the grade level at which a student can dothe work.

3.      Extermely high and low grade equivalent scores are not as dependable indicators of test performance as those near the student’s grade level.

            In addition to these cautions to be observed when interpreting an individual’s grade equivalent scores, a comparison of scores on tests in a test battery requires an additional caution. Growth in basic skill, for example, is uneven. In reading, growth is more rapid than in math, which depends more directly on the skill taught in school. Thus, a difference of a year in grade squivalent scores represents in larger difference in achievement on a reading test than on a math test. In addition, growth in achievement tends to slow down at different times for different skills, and when growth slow down, the differences in achievement between grade equivalent score become smaller. Both the variations in growth of skill from one area to another and the variations in patterns of growth over time contribute to the unevenness of the units on our grade equivalent score scale. In comparing a student’s grade equivalent scores pn different tests from a test battery, it may be wise to look at the norm table to see how the raw scores spread out on each test. A high or low grade equivalent score is based on relatively few raw score points.

Standard Scores

A standrad score describes test performance in terms of how far a raw score is above or below average. It is expressed in units that are computed from the mean and the standard deviation of a set of scores. We are all familiar with the use of the mean as an average. It is obtained by summing the test scores and dividing by the number of scores. The computation for obtaining the standard deviation is shown in Box 13.1, but that does not help us understand its meaning or its use in interpreting standard scores. This cant best be done by describing its properties and showing how it is used as the basic unit for the various types of standard scores.

       The standard deviation (SD or s ) is an important and widely applicable statistic in testing.In addition to its use as a basic unitin standard scores,it also serves as a basis for computing reliability coefficients and the standard error of measurement,as we saw in the chapter 4.

The mean,standard Deviation,and normal curve.The mean and the and standard deviation can probably be best understood in terms of the normal curve,although a normal distribution is not required for computing them.The normal curve is a symmetrical bell-shaped curve based on a precise mathematical equation.Scores distributed according to the normal curve are concentrated near the mean.A sample normal curve is presented in figurure.

        it will be noted in figure 13.1 that the mean falls at the exact center of a normal distribution.Note also that when the normal curve is devided into standard deviation ( SD) Units,Which are equal distances along the baseline of the curve,eachpartion under the curve contains a fixed percentage of cases thus,34 percent of the cases fall between the mean and +1 SD,14 percent between + 1 SD and + 2 SD and 2 percent of the cases fall between the mean and + 1 SD,14 percent be tween + 1 SD and +2 SD and 2 percent between + 2 SD and + 3 SD.Since the

   Interpreting Standardized Achievement test Scores

 

curve is symmetrical,the same percentages,of course, apply to the intervals below the mean.These percentages have been rounded to the nearest whole number,but only a small fraction of a percent ( 0,13 percent) of the cases fall above and below three standard deviations from the mean. Thus,from a practical standpoint, a normal distribution of scores falls between -3 and +3 standard deviations from the mean.

                      To aid in understanding the meaning of standard deviation,a set ofraw scores with a mean of 40 and a standard deviation of 5 has been placed below the baseline of the curve in figure 13.1 Note that the mean raw score of 40 has been placed at the zero point and that the distance of one standard devi ation is 5 raw score points everywhere along the baseline of the curve.Thus,the point one standard

deviation above  the mean aquals 45 ( 40 + 5 ) . in this particular set of scores,then,approximately 68 percent of the scores ( about two thirds )fall between 35 and 45 approximately 96 percent fall between 30 and 50 and approximately 99.7 percent fall between 25 and 55 ( the figure shows 100 percent because numbers are rounded ).

             When the standard deviation is being computed for a set of normally distributed scores we are assentially determining how far we need to go above ( or below ) the mean in raw score points to include 43 percent of the cases.The scores obtained with standardizwed tests typically approximate a normal distribution or are normalized by statistical means and thus permit the types of interpretations we are making here.

 z-scores. A number of standard scores are based on the standard deviation unit.The simplest of these ant the one that is basic to the others is the z-score.This score is above or below the mean,the raw score of 45 in figure 13.1for example,would be assigned a z-score 0f 1.0because it is one standard devi-ation above the mean.the raw score of 30 in figure 13.1 would be given a z-crore of 20 because it is two standarddeviations below the mean.the formula for z-scores is :

 z-score = Raw score – mean
                  standard deviation

For example,z-score for raw score of 47 and 36 in Figure 13.1 would be computed as follows :

z-score = 47-40     = 14  z-Score   36 – 40    = -8
                    5                                        5

Thus, a raw score of 47 is 1.4 standard deviations above the mean and a raw score 0f 47 is 1.4 standard deviation below the mean.

        While usedin research,z-score are seldom used directly in test interpretation because ofthr use of decimal points and minus sign.instead, z-sxcore are converted to otherbtypesof standard score that use only whole numbers and positive values.such score are more convebient to use and avoid the possibility of misinterpretation due to a forgotten minus sign.

         There are a number of different types of standard score used in test in terpretation.All one needs to do is select an arbitrary mean and standard deviation and convert the score into the standard score units.To illustrate,we will describe the procedure for some common standard score.

 T-score T-score have a  mean of 50 and a standard deviation of 10.they are obtained from z-score by multiplying the z-score by 10 and adding the result to 50,as shown in the following formula.

 T- score = 50 + 10 (z-score )

Applying the formula to the various z-scores discussed earlier ( 1.0 -2.0,1.4 – 8 ) we wouldobtain T,scores as follows :

T=50 + 10 (1.0) = 60                             T = 50 + 10 (-2.0) = 30

T = 50 + 10 (1.4)  =64                           T = 50 + 10 ( -8)= 42

   T-score can be easily interpreted because they always have the same mean and standard deviation.A T-score of 60 always means one standard deviation above the mean and a T-score of 30 always means two standard deviations below the mean.Thus,with the use of T-scores,an individual’s per formance on different test can be directly compered,and the scores can be combined or averaged without the distortion of different size standard deviations,which accur the raw scores.
 

           Where a normal distribution can be assumed,T- scores can also be in terpreted in terms of percentile ranks because,in this case,there is a direct relationship between the two as shown in figure 13.2 note that a t- score of 30 is equaivalent to a percentile rank of 2 a t score of 40 is equivalent to a percentile rank of 16 and so on. This relationship makes it possible to use standard scores for those purpose where equal units are needed and to use percentile ranks when interpreting test performance to students and parents.

          because T-scores and percentile ranks both have a mean of 50 and use similar two- digit numbers, the two types of scores are often confused by those inexperienced in test interpretation.thus, it is important to keep in mind that percentile rank indicates the percentage of individuals who fall at or below a given score,while a t-score indicates how many standard deviation units a given score falls above or below the mean.Note in figure 13.2  that although percentile ranks and T-scores have the same mean value of 50 .

                                        

Below the mean percentile ranks are amaller than T-scores and above the mean they are large than T-scores.This is accounted for,of course, by the fact that percentile ranks are crowded together in the center of the distribution and spread out at the ends,while T-scores provide equal units through out the distribution of scores.

Normal curve equaivalent score ( NICE ) .Another standard score that might be confused with both T-scores and percentile ranksis the normal curve equivalent score (NICE). This set of scores also has a mean of 50, but a standard deviation of 21.06.This provides a set of scores with equal units, like the T-score, but the scores range from 1 to 99 percentile ranks also range from 1 to 99, but they do not provide equal units. As can be seen in figure 13,2 percentile ranks are smaller at the middele of the distribution( e.g one SD = 34%) than at the ends ( e.g One SD = 2 %).Thus,when interpreting NICE scores,don’t confuse them with T-scores,which have a move restricded range of scores ( typically 20 to 80 ) ,or with percentile ranks that have the same range of scores ( 1 to 99 ) but are based on unequal units.

Ability score.Publishers of achievement test batteries typically administer a test of learning ability (also called cognitive ability ,school ability,or intelligence ) to the same norm groups as the aschievement battery to make comparisons of learning ability and achievement possible.The scores on these tests are now reported as standard scores with a mean of 100 and standard deviatoion of 16 ( 15 on some tests). these scores are to be interpreted like any other standard score ( see figure 13.2 ) A score of 116 means one standard deviation above the mean ( percentile = 84 ).these scores were originally called deviation iQs, because they replaced the old ratio iQ ( ie, MAx 100 ).
                                                                                     CA
more recently,however to avoid the confusion surrounding iQ scores,they have been given more appropriate names,such as school abilityscores and standard age scores.

Stanine scores. Test scores ccan also be expressed by single-digit standard scores called stanines ( pronounced stay-nines) .The stanine scale divides the distribution of raw scores into nine parts ( the tream stanine was derived from standard nines ). The highest stanine scores is 9,tge lowest is 1 and stanine 5 is located in the center of the distribution.Each stanine,except 9 and 1,includes a band of raw scores one half of a standard deviation wide. Thius,stanines are standard scores with a mean of 5 and a standard deviation of 2 the distribution of stanines and the percentage of cases in each stanine are shown in figure 13.2

         The nine-point scale is simple to interpret to students and parents be nine point scale,where 5 is average.Because each stanine includes aband of raw scores,there is also less ahance that test performance will overwinter preted.When comparing scores on two different tests in a test battery, a diference of two stanines is tyicallysignificant. thus, ininterpreting the following scores for a student, we would conclude that the student is higherin math but there for is no difference between reading and language.

 reading stanine = 5

 language stanine = 4

Math stanine = 7

        Imn addition to the ease of interpretation,stanines provide a simple method for combining and everaging test scores.The conversion of raw scores to stanines puts the scores from different tests on the same standard scores scales, with equal units units. Thus, they have uniform meaning from one part nine 5 and stanine 7 is the same as a difference between a stanine of 4 and a stanine of 6 A difference of two stanines is also the same,if we are reffering to a reading test,a language test a math test or any other test. like other atandard scores,they provide equal units  that can be readily combined.Unlike other standard scores,they are easy to interpret And to explain to others see box 13.2 for suggestions on how to interpret test results.

Criterion- Referenced Interpretation    

To make standardized achievement test batteries more useful for instructional purpose.some test publishers have modified the multiple-choice items to in-clude more real-life situations,added open-ended performance tasks,and made provisions for criterion-referenced interpretation of test performance.

      One of the best known methods of reporting criterion-referenced test results is the percentage-correct score. This simply reports the percentage of test items in a test,or,subset,answered correctly.The reports can be for individual students,the class,the school,or the entire school district.Percentage-correct scores in the national norm sample as one basis for evaluating the schools.

      Scores for individuals can also be presented by clusters of items representing a content area,skill,or objective,with an indication of the level of performance ( e.g,above everage,below everage ). Care must be taken in interpreting these result where small number of items is included in the item clusters.

     Some test batteries provide reports that include standards of reformance.The standards are typically set by panels of educators and the report indicates a student level of performance by categories ranging from lack of mastery to superior performance.

    When making criterion-referenced interpretations of test performance there are a number of questions to keep in mind.

1.      Do the objective the tests were designed to measure match the school’s objectives for these subjects (e.g.routine skills versus reasoning) ?

2.      Are the skills and content measured by the tests appropriate for the grade levels tested?

3.      Did elimination of the easy items from the test,to obtain greater discrimination among students for norm-referenced interpretation,result in inadequate description of what low- achieving students can do?

4.      Was there a sufficient number of test items in each item cluster to permit criterion-referenced interpretation?

5.      Was the procedure in setting performance standars edequate for this type of interpretation?

     Criterion-referenced interpretation of standardized tests can be useful in classroom instruction but they must be cautiously made.

 Summary of points  

1.      standardized achievement tests have been widely used in the schools to determine how student performance compared to that of a sample of students ( i.e,a norm group) at national,regiona,or state level.

2.      standardized tests were carefully constructed  to fit a set of test specifi cations,triedout improved,and administered to a norm group for norm referenced test interpretation.

3.      The most common types of norm-referenced scores used with standardized tests of standard scores.

4.      A percentile rank indicates relative position in a group in terms of the percentage of group members scoring at or below a given score.It should not be confused with the percentage of items answered correctly ( a criterion-referenced interpretation).

5.      Percentile bands are used in reporting test performance to allow for possible error in test scores.The width of the band indicates the amount of error to allow for during interpretation,and it prevents the overin-terpretation of small differences in test scores.

6.      A grade equaivelent score indicates relative test performance in terms of the grade level at which the student’s raw score matches the everage score earned by the norm group.Thus,a grade equaivalent score of 4.5 indicates performance equal to the everage student in the middle of the fourth grade.Grade equivalent score are asy to interpret but they are subject to numerous misinterpretations.

7.      Standard scores are based on the mean ( M) and standard deviation ( SD)of  set of score.To fully understand them it is necessary to understand the meaning of these statistics.

8.      Standard score indicate the number of standard deviations a raw score falls above and below the mean.They are more difficult to understand and interpret to others but they have the advantage of providing equal units.

9.      The standard scores discussed in this chapter have the following means ( M ) and standard deviations ( SD ). The third column shows the score for one standard above the mean.

Z-score                                      0          1                    1

T-scores                                    50       10                  60

NCE scores                                50     21.06               71

Ability scores                            100    16                    116

Stanines                                       5        2                      7

10.  In a normal distribution,any standard score can be converted to per centile rank for easy interpretation.For example,one standard deviation above the mean ( see column 3 ) haw a percentile rank of 84,no matter what type pf standard score ,and is used to express test performance.

11.  Because T-score,NCE scores,and percentile ranks all have a mean of 50 and use similar two-digit numbers,care must be taken not to confuse them when interpreating test performance.

12.  stanines are single-digit score that range from1 to 9.They are easily explained to students and a difference of two stanines typically indicates a significant ( i.e,”real”) difference in test performance

13.  Stanines and percentile ranks using percentile bands are the two types of scores that are favoredwhen interpreting test result to others and judging differences between test scores

14.  Criterion referenced interpretation of test performance have been added to many standardized tests.these include percentage correct scores,the use of performance standard,and interpretation by item clusters representing a content area,skill or objective.

15.  Criterion referenced interpretation of standardized tests require a check on how well the objectives,content,and skills of the test match the local instructional program wheter the construction of the test favors criterion-referenced interpretation whether there is a sufficient number of test items for each types of interpretation,and how the performance standards are determined.

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